[{"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "Let's say we're trying to understand the relationship between people's height and their weight. So what we do is we go to 10 different people and we measure each of their heights and each of their weights. And so on this scatter plot here, each dot represents a person. So for example, this dot over here represents a person whose height was 60 inches or five feet tall. So that's the point 60 comma, and whose weight, which we have on the y-axis, was 125 pounds. And so when you look at this scatter plot, your eyes naturally see some type of a trend. It seems like, generally speaking, as height increases, weight increases as well."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "So for example, this dot over here represents a person whose height was 60 inches or five feet tall. So that's the point 60 comma, and whose weight, which we have on the y-axis, was 125 pounds. And so when you look at this scatter plot, your eyes naturally see some type of a trend. It seems like, generally speaking, as height increases, weight increases as well. But I said generally speaking. You definitely have circumstances where there are taller people who might weigh less. But an interesting question is, can we try to fit a line to this data?"}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "It seems like, generally speaking, as height increases, weight increases as well. But I said generally speaking. You definitely have circumstances where there are taller people who might weigh less. But an interesting question is, can we try to fit a line to this data? And this idea of trying to fit a line as closely as possible to as many of the points as possible is known as linear regression. Now, the most common technique is to try to fit a line that minimizes the squared distance to each of those points. And we're gonna talk more about that in future videos."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "But an interesting question is, can we try to fit a line to this data? And this idea of trying to fit a line as closely as possible to as many of the points as possible is known as linear regression. Now, the most common technique is to try to fit a line that minimizes the squared distance to each of those points. And we're gonna talk more about that in future videos. But for now, we wanna get an intuitive feel for that. So if you were to just eyeball it and look at a line like that, you wouldn't think that it would be a particularly good fit. It looks like most of the data sits above the line."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "And we're gonna talk more about that in future videos. But for now, we wanna get an intuitive feel for that. So if you were to just eyeball it and look at a line like that, you wouldn't think that it would be a particularly good fit. It looks like most of the data sits above the line. Similarly, something like this also doesn't look that great. Here, most of our data points are sitting below the line. But something like this actually looks very good."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "It looks like most of the data sits above the line. Similarly, something like this also doesn't look that great. Here, most of our data points are sitting below the line. But something like this actually looks very good. It looks like it's getting as close as possible to as many of the points as possible. It seems like it's describing this general trend. And so this is the actual regression line."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "But something like this actually looks very good. It looks like it's getting as close as possible to as many of the points as possible. It seems like it's describing this general trend. And so this is the actual regression line. And the equation here, we would write as, we would write y with a little hat over it. And that means that we are trying to estimate a y for a given x. It's not always going to be the actual y for a given x because as we see, sometimes the points aren't sitting on the line."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "And so this is the actual regression line. And the equation here, we would write as, we would write y with a little hat over it. And that means that we are trying to estimate a y for a given x. It's not always going to be the actual y for a given x because as we see, sometimes the points aren't sitting on the line. But we say y hat is equal to, and our y-intercept for this particular regression line, it is negative 140 plus the slope, 14 over three, times x. Now as we can see, for most of these points, given the x value of those points, the estimate that our regression line gives is different than the actual value. And that difference between the actual and the estimate from the regression line is known as the residual."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "It's not always going to be the actual y for a given x because as we see, sometimes the points aren't sitting on the line. But we say y hat is equal to, and our y-intercept for this particular regression line, it is negative 140 plus the slope, 14 over three, times x. Now as we can see, for most of these points, given the x value of those points, the estimate that our regression line gives is different than the actual value. And that difference between the actual and the estimate from the regression line is known as the residual. So let me write that down. So for example, the residual at that point, residual at that point, is going to be equal to, for a given x, the actual y value minus the estimated y value from the regression line for that same x. Or another way to think about it is, for that x value, when x is equal to 60, we're talking about the residual just at that point, it's going to be the actual y value minus our estimate of what the y value is from this regression line for that x value."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "And that difference between the actual and the estimate from the regression line is known as the residual. So let me write that down. So for example, the residual at that point, residual at that point, is going to be equal to, for a given x, the actual y value minus the estimated y value from the regression line for that same x. Or another way to think about it is, for that x value, when x is equal to 60, we're talking about the residual just at that point, it's going to be the actual y value minus our estimate of what the y value is from this regression line for that x value. So pause this video and see if you can calculate this residual, and you can visually imagine it as being this right over here. Well, to actually calculate the residual, you would take our actual value, which is 125, for that x value. Remember, we're calculating the residual for a point."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "Or another way to think about it is, for that x value, when x is equal to 60, we're talking about the residual just at that point, it's going to be the actual y value minus our estimate of what the y value is from this regression line for that x value. So pause this video and see if you can calculate this residual, and you can visually imagine it as being this right over here. Well, to actually calculate the residual, you would take our actual value, which is 125, for that x value. Remember, we're calculating the residual for a point. So it's the actual y there, minus what would be the estimated y there for that x value? Well, we could just go to this equation and say what would y hat be when x is equal to 60? Well, it's going to be equal to, let's see, we have negative 140, plus 14 over three times 60."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "Remember, we're calculating the residual for a point. So it's the actual y there, minus what would be the estimated y there for that x value? Well, we could just go to this equation and say what would y hat be when x is equal to 60? Well, it's going to be equal to, let's see, we have negative 140, plus 14 over three times 60. Let's see, 60 divided by three is 20. 20 times 14 is 280. And so all of this is going to be 140."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "Well, it's going to be equal to, let's see, we have negative 140, plus 14 over three times 60. Let's see, 60 divided by three is 20. 20 times 14 is 280. And so all of this is going to be 140. And so our residual for this point is gonna be 125 minus 140, which is negative 15. And residuals, indeed, can be negative. If your residual is negative, it means for that x value, your data point, your actual y value, is below the estimate."}, {"video_title": "Introduction to residuals and least-squares regression AP Statistics Khan Academy.mp3", "Sentence": "And so all of this is going to be 140. And so our residual for this point is gonna be 125 minus 140, which is negative 15. And residuals, indeed, can be negative. If your residual is negative, it means for that x value, your data point, your actual y value, is below the estimate. If we were to calculate the residual here, or if we were to calculate the residual here, our actual for that x value is above our estimate. So we would get positive residuals. And as you'll see later in your statistics career, the way that we calculate these regression lines is all about minimizing the square of these residuals."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "One fair coin and one double-sided coin. He picks one at random, flips it, and shouts the result. Heads! Now what is the probability that he flipped the fair coin? To answer this question, we need only rewind and grow a tree. The first event, he picks one of two coins. So our tree grows two branches, leading to two equally likely outcomes, fair or unfair."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "Now what is the probability that he flipped the fair coin? To answer this question, we need only rewind and grow a tree. The first event, he picks one of two coins. So our tree grows two branches, leading to two equally likely outcomes, fair or unfair. The next event, he flips the coin. We grow again. If he had the fair coin, we know this flip can result in two equally likely outcomes, heads and tails."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "So our tree grows two branches, leading to two equally likely outcomes, fair or unfair. The next event, he flips the coin. We grow again. If he had the fair coin, we know this flip can result in two equally likely outcomes, heads and tails. While the unfair coin results in two outcomes, both heads. Our tree is finished and we see it has four leaves, representing four equally likely outcomes. The final step, new evidence."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "If he had the fair coin, we know this flip can result in two equally likely outcomes, heads and tails. While the unfair coin results in two outcomes, both heads. Our tree is finished and we see it has four leaves, representing four equally likely outcomes. The final step, new evidence. He says, heads! Whenever we gain evidence, we must trim our tree. We cut any branch leading to tails because we know tails did not occur."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "The final step, new evidence. He says, heads! Whenever we gain evidence, we must trim our tree. We cut any branch leading to tails because we know tails did not occur. And that is it. So the probability that he chose the fair coin is the one fair outcome leading to heads, divided by the three possible outcomes leading to heads, or one third. What happens if he flips again and reports, heads!"}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "We cut any branch leading to tails because we know tails did not occur. And that is it. So the probability that he chose the fair coin is the one fair outcome leading to heads, divided by the three possible outcomes leading to heads, or one third. What happens if he flips again and reports, heads! Remember, after each event, our tree grows. The fair coin leaves result in two equally likely outcomes, heads and tails. The unfair coin leaves result in two equally likely outcomes, heads and heads."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "What happens if he flips again and reports, heads! Remember, after each event, our tree grows. The fair coin leaves result in two equally likely outcomes, heads and tails. The unfair coin leaves result in two equally likely outcomes, heads and heads. After we hear the second, heads! We cut any branches leading to tails. Therefore, the probability the coin is fair after two heads in a row is the one fair outcome leading to heads, divided by all possible outcome leading to heads, or one fifth."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "The unfair coin leaves result in two equally likely outcomes, heads and heads. After we hear the second, heads! We cut any branches leading to tails. Therefore, the probability the coin is fair after two heads in a row is the one fair outcome leading to heads, divided by all possible outcome leading to heads, or one fifth. Notice our confidence in the fair coin is dropping as more heads occur, though realize it will never reach zero. No matter how many flips occur, we can never be 100% certain the coin is unfair. In fact, all conditional probability questions can be solved by growing trees."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "Therefore, the probability the coin is fair after two heads in a row is the one fair outcome leading to heads, divided by all possible outcome leading to heads, or one fifth. Notice our confidence in the fair coin is dropping as more heads occur, though realize it will never reach zero. No matter how many flips occur, we can never be 100% certain the coin is unfair. In fact, all conditional probability questions can be solved by growing trees. Let's do one more to be sure. Bob has three coins. Two are fair."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "In fact, all conditional probability questions can be solved by growing trees. Let's do one more to be sure. Bob has three coins. Two are fair. One is biased, which is weighted to land heads two thirds of the time and tails one third. He chooses a coin at random and flips it. Heads!"}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "Two are fair. One is biased, which is weighted to land heads two thirds of the time and tails one third. He chooses a coin at random and flips it. Heads! Now, what is the probability he chose the biased coin? Let's rewind and build a tree. The first event, choosing the coin, can lead to three equally likely outcomes, fair coin, fair coin, and unfair coin."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "Heads! Now, what is the probability he chose the biased coin? Let's rewind and build a tree. The first event, choosing the coin, can lead to three equally likely outcomes, fair coin, fair coin, and unfair coin. The next event, the coin is flipped. Each fair coin leads to two equally likely leaves, heads and tails. The biased coin leads to three equally likely leaves, two representing heads and one representing tails."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "The first event, choosing the coin, can lead to three equally likely outcomes, fair coin, fair coin, and unfair coin. The next event, the coin is flipped. Each fair coin leads to two equally likely leaves, heads and tails. The biased coin leads to three equally likely leaves, two representing heads and one representing tails. Now, the trick is to always make sure our tree is balanced, meaning an equal amount of leaves growing out of each branch. To do this, we simply scale up the number of branches to the least common multiple. For two and three, this is six."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "The biased coin leads to three equally likely leaves, two representing heads and one representing tails. Now, the trick is to always make sure our tree is balanced, meaning an equal amount of leaves growing out of each branch. To do this, we simply scale up the number of branches to the least common multiple. For two and three, this is six. And finally, we label our leaves. The fair coin now splits into six equally likely leaves, three heads and three tails. For the biased coin, we now have two tail leaves and four head leaves, and that is it."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "For two and three, this is six. And finally, we label our leaves. The fair coin now splits into six equally likely leaves, three heads and three tails. For the biased coin, we now have two tail leaves and four head leaves, and that is it. When Bob shows the result, heads! this new evidence allows us to trim all branches, leading to tails, since tails did not occur. So, the probability that he chose the biased coin, given heads occur?"}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "For the biased coin, we now have two tail leaves and four head leaves, and that is it. When Bob shows the result, heads! this new evidence allows us to trim all branches, leading to tails, since tails did not occur. So, the probability that he chose the biased coin, given heads occur? Well, four leaves can come from the biased coin, divided by all possible leaves. Four divided by ten, or 40%. When in doubt, it's always possible to answer conditional probability questions by Bayes' theorem."}, {"video_title": "Conditional probability explained visually (Bayes' Theorem).mp3", "Sentence": "So, the probability that he chose the biased coin, given heads occur? Well, four leaves can come from the biased coin, divided by all possible leaves. Four divided by ten, or 40%. When in doubt, it's always possible to answer conditional probability questions by Bayes' theorem. It tells us the probability of event A, given some new evidence B. Though if you forgot it, no worries. You need only know how to grow stories with trimmed trees."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "So let's look at an example. Let's say that there is a concert that is happening, and we expect approximately 10,000 people to attend the concert. And we want to randomly sample people at the concert. Maybe we want to do a study on how do people get to the concert? How do people get to the concert? Do they drive and park? Do they ride with a friend?"}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "Maybe we want to do a study on how do people get to the concert? How do people get to the concert? Do they drive and park? Do they ride with a friend? Do they take an Uber or a cab of some kind? And so we want to find a random sample, ideally without bias, to survey people. So there's a couple of ways you could do it."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "Do they ride with a friend? Do they take an Uber or a cab of some kind? And so we want to find a random sample, ideally without bias, to survey people. So there's a couple of ways you could do it. You could try to do a simple random sample. And that might be a case of if you could somehow get the names of all 10,000 people and put them into a big bowl like this, and then let's say you want to sample 100 people. Let's say you want to sample approximately 100 people."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "So there's a couple of ways you could do it. You could try to do a simple random sample. And that might be a case of if you could somehow get the names of all 10,000 people and put them into a big bowl like this, and then let's say you want to sample 100 people. Let's say you want to sample approximately 100 people. You could just mix up all the names that may be on these little pieces of paper, 10,000 of them, and then pull them out and pull out a random sample of 100 of them. That would be a simple random sample. But you could already imagine there might be some logistic difficulties of doing this."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "Let's say you want to sample approximately 100 people. You could just mix up all the names that may be on these little pieces of paper, 10,000 of them, and then pull them out and pull out a random sample of 100 of them. That would be a simple random sample. But you could already imagine there might be some logistic difficulties of doing this. How are you going to get the 10,000 names? You're gonna write them on a piece of paper. That's gonna be a, you'd have to really mix some goods so it's truly random who you're picking out."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "But you could already imagine there might be some logistic difficulties of doing this. How are you going to get the 10,000 names? You're gonna write them on a piece of paper. That's gonna be a, you'd have to really mix some goods so it's truly random who you're picking out. So are there other ways of doing a random sample? And as you can imagine, yes, there are. And that's where systematic random sampling is useful."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "That's gonna be a, you'd have to really mix some goods so it's truly random who you're picking out. So are there other ways of doing a random sample? And as you can imagine, yes, there are. And that's where systematic random sampling is useful. One way to think about systematic random sampling is you're going to randomly sample a subset of the people who are maybe walking into the concert. So let's say people get to the concert and they start forming a line to get into the concert. What you wanna do in systematic random sampling is randomly pick your first person."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "And that's where systematic random sampling is useful. One way to think about systematic random sampling is you're going to randomly sample a subset of the people who are maybe walking into the concert. So let's say people get to the concert and they start forming a line to get into the concert. What you wanna do in systematic random sampling is randomly pick your first person. There's a bunch of ways that you could do that. Let's say you have a random number generator that'll generate a number from one to 100, and that's going to be the first person you survey if that random number generator generates a 37, then you're going to start with the 37th person in line. So you pick that first person randomly, you survey them."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "What you wanna do in systematic random sampling is randomly pick your first person. There's a bunch of ways that you could do that. Let's say you have a random number generator that'll generate a number from one to 100, and that's going to be the first person you survey if that random number generator generates a 37, then you're going to start with the 37th person in line. So you pick that first person randomly, you survey them. And remember, our goal is to sample about 100 people out of 10,000. So we wanna roughly sample one out of every 100 people. And so what you do there is once you have that first person that you're sampling, you then sample every 100th person after that."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "So you pick that first person randomly, you survey them. And remember, our goal is to sample about 100 people out of 10,000. So we wanna roughly sample one out of every 100 people. And so what you do there is once you have that first person that you're sampling, you then sample every 100th person after that. That's called sometimes the sample interval. And the reason why 100 people is because if you sample every 100th person after that, you're going to roughly get 100 people in your sample out of a total of 10,000. So this is going to be after 100, you're going to sample someone else."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "And so what you do there is once you have that first person that you're sampling, you then sample every 100th person after that. That's called sometimes the sample interval. And the reason why 100 people is because if you sample every 100th person after that, you're going to roughly get 100 people in your sample out of a total of 10,000. So this is going to be after 100, you're going to sample someone else. And then after another 100, you're going to sample someone else. Now, the reason why this is useful is you could say, okay, that first person was random, and then every person after that. It doesn't seem like there'd be any bias for why they would be the 100th person after that first person."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "So this is going to be after 100, you're going to sample someone else. And then after another 100, you're going to sample someone else. Now, the reason why this is useful is you could say, okay, that first person was random, and then every person after that. It doesn't seem like there'd be any bias for why they would be the 100th person after that first person. You don't wanna just do the first 100 people because those might be the early birds, the people who maybe disproportionately went parking or planned early or had some bias in some way. So you do wanna make sure that you're getting both the beginning, the middle, and the end of the line, which this thing helps. Now, we have to be careful."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "It doesn't seem like there'd be any bias for why they would be the 100th person after that first person. You don't wanna just do the first 100 people because those might be the early birds, the people who maybe disproportionately went parking or planned early or had some bias in some way. So you do wanna make sure that you're getting both the beginning, the middle, and the end of the line, which this thing helps. Now, we have to be careful. Even systematic random sampling is not foolproof. There's a situation where inadvertently, even this system has bias. Let's say that this is the arena."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "Now, we have to be careful. Even systematic random sampling is not foolproof. There's a situation where inadvertently, even this system has bias. Let's say that this is the arena. This is a top view of the arena right over here. And this is the line of people coming in. And this is where you are standing, and you are counting every 100th person."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "Let's say that this is the arena. This is a top view of the arena right over here. And this is the line of people coming in. And this is where you are standing, and you are counting every 100th person. But maybe, and let's say there's a tree right over here, and maybe there's a road. I'm making this quite elaborate. So maybe there is a road right over here."}, {"video_title": "Systematic random sampling AP Statistics Khan Academy.mp3", "Sentence": "And this is where you are standing, and you are counting every 100th person. But maybe, and let's say there's a tree right over here, and maybe there's a road. I'm making this quite elaborate. So maybe there is a road right over here. And a lot of people, maybe all of the people who are walking or taking a cab are coming from this direction. And maybe all of the people from the parking lot are coming from this direction. And maybe you have a police officer right over here who is doing crowd control, who lets 50 of these people in, followed by 50 of these people in."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "The first time you're exposed to permutations and combinations, it takes a little bit to get your brain around it. So I think it never hurts to do as many examples. But each incremental example, I'm gonna go, I'm gonna review what we've done before, but hopefully go a little bit further. So let's just take another example. And this is in the same vein. In videos after this, I'll start using other examples other than just people sitting in chairs. But let's stick with it for now."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So let's just take another example. And this is in the same vein. In videos after this, I'll start using other examples other than just people sitting in chairs. But let's stick with it for now. So let's say we have six people again. So person A, B, C, D, E, and F. So we have six people. And now let's put them into four chairs."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "But let's stick with it for now. So let's say we have six people again. So person A, B, C, D, E, and F. So we have six people. And now let's put them into four chairs. We can go through this fairly quickly. One, two, three, four chairs. And we've seen this show multiple times."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And now let's put them into four chairs. We can go through this fairly quickly. One, two, three, four chairs. And we've seen this show multiple times. How many ways, how many permutations are there of putting these six people into four chairs? Well, the first chair, if we seat them in order, we might as well, we could say, well, there'd be six possibilities here. Now for each of those six possibilities, there'd be five possibilities of who we put here, because one person's already sitting down."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And we've seen this show multiple times. How many ways, how many permutations are there of putting these six people into four chairs? Well, the first chair, if we seat them in order, we might as well, we could say, well, there'd be six possibilities here. Now for each of those six possibilities, there'd be five possibilities of who we put here, because one person's already sitting down. Now for each of these 30 possibilities of seating these first two people, there'd be four possibilities of who we put in chair number three. And then for each of these, what is this, 120 possibilities, there would be three possibilities of who we put in chair four. And so this six times four times, six times five times four times three is the number of permutations."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Now for each of those six possibilities, there'd be five possibilities of who we put here, because one person's already sitting down. Now for each of these 30 possibilities of seating these first two people, there'd be four possibilities of who we put in chair number three. And then for each of these, what is this, 120 possibilities, there would be three possibilities of who we put in chair four. And so this six times four times, six times five times four times three is the number of permutations. And we've seen in one of the early videos on permutations that, or when we talk about the permutation formula, one way to write this, if we wanted to write it in terms of factorial, we could write this as six factorial, six factorial, which is going to be equal to six times five times four times three times two times one. But we want to get rid of the two times one. So we're gonna divide that."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And so this six times four times, six times five times four times three is the number of permutations. And we've seen in one of the early videos on permutations that, or when we talk about the permutation formula, one way to write this, if we wanted to write it in terms of factorial, we could write this as six factorial, six factorial, which is going to be equal to six times five times four times three times two times one. But we want to get rid of the two times one. So we're gonna divide that. We're gonna divide that. Now what's two times one? Well, two times one is two factorial, and where did we get that?"}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So we're gonna divide that. We're gonna divide that. Now what's two times one? Well, two times one is two factorial, and where did we get that? Well, we wanted the first four, the first four factors of six factorial. So if you want, and that's where the four came from. We wanted the first four factors, and so the way we got two is we said six minus four."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Well, two times one is two factorial, and where did we get that? Well, we wanted the first four, the first four factors of six factorial. So if you want, and that's where the four came from. We wanted the first four factors, and so the way we got two is we said six minus four. Six minus four, that's going to get us what we want to get, that's going to give us the number that we want to get rid of. So we wanted to get rid of two, or the factors we want to get rid of. So that's going to give us two factorial."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "We wanted the first four factors, and so the way we got two is we said six minus four. Six minus four, that's going to get us what we want to get, that's going to give us the number that we want to get rid of. So we wanted to get rid of two, or the factors we want to get rid of. So that's going to give us two factorial. So if we use six minus four factorial, then that's going to give us two factorial, which is two times one, and then these cancel out, and we are all set. And so this is one way, this is, you know, I put in the particular numbers here, but this is a review of the permutations formula, where people say, hey, if I'm saying n, if I'm taking n things, and I want to figure out how many permutations are there of putting them into, let's say, k spots, it's going to be equal to n factorial over n minus k factorial. That's exactly what we did over here, where six is n, and k, or four, is k. Four is k. Actually, let me color code the whole thing so that we see the parallel."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So that's going to give us two factorial. So if we use six minus four factorial, then that's going to give us two factorial, which is two times one, and then these cancel out, and we are all set. And so this is one way, this is, you know, I put in the particular numbers here, but this is a review of the permutations formula, where people say, hey, if I'm saying n, if I'm taking n things, and I want to figure out how many permutations are there of putting them into, let's say, k spots, it's going to be equal to n factorial over n minus k factorial. That's exactly what we did over here, where six is n, and k, or four, is k. Four is k. Actually, let me color code the whole thing so that we see the parallel. Now, all of that is review, but then we went into the world of combinations, and in the world of combinations, we said, okay, permutations make a difference between who's sitting in what chair. So for example, in the permutations world, and this is all review, we've covered this in the first combinations video, in the permutations world, A, B, C, D, and D, A, B, C, these would be two different permutations. It's being counted in whatever number this is."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "That's exactly what we did over here, where six is n, and k, or four, is k. Four is k. Actually, let me color code the whole thing so that we see the parallel. Now, all of that is review, but then we went into the world of combinations, and in the world of combinations, we said, okay, permutations make a difference between who's sitting in what chair. So for example, in the permutations world, and this is all review, we've covered this in the first combinations video, in the permutations world, A, B, C, D, and D, A, B, C, these would be two different permutations. It's being counted in whatever number this is. This is what? This is 30 times 12. This is equal to 360."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "It's being counted in whatever number this is. This is what? This is 30 times 12. This is equal to 360. So this is, each of these, this is one permutation. This is another permutation, and if we keep doing it, we would count up to 360. But we learned in combinations, when we're thinking about combinations, let me write combinations."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "This is equal to 360. So this is, each of these, this is one permutation. This is another permutation, and if we keep doing it, we would count up to 360. But we learned in combinations, when we're thinking about combinations, let me write combinations. So if we're saying N choose K, or how many combinations are there, if we take K things, and we just want to figure out how many combinations, sorry, if we start with N, if we have a pool of N things, and we want to say how many combinations of K things are there, then we would count these as the same combination. So what we really want to do is we want to take the number of permutations there are, we want to take the number of permutations there are, which is equal to N factorial over N minus K factorial, over N minus K factorial, and we want to divide by the number of ways that you could arrange four people. Once again, and this takes, I remember the first time I learned it, it took my brain a little while, so if it's taking a little while to think about it, not a big deal."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "But we learned in combinations, when we're thinking about combinations, let me write combinations. So if we're saying N choose K, or how many combinations are there, if we take K things, and we just want to figure out how many combinations, sorry, if we start with N, if we have a pool of N things, and we want to say how many combinations of K things are there, then we would count these as the same combination. So what we really want to do is we want to take the number of permutations there are, we want to take the number of permutations there are, which is equal to N factorial over N minus K factorial, over N minus K factorial, and we want to divide by the number of ways that you could arrange four people. Once again, and this takes, I remember the first time I learned it, it took my brain a little while, so if it's taking a little while to think about it, not a big deal. It can be confusing at first, but it'll hopefully, if you keep thinking about it, hopefully you will see clarity at some moment. But what we want to do is we want to divide by all of the ways that you could arrange four things, because once again, in the permutations, it's counting all of the different arrangements of four things, but we don't want to count all of those different arrangements of four things. We want to just say, well, they're all one combination."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Once again, and this takes, I remember the first time I learned it, it took my brain a little while, so if it's taking a little while to think about it, not a big deal. It can be confusing at first, but it'll hopefully, if you keep thinking about it, hopefully you will see clarity at some moment. But what we want to do is we want to divide by all of the ways that you could arrange four things, because once again, in the permutations, it's counting all of the different arrangements of four things, but we don't want to count all of those different arrangements of four things. We want to just say, well, they're all one combination. So we want to divide by the number of ways to arrange four things, or the number of ways to arrange K things. So let me write this down. So what is the number of ways, number of ways to arrange K things, K things in K spots?"}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "We want to just say, well, they're all one combination. So we want to divide by the number of ways to arrange four things, or the number of ways to arrange K things. So let me write this down. So what is the number of ways, number of ways to arrange K things, K things in K spots? And I encourage you to pause the video, because this is actually a review from the first permutation video. Well, if you have K spots, let me do it. So this is the first spot, the second spot, third spot, and then you're going to go all the way to the Kth spot."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So what is the number of ways, number of ways to arrange K things, K things in K spots? And I encourage you to pause the video, because this is actually a review from the first permutation video. Well, if you have K spots, let me do it. So this is the first spot, the second spot, third spot, and then you're going to go all the way to the Kth spot. Well, for the first spot, there could be K possibilities. There's K things that could take the first spot. Now, for each of those K possibilities, how many things could be in the second spot?"}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So this is the first spot, the second spot, third spot, and then you're going to go all the way to the Kth spot. Well, for the first spot, there could be K possibilities. There's K things that could take the first spot. Now, for each of those K possibilities, how many things could be in the second spot? Well, it's going to be K minus one, because you already put something in the first spot, and then over here, what's it going to be? K minus two, all the way to the last spot. There's only one thing that could be put in the last spot."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Now, for each of those K possibilities, how many things could be in the second spot? Well, it's going to be K minus one, because you already put something in the first spot, and then over here, what's it going to be? K minus two, all the way to the last spot. There's only one thing that could be put in the last spot. So what is this thing here? K times K minus one times K minus two times K minus three, all the way down to one. Well, this is just equal to K factorial."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "There's only one thing that could be put in the last spot. So what is this thing here? K times K minus one times K minus two times K minus three, all the way down to one. Well, this is just equal to K factorial. The number of ways to arrange K things in K spots, K factorial. The number of ways to arrange four things in four spots, that's four factorial. The number of ways to arrange three things in three spots, it's three factorial."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Well, this is just equal to K factorial. The number of ways to arrange K things in K spots, K factorial. The number of ways to arrange four things in four spots, that's four factorial. The number of ways to arrange three things in three spots, it's three factorial. So we could just divide this. We could just divide this by K factorial, and so this would get us, this would get us N factorial divided by K factorial, K factorial times, times N minus K factorial. N minus K, N minus K, and I'll put the factorial right over there."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "The number of ways to arrange three things in three spots, it's three factorial. So we could just divide this. We could just divide this by K factorial, and so this would get us, this would get us N factorial divided by K factorial, K factorial times, times N minus K factorial. N minus K, N minus K, and I'll put the factorial right over there. And this right over here is the formula, this right over here is the formula for combinations. Sometimes this is also called the binomial coefficient. Some people will call this N choose K. They'll also write it like this."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "N minus K, N minus K, and I'll put the factorial right over there. And this right over here is the formula, this right over here is the formula for combinations. Sometimes this is also called the binomial coefficient. Some people will call this N choose K. They'll also write it like this. N choose K, especially when they're thinking in terms of binomial coefficients. But I got into kind of an abstract tangent here when I started getting into the general formula, but let's go back to our example. So in our example, we saw there was 360 ways of seating six people into four chairs."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Some people will call this N choose K. They'll also write it like this. N choose K, especially when they're thinking in terms of binomial coefficients. But I got into kind of an abstract tangent here when I started getting into the general formula, but let's go back to our example. So in our example, we saw there was 360 ways of seating six people into four chairs. But what if we didn't care about who's sitting in which chairs, and we just want to say, how many ways are there to choose four people from a pool of six? Well, that would be, that would be how many ways are there, so that would be six, how many combinations, if I'm starting with a pool of six, how many combinations are there, how many combinations are there for selecting four? Or another way of thinking about it is, how many ways are there to, from a pool of six items, people in this example, how many ways are there to choose four of them?"}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So in our example, we saw there was 360 ways of seating six people into four chairs. But what if we didn't care about who's sitting in which chairs, and we just want to say, how many ways are there to choose four people from a pool of six? Well, that would be, that would be how many ways are there, so that would be six, how many combinations, if I'm starting with a pool of six, how many combinations are there, how many combinations are there for selecting four? Or another way of thinking about it is, how many ways are there to, from a pool of six items, people in this example, how many ways are there to choose four of them? And that is going to be, you know, we could do it, you know, well, I'll apply the formula first, and then I'll reason through it. And like I always say, I don't, I'm not a huge fan of the formula. Every time I, you know, I revisit it after a few years, I actually just rethink about it as opposed to memorizing it, because memorizing is a good way to not understand what's actually going on."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Or another way of thinking about it is, how many ways are there to, from a pool of six items, people in this example, how many ways are there to choose four of them? And that is going to be, you know, we could do it, you know, well, I'll apply the formula first, and then I'll reason through it. And like I always say, I don't, I'm not a huge fan of the formula. Every time I, you know, I revisit it after a few years, I actually just rethink about it as opposed to memorizing it, because memorizing is a good way to not understand what's actually going on. But if we just applied the formula here, but I really want you to understand what's happening with the formula, it would be six factorial over four factorial, over four factorial, times six minus four factorial. Six, whoops, let me actually, let me just, so this is six minus four factorial, so this part right over here, six minus four, actually, let me write it out, because I know this can be a little bit confusing the first time you see it. So six minus four factorial, factorial, which is equal to, which is equal to six factorial over four factorial, over four factorial, times, this thing right over here is two factorial, times two factorial, which is going to be equal to, we could just write out the factorial, six times five times four times three times two times one, over four, four times three times two times one, times, times two times one, and of course, that's going to cancel with that, and then the one really doesn't change the value, so let me get rid of this one here, and then let's see, this three can cancel with this three, this four could cancel with this four, and then it's six divided by two is going to be three, and so we are just left with three times five, so we are left with, we are left with, there's 15 combinations."}, {"video_title": "Combination formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Every time I, you know, I revisit it after a few years, I actually just rethink about it as opposed to memorizing it, because memorizing is a good way to not understand what's actually going on. But if we just applied the formula here, but I really want you to understand what's happening with the formula, it would be six factorial over four factorial, over four factorial, times six minus four factorial. Six, whoops, let me actually, let me just, so this is six minus four factorial, so this part right over here, six minus four, actually, let me write it out, because I know this can be a little bit confusing the first time you see it. So six minus four factorial, factorial, which is equal to, which is equal to six factorial over four factorial, over four factorial, times, this thing right over here is two factorial, times two factorial, which is going to be equal to, we could just write out the factorial, six times five times four times three times two times one, over four, four times three times two times one, times, times two times one, and of course, that's going to cancel with that, and then the one really doesn't change the value, so let me get rid of this one here, and then let's see, this three can cancel with this three, this four could cancel with this four, and then it's six divided by two is going to be three, and so we are just left with three times five, so we are left with, we are left with, there's 15 combinations. There's 360 permutations for putting six people into four chairs, but there's only 15 combinations, because we're no longer counting all of the different arrangements for the same four people in the four chairs. We're saying, hey, if it's the same four people, that is now one combination, and you can see how many ways are there to arrange four people into four chairs? Well, that's the four factorial part right over here, the four factorial part right over here, which is four times three times two times one, which is 24, so we essentially just took the 360 divided by 24 to get 15, but once again, I don't want to, I don't think I can stress this enough."}, {"video_title": "Example Different ways to pick officers Precalculus Khan Academy.mp3", "Sentence": "A club of nine people wants to choose a board of three officers, president, vice president, and secretary. Assuming the officers are chosen at random, what is the probability that the officers are Marsha for president, Sabitha for vice president, and Robert for secretary? So to think about the probability of Marsha, so let me write this president, president is equal to Marsha, or vice president is equal to Sabitha, and secretary is equal to Robert. This is going to be, this right here is one possible outcome, one specific outcome, so it's one outcome out of the total number of outcomes over the total number of possibilities. Now what is the total number of possibilities? Well to think about that, let's just think about the three positions. You have president, you have vice president, and you have secretary."}, {"video_title": "Example Different ways to pick officers Precalculus Khan Academy.mp3", "Sentence": "This is going to be, this right here is one possible outcome, one specific outcome, so it's one outcome out of the total number of outcomes over the total number of possibilities. Now what is the total number of possibilities? Well to think about that, let's just think about the three positions. You have president, you have vice president, and you have secretary. Now let's just assume that we're going to fill the slot of president first. We don't have to do president first, but we're just going to pick here. So if we're just picking president first, we haven't assigned anyone to any officers just yet, so we have nine people to choose from."}, {"video_title": "Example Different ways to pick officers Precalculus Khan Academy.mp3", "Sentence": "You have president, you have vice president, and you have secretary. Now let's just assume that we're going to fill the slot of president first. We don't have to do president first, but we're just going to pick here. So if we're just picking president first, we haven't assigned anyone to any officers just yet, so we have nine people to choose from. So there are nine possibilities here. Now when we go to selecting our vice president, we would have already assigned one person to the president. So we only have eight people to pick from."}, {"video_title": "Example Different ways to pick officers Precalculus Khan Academy.mp3", "Sentence": "So if we're just picking president first, we haven't assigned anyone to any officers just yet, so we have nine people to choose from. So there are nine possibilities here. Now when we go to selecting our vice president, we would have already assigned one person to the president. So we only have eight people to pick from. And when we assign our secretary, we would have already assigned our president and vice president, so we're only going to have seven people to pick from. So the total permutations here, or the total number of possibilities, or the total number of ways to pick president, vice president, and secretary from nine people is going to be 9 times 8 times 7, which is, let's see, 9 times 8 is 72, 72 times 7, 2 times 7 is 14, 7 times 7 is 49, plus 1 is 50. So there's 504 possibilities."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Before applying to law school in the US, students need to take an exam called the LSAT. Before applying to medical school, students need to take an exam called the MCAT. Here are some summary statistics for each exam. So the LSAT, the mean score is 151 with a standard deviation of 10. And the MCAT, the mean score is 25.1 with a standard deviation of 6.4. Juwan took both exams. He scored 172 on the LSAT and 37 on the MCAT."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So the LSAT, the mean score is 151 with a standard deviation of 10. And the MCAT, the mean score is 25.1 with a standard deviation of 6.4. Juwan took both exams. He scored 172 on the LSAT and 37 on the MCAT. Which exam did he do relatively better on? So pause this video and see if you can figure it out. So the way I would think about it is, you can't just look at the absolute score because they are on different scales and they have different distributions."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "He scored 172 on the LSAT and 37 on the MCAT. Which exam did he do relatively better on? So pause this video and see if you can figure it out. So the way I would think about it is, you can't just look at the absolute score because they are on different scales and they have different distributions. But we can use this information, if we assume it's a normal distribution or relatively close to a normal distribution with a mean centered at this mean, we can think about, well, how many standard deviations from the mean did he score in each of these situations? So in both cases, he scored above the mean, but how many standard deviations above the mean? So let's see if we can figure that out."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So the way I would think about it is, you can't just look at the absolute score because they are on different scales and they have different distributions. But we can use this information, if we assume it's a normal distribution or relatively close to a normal distribution with a mean centered at this mean, we can think about, well, how many standard deviations from the mean did he score in each of these situations? So in both cases, he scored above the mean, but how many standard deviations above the mean? So let's see if we can figure that out. So on the LSAT, let's see, let me write this down. On the LSAT, he scored 172. So how many standard deviations is that going to be?"}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So let's see if we can figure that out. So on the LSAT, let's see, let me write this down. On the LSAT, he scored 172. So how many standard deviations is that going to be? Well, let's take 172, his score, minus the mean. So this is the absolute number that he scored above the mean. And now let's divide that by the standard deviation."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So how many standard deviations is that going to be? Well, let's take 172, his score, minus the mean. So this is the absolute number that he scored above the mean. And now let's divide that by the standard deviation. So on the LSAT, this is what? This is going to be 21 divided by 10. So this is 2.1 standard deviations."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And now let's divide that by the standard deviation. So on the LSAT, this is what? This is going to be 21 divided by 10. So this is 2.1 standard deviations. Deviations above the mean, above the mean. You could view this as a z-score. It's a z-score of 2.1."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So this is 2.1 standard deviations. Deviations above the mean, above the mean. You could view this as a z-score. It's a z-score of 2.1. We are 2.1 above the mean in this situation. Now let's think about how he did on the MCAT. On the MCAT, he scored a 37."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "It's a z-score of 2.1. We are 2.1 above the mean in this situation. Now let's think about how he did on the MCAT. On the MCAT, he scored a 37. He scored a 37. The mean is a 25.1. And there is a standard deviation of 6.4."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "On the MCAT, he scored a 37. He scored a 37. The mean is a 25.1. And there is a standard deviation of 6.4. So let's see, 37.1 minus 25 would be 12, but now it's gonna be 11.9. 11.9 divided by 6.4. So without even looking at this, this is going to be approximately, well, this is gonna be a little bit less than two."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And there is a standard deviation of 6.4. So let's see, 37.1 minus 25 would be 12, but now it's gonna be 11.9. 11.9 divided by 6.4. So without even looking at this, this is going to be approximately, well, this is gonna be a little bit less than two. This is going to be less than two. So based on this information, and we could figure out the exact number here. In fact, let me get my calculator out."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So without even looking at this, this is going to be approximately, well, this is gonna be a little bit less than two. This is going to be less than two. So based on this information, and we could figure out the exact number here. In fact, let me get my calculator out. So you get the calculator. So if we do 11.9 divided by 6.4, that's gonna get us to 1 point, I'll just say approximately 1.86. So approximately 1.86."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "In fact, let me get my calculator out. So you get the calculator. So if we do 11.9 divided by 6.4, that's gonna get us to 1 point, I'll just say approximately 1.86. So approximately 1.86. So relatively speaking, he did slightly better on the LSAT. He did more standard deviations, although this is close. I would say they're comparable."}, {"video_title": "Comparing with z-scores Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So approximately 1.86. So relatively speaking, he did slightly better on the LSAT. He did more standard deviations, although this is close. I would say they're comparable. He did roughly two standard deviations if we were around to the nearest standard deviation, but if you wanted to get precise, he did a little bit better, relatively speaking, on the LSAT. He did 2.1 standard deviations here, while over here he did 1.86 or 1.9 standard deviations. But in everyday language, you would probably say, well, this is comparable."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "How many different possibilities are there? And to make that a little bit tangible, let's have an example with, say, a sofa. My sofa can seat exactly three people. I have seat number one on the left of the sofa, seat number two in the middle of the sofa, and seat number three on the right of the sofa. And let's say we're going to have three people who are going to sit in these three seats, person A, person B, and person C. How many different ways can these three people sit in these three seats? Pause this video and see if you can figure it out on your own. Well, there are several ways to approach this."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "I have seat number one on the left of the sofa, seat number two in the middle of the sofa, and seat number three on the right of the sofa. And let's say we're going to have three people who are going to sit in these three seats, person A, person B, and person C. How many different ways can these three people sit in these three seats? Pause this video and see if you can figure it out on your own. Well, there are several ways to approach this. One way is to just try to think through all of the possibilities. You could do it systematically. You could say, all right, if I have person A in seat number one, then I could have person B in seat number two and person C in seat number three."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "Well, there are several ways to approach this. One way is to just try to think through all of the possibilities. You could do it systematically. You could say, all right, if I have person A in seat number one, then I could have person B in seat number two and person C in seat number three. And I could think of another situation. If I have person A in seat number one, I could then swap B and C, so it could look like that. And that's all of the situations, all of the permutations where I have A in seat number one."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "You could say, all right, if I have person A in seat number one, then I could have person B in seat number two and person C in seat number three. And I could think of another situation. If I have person A in seat number one, I could then swap B and C, so it could look like that. And that's all of the situations, all of the permutations where I have A in seat number one. So now let's put someone else in seat number one. So now let's put B in seat number one, and I could put A in the middle and C on the right. Or I could put B in seat number one and then swap A and C. So C and then A."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "And that's all of the situations, all of the permutations where I have A in seat number one. So now let's put someone else in seat number one. So now let's put B in seat number one, and I could put A in the middle and C on the right. Or I could put B in seat number one and then swap A and C. So C and then A. And then if I put C in seat number one, well, I could put A in the middle and B on the right. Or with C in seat number one, I could put B in the middle and A on the right. And these are actually all of the permutations."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "Or I could put B in seat number one and then swap A and C. So C and then A. And then if I put C in seat number one, well, I could put A in the middle and B on the right. Or with C in seat number one, I could put B in the middle and A on the right. And these are actually all of the permutations. And you can see that there are one, two, three, four, five, six. Now this wasn't too bad, and in general, if you're thinking about permutations of six things or three things in three spaces, you can do it by hand. But it could get very complicated if I said, hey, I have 100 seats and I have 100 people that are going to sit in them."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "And these are actually all of the permutations. And you can see that there are one, two, three, four, five, six. Now this wasn't too bad, and in general, if you're thinking about permutations of six things or three things in three spaces, you can do it by hand. But it could get very complicated if I said, hey, I have 100 seats and I have 100 people that are going to sit in them. How do I figure it out mathematically? Well, the way that you would do it, and this is going to be a technique that you can use for really any number of people in any number of seats, is to really just build off of what we just did here. What we did here is we started with seat number one."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "But it could get very complicated if I said, hey, I have 100 seats and I have 100 people that are going to sit in them. How do I figure it out mathematically? Well, the way that you would do it, and this is going to be a technique that you can use for really any number of people in any number of seats, is to really just build off of what we just did here. What we did here is we started with seat number one. And we said, all right, how many different possibilities are, how many different people could sit in seat number one, assuming no one has sat down before? Well, three different people could sit in seat number one. You can see it right over here."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "What we did here is we started with seat number one. And we said, all right, how many different possibilities are, how many different people could sit in seat number one, assuming no one has sat down before? Well, three different people could sit in seat number one. You can see it right over here. This is where A is sitting in seat number one. This is where B is sitting in seat number one. And this is where C is sitting in seat number one."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "You can see it right over here. This is where A is sitting in seat number one. This is where B is sitting in seat number one. And this is where C is sitting in seat number one. Now for each of those three possibilities, how many people can sit in seat number two? Well, we saw when A sits in seat number one, there's two different possibilities for seat number two. When B sits in seat number one, there's two different possibilities for seat number two."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "And this is where C is sitting in seat number one. Now for each of those three possibilities, how many people can sit in seat number two? Well, we saw when A sits in seat number one, there's two different possibilities for seat number two. When B sits in seat number one, there's two different possibilities for seat number two. When C sits in seat number one, this is a tongue twister, there's two different possibilities for seat number two. And so you're gonna have two different possibilities here. Another way to think about it is, one person has already sat down here, there's three different ways to getting that, and so there's two people left who could sit in the second seat."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "When B sits in seat number one, there's two different possibilities for seat number two. When C sits in seat number one, this is a tongue twister, there's two different possibilities for seat number two. And so you're gonna have two different possibilities here. Another way to think about it is, one person has already sat down here, there's three different ways to getting that, and so there's two people left who could sit in the second seat. And we saw that right over here where we really wrote out the permutations. And so how many different permutations are there for seat number one and seat number two? Well, you would multiply."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "Another way to think about it is, one person has already sat down here, there's three different ways to getting that, and so there's two people left who could sit in the second seat. And we saw that right over here where we really wrote out the permutations. And so how many different permutations are there for seat number one and seat number two? Well, you would multiply. For each of these three, you have two, for each of these three in seat number one, you have two in seat number two. And then what about seat number three? Well, if you know who's in seat number one and seat number two, there's only one person who can be in seat number three."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "Well, you would multiply. For each of these three, you have two, for each of these three in seat number one, you have two in seat number two. And then what about seat number three? Well, if you know who's in seat number one and seat number two, there's only one person who can be in seat number three. And another way to think about it, if two people have already sat down, there's only one person who could be in seat number three. And so mathematically, what we could do is just say three times two times one. And you might recognize the mathematical operation factorial which literally just means, hey, start with that number and then keep multiplying it by the numbers one less than that and then one less than that all the way until you get to one."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "Well, if you know who's in seat number one and seat number two, there's only one person who can be in seat number three. And another way to think about it, if two people have already sat down, there's only one person who could be in seat number three. And so mathematically, what we could do is just say three times two times one. And you might recognize the mathematical operation factorial which literally just means, hey, start with that number and then keep multiplying it by the numbers one less than that and then one less than that all the way until you get to one. And this is three factorial which is going to be equal to six which is exactly what we've got here. And to appreciate the power of this, let's extend our example. Let's say that we have five seats, one, two, three, four, five, and we have five people, person A, B, C, D, and E. How many different ways can these five people sit in these five seats?"}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "And you might recognize the mathematical operation factorial which literally just means, hey, start with that number and then keep multiplying it by the numbers one less than that and then one less than that all the way until you get to one. And this is three factorial which is going to be equal to six which is exactly what we've got here. And to appreciate the power of this, let's extend our example. Let's say that we have five seats, one, two, three, four, five, and we have five people, person A, B, C, D, and E. How many different ways can these five people sit in these five seats? Pause this video and figure it out. Well, you might immediately say, well, that's going to be five factorial which is going to be equal to five times four times three times two times one. Five times four is 20."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "Let's say that we have five seats, one, two, three, four, five, and we have five people, person A, B, C, D, and E. How many different ways can these five people sit in these five seats? Pause this video and figure it out. Well, you might immediately say, well, that's going to be five factorial which is going to be equal to five times four times three times two times one. Five times four is 20. 20 times three is 60. And then 60 times two is 120. And then 120 times one is equal to 120."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "Five times four is 20. 20 times three is 60. And then 60 times two is 120. And then 120 times one is equal to 120. And once again, that makes a lot of sense. There's five different, if no one sat down, there's five different possibilities for seat number one. And then for each of those possibilities, there's four people who could sit in seat number two."}, {"video_title": "Factorial and counting seat arrangements Probability and Statistics Khan Academy.mp3", "Sentence": "And then 120 times one is equal to 120. And once again, that makes a lot of sense. There's five different, if no one sat down, there's five different possibilities for seat number one. And then for each of those possibilities, there's four people who could sit in seat number two. And then for each of those 20 possibilities in seat numbers one and two, well, there's going to be three people who could sit in seat number three. And for each of these 60 possibilities, there's two people who can sit in seat number four. And then once you know who's in the first four seats, you know who has to sit in that fifth seat."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So given that definition of a random variable, what we're going to try to do in this video is think about the probability distribution. So what's the probability of the different possible outcomes or the different possible values for this random variable? And we'll plot them to see how that distribution is spread out amongst those possible outcomes. So let's think about all of the different values that you could get when you flip a fair coin three times. So you could get all heads. Heads, heads, heads. You could get heads, heads, tails."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So let's think about all of the different values that you could get when you flip a fair coin three times. So you could get all heads. Heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. You could have tails, heads, head."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. You could have tails, heads, head. You could have tails, head, tails. You could have tails, tails, heads. And then you could have all tails."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "You could have tails, heads, head. You could have tails, head, tails. You could have tails, tails, heads. And then you could have all tails. So when you do the actual experiment, there's eight equally likely outcomes here. But which of them, how would these relate to the value of this random variable? So let's think about what's the probability."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "And then you could have all tails. So when you do the actual experiment, there's eight equally likely outcomes here. But which of them, how would these relate to the value of this random variable? So let's think about what's the probability. There is a situation where you have zero heads. So we could say, what's the probability that our random variable X is equal to 0? Well, that's this situation right over here where you have zero heads."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So let's think about what's the probability. There is a situation where you have zero heads. So we could say, what's the probability that our random variable X is equal to 0? Well, that's this situation right over here where you have zero heads. It is one out of the eight equally likely outcomes. So that's going to be 1 over 8. What's the probability that our random variable capital X is equal to 1?"}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "Well, that's this situation right over here where you have zero heads. It is one out of the eight equally likely outcomes. So that's going to be 1 over 8. What's the probability that our random variable capital X is equal to 1? Well, let's see. Which of these outcomes gets us exactly one head? We have this one right over here."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "What's the probability that our random variable capital X is equal to 1? Well, let's see. Which of these outcomes gets us exactly one head? We have this one right over here. We have that one right over there. We have this one right over there. And I think that's all of them."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "We have this one right over here. We have that one right over there. We have this one right over there. And I think that's all of them. So three out of the eight equally likely outcomes get us to one head, which is the same thing as saying that our random variable equals 1. So this has a 3 8's probability. Now, what's the probability?"}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "And I think that's all of them. So three out of the eight equally likely outcomes get us to one head, which is the same thing as saying that our random variable equals 1. So this has a 3 8's probability. Now, what's the probability? I think you're maybe getting the hang for it at this point. What's the probability that our random variable X is going to be equal to 2? Well, for X to be equal to 2, that means we got two heads when we flipped the coin three times."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "Now, what's the probability? I think you're maybe getting the hang for it at this point. What's the probability that our random variable X is going to be equal to 2? Well, for X to be equal to 2, that means we got two heads when we flipped the coin three times. So this outcome meets that constraint. This outcome would get our random variable to be equal to 2. And this outcome would make our random variable equal to 2."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "Well, for X to be equal to 2, that means we got two heads when we flipped the coin three times. So this outcome meets that constraint. This outcome would get our random variable to be equal to 2. And this outcome would make our random variable equal to 2. And this is three out of the eight equally likely outcomes. So this has a 3 8's probability. And then finally, we could say, what is the probability that our random variable X is equal to 3?"}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "And this outcome would make our random variable equal to 2. And this is three out of the eight equally likely outcomes. So this has a 3 8's probability. And then finally, we could say, what is the probability that our random variable X is equal to 3? Well, how does our random variable X equal 3? Well, we would have to get three heads when we flip the coin. So there's only one out of the eight equally likely outcomes that meets that constraint."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "And then finally, we could say, what is the probability that our random variable X is equal to 3? Well, how does our random variable X equal 3? Well, we would have to get three heads when we flip the coin. So there's only one out of the eight equally likely outcomes that meets that constraint. So it's a 1 8 probability. So now we just have to think about how we plot this to really see how it's distributed. So let me draw over here on the vertical axis."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So there's only one out of the eight equally likely outcomes that meets that constraint. So it's a 1 8 probability. So now we just have to think about how we plot this to really see how it's distributed. So let me draw over here on the vertical axis. I'll draw this will be the probability. And it's going to be between 0 and 1. You can have a probability larger than 1."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So let me draw over here on the vertical axis. I'll draw this will be the probability. And it's going to be between 0 and 1. You can have a probability larger than 1. So just like this. So let's see. If this is 1 right over here."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "You can have a probability larger than 1. So just like this. So let's see. If this is 1 right over here. And let's see. Everything here, it looks like it's an eighth. So let's put everything in terms of eighths."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "If this is 1 right over here. And let's see. Everything here, it looks like it's an eighth. So let's put everything in terms of eighths. So that's half. This is a fourth. That's a fourth."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So let's put everything in terms of eighths. So that's half. This is a fourth. That's a fourth. That's not quite a fourth. This is a fourth right over here. And then we can do it in terms of eighths."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "That's a fourth. That's not quite a fourth. This is a fourth right over here. And then we can do it in terms of eighths. So that's a pretty good rough approximation. And then over here, we could have the outcomes. And so outcomes, I'll say outcomes for, or let's write this so value."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "And then we can do it in terms of eighths. So that's a pretty good rough approximation. And then over here, we could have the outcomes. And so outcomes, I'll say outcomes for, or let's write this so value. So value for X. So X could be 0, 1. Actually, let me do those same colors."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "And so outcomes, I'll say outcomes for, or let's write this so value. So value for X. So X could be 0, 1. Actually, let me do those same colors. X could be 0. X could be 1. X could be 2."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "Actually, let me do those same colors. X could be 0. X could be 1. X could be 2. X could be equal to 2. And X could be equal to 3. These are the possible values for X."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "X could be 2. X could be equal to 2. And X could be equal to 3. These are the possible values for X. And now we're just going to plot the probability. The probability that X has a value of 0 is 1 eighth. So I'll make a little bar right over here that goes up to 1 eighth."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "These are the possible values for X. And now we're just going to plot the probability. The probability that X has a value of 0 is 1 eighth. So I'll make a little bar right over here that goes up to 1 eighth. So actually, let me draw it like this. So this is 1 eighth right over here. The probability that X equals 1 is 3 eighths."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So I'll make a little bar right over here that goes up to 1 eighth. So actually, let me draw it like this. So this is 1 eighth right over here. The probability that X equals 1 is 3 eighths. So that's 2 eighths, 3 eighths. Gets us right over. Let me do that in that purple color."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "The probability that X equals 1 is 3 eighths. So that's 2 eighths, 3 eighths. Gets us right over. Let me do that in that purple color. So probability of 1, that's 3 eighths. That's right over there. That's 3 eighths."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "Let me do that in that purple color. So probability of 1, that's 3 eighths. That's right over there. That's 3 eighths. So let me draw that bar. Just like that. The probability that X equals 2 is also 3 eighths."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "That's 3 eighths. So let me draw that bar. Just like that. The probability that X equals 2 is also 3 eighths. So that's going to be that same level, just like that. And then the probability that X equals 3, well, that's 1 eighth. So it's going to be the same height as this thing right over here."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "The probability that X equals 2 is also 3 eighths. So that's going to be that same level, just like that. And then the probability that X equals 3, well, that's 1 eighth. So it's going to be the same height as this thing right over here. So actually, I'm using the wrong color. So it's going to look like this. It's going to look like this."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So it's going to be the same height as this thing right over here. So actually, I'm using the wrong color. So it's going to look like this. It's going to look like this. And actually, let me just write this a little bit neater. I can move that 3. So cut and paste."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "It's going to look like this. And actually, let me just write this a little bit neater. I can move that 3. So cut and paste. Let me move that 3 a little bit closer in, just so it looks a little bit neater. And I can move that 2 in, actually, as well. So cut and paste."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So cut and paste. Let me move that 3 a little bit closer in, just so it looks a little bit neater. And I can move that 2 in, actually, as well. So cut and paste. So I can move that 2. And there you have it. We have made a probability distribution for the random variable X."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "So cut and paste. So I can move that 2. And there you have it. We have made a probability distribution for the random variable X. And the random variable X can only take on these discrete values. It can't take on the value half or the value pi or anything like that. And so what we've just done here is we've just constructed a discrete probability distribution."}, {"video_title": "Constructing a probability distribution for random variable Khan Academy.mp3", "Sentence": "We have made a probability distribution for the random variable X. And the random variable X can only take on these discrete values. It can't take on the value half or the value pi or anything like that. And so what we've just done here is we've just constructed a discrete probability distribution. Let me write that down. So this right over here is a discrete. The random variable only takes on discrete values."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Let's do a little bit of probability with playing cards. And for the sake of this video, we're going to assume that our deck has no jokers in it. You could do the same problems with the joker. You'll just get slightly different numbers. So with that out of the way, let's first just think about how many cards we have in a standard playing deck. So you have four suits. So you have four suits."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "You'll just get slightly different numbers. So with that out of the way, let's first just think about how many cards we have in a standard playing deck. So you have four suits. So you have four suits. And the suits are the spades, the diamonds, the clubs, and the hearts. You have four suits. And then in each of those suits, you have 13 different types of cards."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So you have four suits. And the suits are the spades, the diamonds, the clubs, and the hearts. You have four suits. And then in each of those suits, you have 13 different types of cards. Or sometimes it's called the rank. So each suit has 13 types of cards. You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, 10."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "And then in each of those suits, you have 13 different types of cards. Or sometimes it's called the rank. So each suit has 13 types of cards. You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, 10. And then you have the jack, the king, and the queen. And that is 13 cards. So for each suit, you can have any of these."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "You have the ace, then you have the two, the three, the four, the five, the six, seven, eight, nine, 10. And then you have the jack, the king, and the queen. And that is 13 cards. So for each suit, you can have any of these. For any of these, you can have any of the suits. So you could have a jack of diamonds, a jack of clubs, a jack of spades, or a jack of hearts. So if you just multiply these two things, you could take a deck of playing cards and actually count them, take out the jokers and count them."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So for each suit, you can have any of these. For any of these, you can have any of the suits. So you could have a jack of diamonds, a jack of clubs, a jack of spades, or a jack of hearts. So if you just multiply these two things, you could take a deck of playing cards and actually count them, take out the jokers and count them. But if you just multiply this, you have four suits. Each of those suits have 13 types. So you're going to have 4 times 13 cards."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So if you just multiply these two things, you could take a deck of playing cards and actually count them, take out the jokers and count them. But if you just multiply this, you have four suits. Each of those suits have 13 types. So you're going to have 4 times 13 cards. Or you're going to have 52 cards in a standard playing deck. Another way you could say it, you're like, look, I'm going to have these ranks or types. And each of those come in four different suits, 13 times 4."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So you're going to have 4 times 13 cards. Or you're going to have 52 cards in a standard playing deck. Another way you could say it, you're like, look, I'm going to have these ranks or types. And each of those come in four different suits, 13 times 4. Once again, you would have gotten 52 cards. Now with that out of the way, let's think about the probabilities of different events. So let's say I shuffle that deck."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "And each of those come in four different suits, 13 times 4. Once again, you would have gotten 52 cards. Now with that out of the way, let's think about the probabilities of different events. So let's say I shuffle that deck. I shuffle it really, really well. And then I randomly pick a card from that deck. And I want to think about what is the probability that I pick a jack."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So let's say I shuffle that deck. I shuffle it really, really well. And then I randomly pick a card from that deck. And I want to think about what is the probability that I pick a jack. Well, how many equally likely events are there? Well, I could pick any one of those 52 cards. So there's 52 possibilities for when I pick that card."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "And I want to think about what is the probability that I pick a jack. Well, how many equally likely events are there? Well, I could pick any one of those 52 cards. So there's 52 possibilities for when I pick that card. And how many of those 52 possibilities are jacks? Well, you have the jack of spades, the jack of diamonds, the jack of clubs, and the jack of hearts. There's four jacks in that deck."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So there's 52 possibilities for when I pick that card. And how many of those 52 possibilities are jacks? Well, you have the jack of spades, the jack of diamonds, the jack of clubs, and the jack of hearts. There's four jacks in that deck. So it is 4 over 52. These are both divisible by 4. 4 divided by 4 is 1."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "There's four jacks in that deck. So it is 4 over 52. These are both divisible by 4. 4 divided by 4 is 1. 52 divided by 4 is 13. Now let's think about the probability. So we're going to start over."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "4 divided by 4 is 1. 52 divided by 4 is 13. Now let's think about the probability. So we're going to start over. I'm going to put that jack back in. I'm going to reshuffle the deck. So once again, I still have 52 cards."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So we're going to start over. I'm going to put that jack back in. I'm going to reshuffle the deck. So once again, I still have 52 cards. So what's the probability that I get a hearts? What's the probability that I just randomly pick a card from a shuffled deck and it is a hearts? Its suit is a heart."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So once again, I still have 52 cards. So what's the probability that I get a hearts? What's the probability that I just randomly pick a card from a shuffled deck and it is a hearts? Its suit is a heart. Well, once again, there's 52 possible cards I could pick from, 52 possible equally likely events that we're dealing with. And how many of those have our hearts? Well, essentially 13 of them are hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Its suit is a heart. Well, once again, there's 52 possible cards I could pick from, 52 possible equally likely events that we're dealing with. And how many of those have our hearts? Well, essentially 13 of them are hearts. For each of those suits, you have 13 types. So there are 13 hearts in that deck. There are 13 diamonds in that deck."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Well, essentially 13 of them are hearts. For each of those suits, you have 13 types. So there are 13 hearts in that deck. There are 13 diamonds in that deck. There are 13 spades in that deck. There are 13 clubs in that deck. So 13 of the 52 would result in hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "There are 13 diamonds in that deck. There are 13 spades in that deck. There are 13 clubs in that deck. So 13 of the 52 would result in hearts. And both of these are divisible by 13. This is the same thing as 1 fourth. 1 and 4 times, I will pick it out or I have a 1 and 4 probability of getting a hearts when I go to that, when I randomly pick a card from that shuffle deck."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So 13 of the 52 would result in hearts. And both of these are divisible by 13. This is the same thing as 1 fourth. 1 and 4 times, I will pick it out or I have a 1 and 4 probability of getting a hearts when I go to that, when I randomly pick a card from that shuffle deck. Now let's do something that's a little bit more interesting. Or maybe it's a little obvious. What's the probability that I pick something that is a jack?"}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "1 and 4 times, I will pick it out or I have a 1 and 4 probability of getting a hearts when I go to that, when I randomly pick a card from that shuffle deck. Now let's do something that's a little bit more interesting. Or maybe it's a little obvious. What's the probability that I pick something that is a jack? I'll just write J. It's a jack and it is a hearts. Well, if you're reasonably familiar with cards, you'll know that there's actually only one card that is both a jack and a heart."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "What's the probability that I pick something that is a jack? I'll just write J. It's a jack and it is a hearts. Well, if you're reasonably familiar with cards, you'll know that there's actually only one card that is both a jack and a heart. It is literally the jack of hearts. So we're saying, what is the probability that we pick the exact card, the jack of hearts? Well, there's only one event, one card, that meets this criteria right over here."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Well, if you're reasonably familiar with cards, you'll know that there's actually only one card that is both a jack and a heart. It is literally the jack of hearts. So we're saying, what is the probability that we pick the exact card, the jack of hearts? Well, there's only one event, one card, that meets this criteria right over here. And there's 52 possible cards. So there's a 1 in 52 chance that I pick the jack of hearts, something that is both a jack and it's a heart. Now let's do something a little bit more interesting."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Well, there's only one event, one card, that meets this criteria right over here. And there's 52 possible cards. So there's a 1 in 52 chance that I pick the jack of hearts, something that is both a jack and it's a heart. Now let's do something a little bit more interesting. What is the probability? You might want to pause this and think about this a little bit before I give you the answer. What is the probability of, so I once again, I have a deck of 52 cards."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Now let's do something a little bit more interesting. What is the probability? You might want to pause this and think about this a little bit before I give you the answer. What is the probability of, so I once again, I have a deck of 52 cards. I shuffle it, randomly pick a card from that deck. What is the probability that that card that I pick from that deck is a jack or a heart? So it could be the jack of hearts or it could be the jack of diamonds or it could be the jack of spades or it could be the queen of hearts or it could be the two of hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "What is the probability of, so I once again, I have a deck of 52 cards. I shuffle it, randomly pick a card from that deck. What is the probability that that card that I pick from that deck is a jack or a heart? So it could be the jack of hearts or it could be the jack of diamonds or it could be the jack of spades or it could be the queen of hearts or it could be the two of hearts. So what is the probability of this? And this is a little bit more of an interesting thing because we know, first of all, that there are 52 possibilities. There are 52 possibilities."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So it could be the jack of hearts or it could be the jack of diamonds or it could be the jack of spades or it could be the queen of hearts or it could be the two of hearts. So what is the probability of this? And this is a little bit more of an interesting thing because we know, first of all, that there are 52 possibilities. There are 52 possibilities. But how many of those possibilities meet the criteria, meet these conditions that it is a jack or a heart? And to understand that, I'll draw a Venn diagram. Sounds kind of fancy, but nothing fancy here."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "There are 52 possibilities. But how many of those possibilities meet the criteria, meet these conditions that it is a jack or a heart? And to understand that, I'll draw a Venn diagram. Sounds kind of fancy, but nothing fancy here. So imagine that this rectangle I'm drawing here represents all of the outcomes. So if you want, you can imagine it has an area of 52. So this is 52 possible outcomes."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Sounds kind of fancy, but nothing fancy here. So imagine that this rectangle I'm drawing here represents all of the outcomes. So if you want, you can imagine it has an area of 52. So this is 52 possible outcomes. Now, how many of those outcomes result in a jack? So we already learned, it's one out of 13 of those outcomes result in a jack. So I could draw a little circle here where that area, and I'm approximating, that represents the probability of a jack."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So this is 52 possible outcomes. Now, how many of those outcomes result in a jack? So we already learned, it's one out of 13 of those outcomes result in a jack. So I could draw a little circle here where that area, and I'm approximating, that represents the probability of a jack. So it should be roughly 1 13th or 4 52nds of this area right over here. So I'll just draw it like this. So this right over here is the probability of a jack."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So I could draw a little circle here where that area, and I'm approximating, that represents the probability of a jack. So it should be roughly 1 13th or 4 52nds of this area right over here. So I'll just draw it like this. So this right over here is the probability of a jack. The probability of the jack. It is four, there's four possible cards out of the 52. So that is four 52nds or one out of 13."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So this right over here is the probability of a jack. The probability of the jack. It is four, there's four possible cards out of the 52. So that is four 52nds or one out of 13. 1 13. Now, what's the probability of getting a hearts? Well, I'll draw another little circle here that represents that."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So that is four 52nds or one out of 13. 1 13. Now, what's the probability of getting a hearts? Well, I'll draw another little circle here that represents that. 13 out of 52. 13 out of these 52 cards represent a heart. And actually, one of them represents both a heart and a jack."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Well, I'll draw another little circle here that represents that. 13 out of 52. 13 out of these 52 cards represent a heart. And actually, one of them represents both a heart and a jack. So I'm actually going to overlap them, and hopefully this will make sense in a second. So there's actually 13 cards that are a heart. So this is the number of hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "And actually, one of them represents both a heart and a jack. So I'm actually going to overlap them, and hopefully this will make sense in a second. So there's actually 13 cards that are a heart. So this is the number of hearts. Number of hearts. And actually, let me write this top thing that way as well. That makes it a little bit clearer that we're actually looking at."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So this is the number of hearts. Number of hearts. And actually, let me write this top thing that way as well. That makes it a little bit clearer that we're actually looking at. So let me clear that. So the number of jacks. Number of jacks."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "That makes it a little bit clearer that we're actually looking at. So let me clear that. So the number of jacks. Number of jacks. And of course, this overlap right here is the number of jacks and hearts. The number of items out of this 52 that are both a jack and a heart. It is in both sets here."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Number of jacks. And of course, this overlap right here is the number of jacks and hearts. The number of items out of this 52 that are both a jack and a heart. It is in both sets here. It is in this green circle, and it is in this orange circle. So this right over here, let me do that in yellow since I did that problem in yellow. This right over here is the number of jacks and hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "It is in both sets here. It is in this green circle, and it is in this orange circle. So this right over here, let me do that in yellow since I did that problem in yellow. This right over here is the number of jacks and hearts. So let me draw a little arrow there. It's getting a little cluttered. Maybe I should have drawn a little bit bigger."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "This right over here is the number of jacks and hearts. So let me draw a little arrow there. It's getting a little cluttered. Maybe I should have drawn a little bit bigger. Number of jacks and hearts. Number of jacks and hearts. And that's an overlap over there."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Maybe I should have drawn a little bit bigger. Number of jacks and hearts. Number of jacks and hearts. And that's an overlap over there. So what is the probability of getting a jack or a heart? So if you think about it, the probability is going to be the number of events that meet these conditions over the total number of events. We already know the total number of events are 52, but how many meet these conditions?"}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "And that's an overlap over there. So what is the probability of getting a jack or a heart? So if you think about it, the probability is going to be the number of events that meet these conditions over the total number of events. We already know the total number of events are 52, but how many meet these conditions? So it's going to be the number, it's going to be, you could say, well, look, the green circle right there says the number that gives us a jack, and the orange circle tells us that the number that gives us a heart. So you might want to say, well, why don't we add up the green and the orange? But if you did that, you would be double counting."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "We already know the total number of events are 52, but how many meet these conditions? So it's going to be the number, it's going to be, you could say, well, look, the green circle right there says the number that gives us a jack, and the orange circle tells us that the number that gives us a heart. So you might want to say, well, why don't we add up the green and the orange? But if you did that, you would be double counting. Because if you added up, if you just did four, if you did four plus 13, what are we saying? We're saying that there are four jacks, and we're saying that there are 13 hearts. But in both of these, when we do it this way, in both cases, we are counting the jack of hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "But if you did that, you would be double counting. Because if you added up, if you just did four, if you did four plus 13, what are we saying? We're saying that there are four jacks, and we're saying that there are 13 hearts. But in both of these, when we do it this way, in both cases, we are counting the jack of hearts. We're putting the jack of hearts here, and we're putting the jack of hearts here. So we're counting the jack of hearts twice, even though there's only one card there. So you would have to subtract out where they're common."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "But in both of these, when we do it this way, in both cases, we are counting the jack of hearts. We're putting the jack of hearts here, and we're putting the jack of hearts here. So we're counting the jack of hearts twice, even though there's only one card there. So you would have to subtract out where they're common. You would have to subtract out the item that is both a jack and a heart. So you would subtract out a one. Another way to think about it is, you really want to figure out the total area here."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So you would have to subtract out where they're common. You would have to subtract out the item that is both a jack and a heart. So you would subtract out a one. Another way to think about it is, you really want to figure out the total area here. You want to figure out the total area here. You want to figure out this total area. And let me zoom in, and I'll generalize it a little bit."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "Another way to think about it is, you really want to figure out the total area here. You want to figure out the total area here. You want to figure out this total area. And let me zoom in, and I'll generalize it a little bit. So if you have one circle like that, and then you have another overlapping circle like that, and you wanted to figure out the total area of the circles combined, you would look at the area of this circle, you would look at the area of this circle, and then you could add it to the area of this circle. But when you do that, you'll see that when you add the two areas, you're counting this area twice. So in order to only count that area once, you have to subtract that area from the sum."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "And let me zoom in, and I'll generalize it a little bit. So if you have one circle like that, and then you have another overlapping circle like that, and you wanted to figure out the total area of the circles combined, you would look at the area of this circle, you would look at the area of this circle, and then you could add it to the area of this circle. But when you do that, you'll see that when you add the two areas, you're counting this area twice. So in order to only count that area once, you have to subtract that area from the sum. So if this area has A, this area is B, and the intersection where they overlap is C, the combined area is going to be A plus B minus where they overlap, minus C. So that's the same thing over here. We're counting all the jacks, and that includes the jack of hearts. We're counting all the hearts, and that includes the jack of hearts."}, {"video_title": "Probability with playing cards and Venn diagrams Probability and Statistics Khan Academy.mp3", "Sentence": "So in order to only count that area once, you have to subtract that area from the sum. So if this area has A, this area is B, and the intersection where they overlap is C, the combined area is going to be A plus B minus where they overlap, minus C. So that's the same thing over here. We're counting all the jacks, and that includes the jack of hearts. We're counting all the hearts, and that includes the jack of hearts. So we counted the jack of hearts twice, so we have to subtract one out of that. So it's gonna be four plus 13 minus one, or this is going to be 1650 seconds, and both of these things are divisible by four, so this is going to be the same thing as, divide 16 by four, you get four. 52 divided by four is 13."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "After randomization, each child was asked to watch a cartoon in a private room, containing a large bowl of Goldfish crackers. The cartoon included two commercial breaks. The first group watched food commercials, mostly snacks, while the second group watched non-food commercials, games and entertainment products. Once the child finished watching the cartoon, the conductors of the experiment weighed the cracker bowls to measure how many grams of crackers the child ate. They found that the mean amount of crackers eaten by the children who watched food commercials is 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. So let's just think about what happens up to this point. They took 500 children and then they randomly assigned them to two different groups."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "Once the child finished watching the cartoon, the conductors of the experiment weighed the cracker bowls to measure how many grams of crackers the child ate. They found that the mean amount of crackers eaten by the children who watched food commercials is 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. So let's just think about what happens up to this point. They took 500 children and then they randomly assigned them to two different groups. So you have group 1 over here and you have group 2. So let's say that this right over here is the first group. The first group watched food commercials."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "They took 500 children and then they randomly assigned them to two different groups. So you have group 1 over here and you have group 2. So let's say that this right over here is the first group. The first group watched food commercials. So this is group number 1. So they watched food commercials. We could call this the treatment group."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "The first group watched food commercials. So this is group number 1. So they watched food commercials. We could call this the treatment group. We're trying to see what's the effect of watching food commercials. And then they tell us the second group watched non-food commercials. So this is the control group."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "We could call this the treatment group. We're trying to see what's the effect of watching food commercials. And then they tell us the second group watched non-food commercials. So this is the control group. So number 2, this is non-food commercials. So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "So this is the control group. So number 2, this is non-food commercials. So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance?"}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance?"}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times. It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy (2).mp3", "Sentence": "It happened less frequently than 1 in 100 times. It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times. So it seems like this was very, it's very unlikely that this was purely due to chance. If this was just a chance event, this would only happen roughly 1 in 150 times. But the fact that this happened in your experiment makes you feel pretty confident that your experiment is significant."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "Let's say that we are trying to understand a relationship in a classroom of 200 students between the amount of time studied and the percent correct. And so what we could do is we could set up some buckets of time studied and some buckets of percent correct, and then we could survey the students and or look at the data from the scores on the test. And then we can place students in these buckets. So what you see right over here, this is a two-way table, and you can also view this as a joint distribution along these two dimensions. So one way to read this is that 20 out of the 200 total students got between 60 and 79% on the test and studied between 21 and 40 minutes. So there's all sorts of interesting things that we could try to glean from this, but what we're going to focus on this video is two more types of distributions other than the joint distribution that we see in this data. One type is a marginal distribution."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "So what you see right over here, this is a two-way table, and you can also view this as a joint distribution along these two dimensions. So one way to read this is that 20 out of the 200 total students got between 60 and 79% on the test and studied between 21 and 40 minutes. So there's all sorts of interesting things that we could try to glean from this, but what we're going to focus on this video is two more types of distributions other than the joint distribution that we see in this data. One type is a marginal distribution. And a marginal distribution is just focusing on one of these dimensions. And one way to think about it is you can determine it by looking at the margin. So for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "One type is a marginal distribution. And a marginal distribution is just focusing on one of these dimensions. And one way to think about it is you can determine it by looking at the margin. So for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows. So these counts right over here give you the marginal distribution of the percent correct. 40 out of the 200 got between 80 and 100. 60 out of the 200 got between 60 and 79, so on and so forth."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "So for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows. So these counts right over here give you the marginal distribution of the percent correct. 40 out of the 200 got between 80 and 100. 60 out of the 200 got between 60 and 79, so on and so forth. Now a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would divide each of these counts by the total, which is 200. So 40 over 200, that would be 20%."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "60 out of the 200 got between 60 and 79, so on and so forth. Now a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would divide each of these counts by the total, which is 200. So 40 over 200, that would be 20%. 60 out of 200, that'd be 30%. 70 out of 200, that would be 35%. 20 out of 200 is 10%."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "So 40 over 200, that would be 20%. 60 out of 200, that'd be 30%. 70 out of 200, that would be 35%. 20 out of 200 is 10%. And 10 out of 200 is 5%. So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. So you could say 10% got between a 20 and a 39."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "20 out of 200 is 10%. And 10 out of 200 is 5%. So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. So you could say 10% got between a 20 and a 39. Now you could also think about marginal distributions the other way. You could think about the marginal distribution for the time studied in the class. And so then you would look at these counts right over here."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "So you could say 10% got between a 20 and a 39. Now you could also think about marginal distributions the other way. You could think about the marginal distribution for the time studied in the class. And so then you would look at these counts right over here. You'd say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent correct anymore. A total of 30 studied between 21 and 40 minutes."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "And so then you would look at these counts right over here. You'd say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent correct anymore. A total of 30 studied between 21 and 40 minutes. And likewise, you could write these as percentages. This would be 7%. This would be 15%."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "A total of 30 studied between 21 and 40 minutes. And likewise, you could write these as percentages. This would be 7%. This would be 15%. This would be 43%. And this would be 35% right over there. Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "This would be 15%. This would be 43%. And this would be 35% right over there. Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution. Conditional distribution. And this is the distribution of one variable given something true about the other variable. So for example, an example of a conditional distribution would be the distribution, distribution of percent correct, correct, given that students, students, students study between, let's say, 41 and 60 minutes."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "Now another idea that you might sometimes see when people are trying to interpret a joint distribution like this or get more information or more realizations from it is to think about something known as a conditional distribution. Conditional distribution. And this is the distribution of one variable given something true about the other variable. So for example, an example of a conditional distribution would be the distribution, distribution of percent correct, correct, given that students, students, students study between, let's say, 41 and 60 minutes. Between 41 and 60 minutes. Well, to think about that, you would first look at your condition. Okay, let's look at the students who have studied between 41 and 60 minutes."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "So for example, an example of a conditional distribution would be the distribution, distribution of percent correct, correct, given that students, students, students study between, let's say, 41 and 60 minutes. Between 41 and 60 minutes. Well, to think about that, you would first look at your condition. Okay, let's look at the students who have studied between 41 and 60 minutes. And so that would be this column right over here. And then that column, the information in it, can give you your conditional distribution. Now an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distribution is to think in terms of percentages."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "Okay, let's look at the students who have studied between 41 and 60 minutes. And so that would be this column right over here. And then that column, the information in it, can give you your conditional distribution. Now an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distribution is to think in terms of percentages. So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this. Let me get a little bit more space. So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is calculate the percentage that fall into each of these buckets, given that we're studying between 41 and 60 minutes."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "Now an important thing to realize is a marginal distribution can be represented as counts for the various buckets or percentages, while the standard practice for conditional distribution is to think in terms of percentages. So the conditional distribution of the percent correct given that students study between 41 and 60 minutes, it would look something like this. Let me get a little bit more space. So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is calculate the percentage that fall into each of these buckets, given that we're studying between 41 and 60 minutes. So this first one, 80 to 100, it would be 16 out of the 86 students. So we would write 16 out of 86, which is equal to, 16 divided by 86 is equal to, I'll just round to one decimal place, it's roughly 18.6%. 18.6, approximately equal to 18.6%."}, {"video_title": "Marginal and conditional distributions Analyzing categorical data AP Statistics Khan Academy.mp3", "Sentence": "So if we set up the various categories, 80 to 100, 60 to 79, 40 to 59, continue it over here, 20 to 39, and zero to 19, what we'd wanna do is calculate the percentage that fall into each of these buckets, given that we're studying between 41 and 60 minutes. So this first one, 80 to 100, it would be 16 out of the 86 students. So we would write 16 out of 86, which is equal to, 16 divided by 86 is equal to, I'll just round to one decimal place, it's roughly 18.6%. 18.6, approximately equal to 18.6%. And then to get the full conditional distribution, we would keep doing that. We would figure out the percentage, 60 to 79, that would be 30 out of 86. 30 out of 86, whatever percentage that is, and so on and so forth, in order to get that entire distribution."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "So they all agree to put in their salaries into a computer, and so these are their salaries, they're measured in thousands, so one makes 35,000, 50,000, 50,000, 50,000, 56,000, two make 60,000, one make 75,000, and one makes 250,000, so she's doing very well for herself. And the computer spits out a bunch of parameters based on this data here. So it spits out two typical measures of central tendency. The mean is roughly 76.2, the computer would calculate it by adding up all of these numbers, these nine numbers, and then dividing by nine. And the median is 56. And median is quite easy to calculate, you just order the numbers and you take the middle number here, which is 56. Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure?"}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "The mean is roughly 76.2, the computer would calculate it by adding up all of these numbers, these nine numbers, and then dividing by nine. And the median is 56. And median is quite easy to calculate, you just order the numbers and you take the middle number here, which is 56. Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure? All right, so let's think about this a little bit. I'm gonna plot it on a line here, I'm gonna plot my data so we get a better sense, so we just don't see them, so we just don't see things as numbers, but we see where those numbers sit relative to each other. So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "Now what I want you to do is pause this video and think about for this data set, for this population of salaries, which measure of central tendency is a better measure? All right, so let's think about this a little bit. I'm gonna plot it on a line here, I'm gonna plot my data so we get a better sense, so we just don't see them, so we just don't see things as numbers, but we see where those numbers sit relative to each other. So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough. I could draw this a little bit neater, but 60, 70, 80, 90, actually let me just clean this up a little bit more too, this one right over here would be a little bit closer. So this one, let me just put it right around here, so that's 40, and then this would be 30, 20, 10. Okay, that's pretty good."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "So let's say this is zero, let's say this is, let's see, one, two, three, four, five, so this would be 250, this is 50, 100, 150, 200, 200, and let's see, let's say if this is 50, then this would be roughly 40 right here, and I just wanna get rough, so this would be about 60, 70, 80, 90, close enough. I could draw this a little bit neater, but 60, 70, 80, 90, actually let me just clean this up a little bit more too, this one right over here would be a little bit closer. So this one, let me just put it right around here, so that's 40, and then this would be 30, 20, 10. Okay, that's pretty good. So let's plot this data. So one student makes 35,000, so that is right over there. Two make 50, or three make 50,000, so one, two, and three, I'll put it like that."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "Okay, that's pretty good. So let's plot this data. So one student makes 35,000, so that is right over there. Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. One makes 56,000, which would put them right over here. One makes 60,000, or actually two make 60,000, so it's like that. One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "Two make 50, or three make 50,000, so one, two, and three, I'll put it like that. One makes 56,000, which would put them right over here. One makes 60,000, or actually two make 60,000, so it's like that. One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there. And then when we calculate the mean, as 76.2 is our measure of central tendency, 76.2 is right over there. So is this a good measure of central tendency? Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "One makes 75,000, so that's 60, 70, 75,000, this one's gonna be right around there, and then one makes 250,000, so one's salary is all the way around there. And then when we calculate the mean, as 76.2 is our measure of central tendency, 76.2 is right over there. So is this a good measure of central tendency? Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one. And the reason is is that you have this one, that our data is skewed significantly by this data point at $250,000. It is so far from the rest of the distribution, from the rest of the data, that it has skewed the mean. And this is something that you see in general."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "Well, to me, it doesn't feel that good, because our measure of central tendency is higher than all of the data points except for one. And the reason is is that you have this one, that our data is skewed significantly by this data point at $250,000. It is so far from the rest of the distribution, from the rest of the data, that it has skewed the mean. And this is something that you see in general. If you have data that is skewed, and especially things like salary data, where someone might make, most people are making 50, 60, $70,000, but someone might make $2 million, and so that will skew the average, or skew the mean, I should say, when you add them all up and divide by the number of data points you have. In this case, especially when you have data points that would skew the mean, median is much more robust. The median at 56 sits right over here, which seems to be much more indicative for central tendency."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "And this is something that you see in general. If you have data that is skewed, and especially things like salary data, where someone might make, most people are making 50, 60, $70,000, but someone might make $2 million, and so that will skew the average, or skew the mean, I should say, when you add them all up and divide by the number of data points you have. In this case, especially when you have data points that would skew the mean, median is much more robust. The median at 56 sits right over here, which seems to be much more indicative for central tendency. And think about it. Even if you made this, instead of 250,000, if you made this 250,000,000, which would be $250 million, which is a ginormous amount of money to make, it would skew the mean incredibly, but it actually would not even change the median, because the median, it doesn't matter how high this number gets. This could be a trillion dollars."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "The median at 56 sits right over here, which seems to be much more indicative for central tendency. And think about it. Even if you made this, instead of 250,000, if you made this 250,000,000, which would be $250 million, which is a ginormous amount of money to make, it would skew the mean incredibly, but it actually would not even change the median, because the median, it doesn't matter how high this number gets. This could be a trillion dollars. This could be a quadrillion dollars. The median is going to stay the same. So the median is much more robust if you have a skewed data set."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "This could be a trillion dollars. This could be a quadrillion dollars. The median is going to stay the same. So the median is much more robust if you have a skewed data set. Mean makes a little bit more sense if you have a symmetric data set, or if you have things that are, where things are roughly above and below the mean, or things aren't skewed incredibly in one direction, especially by a handful of data points like we have right over here. So in this example, the median is a much better measure of central tendency. And so what about spread?"}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "So the median is much more robust if you have a skewed data set. Mean makes a little bit more sense if you have a symmetric data set, or if you have things that are, where things are roughly above and below the mean, or things aren't skewed incredibly in one direction, especially by a handful of data points like we have right over here. So in this example, the median is a much better measure of central tendency. And so what about spread? Well, you might immediately say, well, Sal, you already told us that the mean is not so good, and the standard deviation is based on the mean. You take each of these data points, find their distance from the mean, square that number, add up those squared distances, divide by the number of data points if we're taking the population standard deviation, and then you take the square root of the whole thing. And so since this is based on the mean, which isn't a good measure of central tendency in this situation, and this is also going to skew that standard deviation, this is going to be, this is a lot larger than if you look at the actual, you want an indication of the spread."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "And so what about spread? Well, you might immediately say, well, Sal, you already told us that the mean is not so good, and the standard deviation is based on the mean. You take each of these data points, find their distance from the mean, square that number, add up those squared distances, divide by the number of data points if we're taking the population standard deviation, and then you take the square root of the whole thing. And so since this is based on the mean, which isn't a good measure of central tendency in this situation, and this is also going to skew that standard deviation, this is going to be, this is a lot larger than if you look at the actual, you want an indication of the spread. Yes, you have this one data point that's way far away from either the mean or the median, depending on how you want to think about it, but most of the data points seem much closer. And so for that situation, not only are we using the median, but the interquartile range is once again more robust. How do we calculate the interquartile range?"}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "And so since this is based on the mean, which isn't a good measure of central tendency in this situation, and this is also going to skew that standard deviation, this is going to be, this is a lot larger than if you look at the actual, you want an indication of the spread. Yes, you have this one data point that's way far away from either the mean or the median, depending on how you want to think about it, but most of the data points seem much closer. And so for that situation, not only are we using the median, but the interquartile range is once again more robust. How do we calculate the interquartile range? Well, you take the median, and then you take the bottom group of numbers and calculate the median of those. So that's 50 right over here. And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "How do we calculate the interquartile range? Well, you take the median, and then you take the bottom group of numbers and calculate the median of those. So that's 50 right over here. And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5. If this looks unfamiliar, we have many videos on interquartile range and calculating standard deviation and median and mean. This is just a little bit of a review. And then the difference between these two is 17.5."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "And then you take the top group of numbers, the upper group of numbers, and the median there, 60 and 75, is 67.5. If this looks unfamiliar, we have many videos on interquartile range and calculating standard deviation and median and mean. This is just a little bit of a review. And then the difference between these two is 17.5. And notice, this distance between these two, this 17.5, this isn't going to change even if this is $250 billion. So once again, it is both of these measures are more robust when you have a skewed data set. So the big takeaway here is mean and standard deviation, they're not bad."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "And then the difference between these two is 17.5. And notice, this distance between these two, this 17.5, this isn't going to change even if this is $250 billion. So once again, it is both of these measures are more robust when you have a skewed data set. So the big takeaway here is mean and standard deviation, they're not bad. If you have a roughly symmetric data set, if you don't have any significant outliers, things that really skew the data set, mean and standard deviation can be quite solid. But if you're looking at something that could get really skewed by a handful of data points, median might be a median in interquartile range. Median for central tendency, interquartile range for spread around that central tendency."}, {"video_title": "Mean and standard deviation versus median and IQR AP Statistics Khan Academy.mp3", "Sentence": "So the big takeaway here is mean and standard deviation, they're not bad. If you have a roughly symmetric data set, if you don't have any significant outliers, things that really skew the data set, mean and standard deviation can be quite solid. But if you're looking at something that could get really skewed by a handful of data points, median might be a median in interquartile range. Median for central tendency, interquartile range for spread around that central tendency. And that's why you'll see when people talk about salaries, they'll often talk about median because you could have some skewed salaries, especially on the upside. When you talk about things like home prices, you'll see median often measured more typically than mean because home prices in a neighborhood or in a city, a lot of the houses might be in the $200,000, $300,000 range but maybe there's one ginormous mansion that is $100 million. And if you calculated mean, that would skew and give a false impression of the average or the central tendency of prices in that city."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Each box of cereal has one prize, and each prize is equally likely to appear in any given box. Amanda wonders how many boxes it takes, on average, to get all six prizes. So there's several ways to approach this for Amanda. She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "She could try to figure out a mathematical way to determine what is the expected number of boxes she would need to collect, on average, to get all six prizes. Or she could run some random numbers to simulate collecting box after box after box and figure out multiple trials on how many boxes does it take to win all six prizes. So for example, she could say, alright, each box is gonna have one of six prizes. So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So there could be, she could assign a number for each of the prizes, one, two, three, four, five, six. And then she could have a computer generate a random string of numbers, maybe something that looks like this. And the general method, she could start at the left here, and each new number she gets, she can say, hey, this is like getting a cereal box, and then it's going to tell me which prize I got. So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So she starts her first experiment, she'll start here at the left, and she'll say, okay, the first cereal box of this experiment, of this simulation, I got prize number one. And she'll keep going, the next one she gets prize number five, then the third one she gets prize number six, then the fourth one she gets prize number six again, and she will keep going until she gets all six prizes. You might say, well look, there are numbers here that aren't one through six. There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "There's zero, there's seven, there's eight or nine. Well, for those numbers, she could just ignore them. She could just pretend like they aren't there, and just keep going past them. So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here. So this is the experiment."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So why don't you pause this video, and do it for the first experiment. On this first experiment, using these numbers, assuming that this is the first box that you are getting in your simulation, how many boxes would you need in order to get all six prizes? So let's, let me make a table here. So this is the experiment. And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So this is the experiment. And then in the second column, I'm gonna say number of boxes. Number of boxes you would have to get in that simulation. So maybe I'll do the first one in this blue color. So we're in the first simulation. So one box, we got the one. Actually, maybe I'll check things off."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So maybe I'll do the first one in this blue color. So we're in the first simulation. So one box, we got the one. Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one. I'll check that off."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Actually, maybe I'll check things off. So we have to get a one, a two, a three, a four, a five, and a six. So let's see, we have a one. I'll check that off. We have a five. I'll check that off. We get a six."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "I'll check that off. We have a five. I'll check that off. We get a six. I'll check that off. Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "We get a six. I'll check that off. Well, the next box, we got another six. We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two. Then the next box, we get a four. Then the next box, the number's a seven."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "We've already have that prize, but we're gonna keep getting boxes. Then the next box, we get a two. Then the next box, we get a four. Then the next box, the number's a seven. So we will just ignore this right over here. The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Then the next box, the number's a seven. So we will just ignore this right over here. The box after that, we get a six, but we already have that prize. Then we ignore the next box, a zero. That doesn't give us a prize. We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Then we ignore the next box, a zero. That doesn't give us a prize. We assume that that didn't even happen. And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "And then we would go to the number three, which is the last prize that we need. So how many boxes did we have to go through? Well, we would only count the valid ones, the ones that gave a valid prize between the numbers one through six, including one and six. So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So let's see. We went through one, two, three, four, five, six, seven, eight boxes in the first experiment. So experiment number one, it took us eight boxes to get all six prizes. Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Let's do another experiment, because this doesn't tell us that, on average, she would expect eight boxes. This just meant that on this first experiment, it took eight boxes. If you wanted to figure out, on average, you wanna do many experiments. And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment. And remember, it's important that these are truly random numbers. And so we will now start at the first valid number."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "And the more experiments you do, the better that that average is going to, the more likely that your average is going to predict what it actually takes, on average, to get all six prizes. So now let's do our second experiment. And remember, it's important that these are truly random numbers. And so we will now start at the first valid number. So we have a two. So this is our second experiment. We got a two."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "And so we will now start at the first valid number. So we have a two. So this is our second experiment. We got a two. We got a one. We can ignore this eight. Then we get a two again."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "We got a two. We got a one. We can ignore this eight. Then we get a two again. We already have that prize. Ignore the nine. Five, that's a prize we need in this experiment."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Then we get a two again. We already have that prize. Ignore the nine. Five, that's a prize we need in this experiment. Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Five, that's a prize we need in this experiment. Nine, we can ignore. And then four, haven't gotten that prize yet in this experiment. Three, haven't gotten that prize yet in this experiment. One, we already got that prize. Three, we already got that prize. Three, already got that prize."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Three, haven't gotten that prize yet in this experiment. One, we already got that prize. Three, we already got that prize. Three, already got that prize. Two, two, already got those prizes. Zero, we already got all of these prizes over here. We can ignore the zero."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Three, already got that prize. Two, two, already got those prizes. Zero, we already got all of these prizes over here. We can ignore the zero. Already got that prize. And finally, we get prize number six. So how many boxes did we need in that second experiment?"}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "We can ignore the zero. Already got that prize. And finally, we get prize number six. So how many boxes did we need in that second experiment? Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So how many boxes did we need in that second experiment? Well, let's see. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17 boxes. So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going. Let's do this one more time. This is strangely fun."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "So in experiment two, I needed 17, or Amanda needed 17 boxes. And she can keep going. Let's do this one more time. This is strangely fun. So experiment three. Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "This is strangely fun. So experiment three. Now remember, we only wanna look at the valid numbers. We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize. These are all invalid. In fact, and then we go to five, we get that prize."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "We'll ignore the invalid numbers, the ones that don't give us a valid prize number. So four, we get that prize. These are all invalid. In fact, and then we go to five, we get that prize. Five, we already have it. We get the two prize. Seven and eight are invalid."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "In fact, and then we go to five, we get that prize. Five, we already have it. We get the two prize. Seven and eight are invalid. Seven's invalid. Six, we get that prize. Seven's invalid."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Seven and eight are invalid. Seven's invalid. Six, we get that prize. Seven's invalid. One, we got that prize. One, we already got it. Nine's invalid."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Seven's invalid. One, we got that prize. One, we already got it. Nine's invalid. Two, we already got it. Nine is invalid. One, we already got the one prize."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Nine's invalid. Two, we already got it. Nine is invalid. One, we already got the one prize. And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through? Let's see."}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "One, we already got the one prize. And then finally, we get prize number three, which was the missing prize. So how many boxes, valid boxes, did we have to go through? Let's see. One, two, three, four, five, six, seven, eight, nine, 10. 10. So with only three experiments, what was our average?"}, {"video_title": "Random number list to run experiment Probability AP Statistics Khan Academy.mp3", "Sentence": "Let's see. One, two, three, four, five, six, seven, eight, nine, 10. 10. So with only three experiments, what was our average? Well, with these three experiments, our average is going to be eight plus 17 plus 10 over three. So let's see, this is 2535 over three, which is equal to 11 2 3rds. Now, do we know that this is the true theoretical expected number of boxes that you would need to get?"}, {"video_title": "Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3", "Sentence": "First-year students at a certain large university are required to live on campus in one of the 24 available residence halls. After their first year, students have the option to live away from campus, but many choose to continue living in the residence halls. Estella oversees 12 of these residence halls. Her department surveyed a large, simple, random sample of first-year students who live in those 12 residence halls about their overall satisfaction with campus living. Estella can safely generalize the results of the survey to which population. So pause this video and see if you can figure it out. Alright, so let's do this together."}, {"video_title": "Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3", "Sentence": "Her department surveyed a large, simple, random sample of first-year students who live in those 12 residence halls about their overall satisfaction with campus living. Estella can safely generalize the results of the survey to which population. So pause this video and see if you can figure it out. Alright, so let's do this together. So Estella has done, it's a large, simple, random sample of first-year students. So let's see. Choice A is only those students who were surveyed."}, {"video_title": "Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3", "Sentence": "Alright, so let's do this together. So Estella has done, it's a large, simple, random sample of first-year students. So let's see. Choice A is only those students who were surveyed. Well no, this was a simple, random sample and it was a large sample, so it's meant to be indicative of all first-year students. You can generalize more than just making statements about just the students who were surveyed. All first-year students, but only those who live in these 12 residence halls."}, {"video_title": "Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3", "Sentence": "Choice A is only those students who were surveyed. Well no, this was a simple, random sample and it was a large sample, so it's meant to be indicative of all first-year students. You can generalize more than just making statements about just the students who were surveyed. All first-year students, but only those who live in these 12 residence halls. Yeah, I think this one looks fair, because you can't generalize to people who don't live in those residence halls. Maybe Estella oversees the 12 best residence halls or the 12 worst residence halls. And so you wouldn't get, if that were the case, you would not be able to generalize beyond that."}, {"video_title": "Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3", "Sentence": "All first-year students, but only those who live in these 12 residence halls. Yeah, I think this one looks fair, because you can't generalize to people who don't live in those residence halls. Maybe Estella oversees the 12 best residence halls or the 12 worst residence halls. And so you wouldn't get, if that were the case, you would not be able to generalize beyond that. Or these might be the 12 that are closest to campus or the 12 that are furthest from campus. So you can only generalize to people who live in those residence halls. All students, first-year or not, but only those who live in these 12 residence halls."}, {"video_title": "Generalizabilty of survey results example Study design AP Statistics Khan Academy.mp3", "Sentence": "And so you wouldn't get, if that were the case, you would not be able to generalize beyond that. Or these might be the 12 that are closest to campus or the 12 that are furthest from campus. So you can only generalize to people who live in those residence halls. All students, first-year or not, but only those who live in these 12 residence halls. Well the issue here is that a second-year student or third-year student might just have a different perspective, even if they live in that same building, and we did a large, random sample of first-year students. We didn't do a large, random sample of all people in those 12 residence halls, so we'll rule that out. All first-year students at the entire university, but not students beyond their first year."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "For a year, James records whether each day is sunny, cloudy, rainy, or snowy, as well as whether this train arrives on time or is delayed. His results are displayed in the table below. Alright, this is interesting. These columns, on time, delayed, and the total. So for example, when it was sunny, there's a total of 170 sunny days that year, 167 of which the train was on time, three of which the train was delayed. And we can look at that by the different types of weather conditions. And then they say, for these days, are the events delayed and snowy independent?"}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "These columns, on time, delayed, and the total. So for example, when it was sunny, there's a total of 170 sunny days that year, 167 of which the train was on time, three of which the train was delayed. And we can look at that by the different types of weather conditions. And then they say, for these days, are the events delayed and snowy independent? So to think about this, and remember, we're only going to be able to figure out experimental probabilities, and you should always view experimental probabilities as somewhat suspect. The more experiments you're able to take, the more likely it is to approximate the true theoretical probability, but there's always some chance that they might be different or even quite different. Let's use this data to try to calculate the experimental probability."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "And then they say, for these days, are the events delayed and snowy independent? So to think about this, and remember, we're only going to be able to figure out experimental probabilities, and you should always view experimental probabilities as somewhat suspect. The more experiments you're able to take, the more likely it is to approximate the true theoretical probability, but there's always some chance that they might be different or even quite different. Let's use this data to try to calculate the experimental probability. So the key question here is, what is the probability that the train is delayed? And then we want to think about, what is the probability that the train is delayed given that it is snowy? If we knew the theoretical probabilities, and if they were exactly the same, if the probability of being delayed was exactly the same as the probability of being delayed given snowy, then being delayed or being snowy would be independent."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "Let's use this data to try to calculate the experimental probability. So the key question here is, what is the probability that the train is delayed? And then we want to think about, what is the probability that the train is delayed given that it is snowy? If we knew the theoretical probabilities, and if they were exactly the same, if the probability of being delayed was exactly the same as the probability of being delayed given snowy, then being delayed or being snowy would be independent. But if we knew the theoretical probabilities and the probability of being delayed given snowy were different than the probability of being delayed, then we would not say that these are independent variables. Now, we don't know the theoretical probabilities. We're just going to calculate the experimental probabilities."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "If we knew the theoretical probabilities, and if they were exactly the same, if the probability of being delayed was exactly the same as the probability of being delayed given snowy, then being delayed or being snowy would be independent. But if we knew the theoretical probabilities and the probability of being delayed given snowy were different than the probability of being delayed, then we would not say that these are independent variables. Now, we don't know the theoretical probabilities. We're just going to calculate the experimental probabilities. And we do have a good number of experiments here. So if these are quite different, I would feel confident saying that they are dependent. If they are pretty close with the experimental probability, I would say that it would be hard to make the statement that they are dependent and that you would probably lean towards independence."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "We're just going to calculate the experimental probabilities. And we do have a good number of experiments here. So if these are quite different, I would feel confident saying that they are dependent. If they are pretty close with the experimental probability, I would say that it would be hard to make the statement that they are dependent and that you would probably lean towards independence. But let's calculate this. What is the probability that the train is just delayed? Pause this video and try to figure that out."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "If they are pretty close with the experimental probability, I would say that it would be hard to make the statement that they are dependent and that you would probably lean towards independence. But let's calculate this. What is the probability that the train is just delayed? Pause this video and try to figure that out. Well, let's see. If we just think in general, we have a total of 365 trials or 365 experiments. And of them, the train was delayed 35 times."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "Pause this video and try to figure that out. Well, let's see. If we just think in general, we have a total of 365 trials or 365 experiments. And of them, the train was delayed 35 times. Now, what's the probability that the train is delayed given that it is snowy? Pause the video and try to figure that out. Well, let's see."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "And of them, the train was delayed 35 times. Now, what's the probability that the train is delayed given that it is snowy? Pause the video and try to figure that out. Well, let's see. We have a total of 20 snowy days. And we are delayed 12 of those 20 snowy days. And so this is going to be a probability."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "Well, let's see. We have a total of 20 snowy days. And we are delayed 12 of those 20 snowy days. And so this is going to be a probability. 12 20ths is the same thing as, if we multiply both the numerator and the denominator by five, this is a 60% probability, or I could say a 0.6 probability of being delayed when it is snowy. This is, of course, an experimental probability, which is much higher than this. This is less than 10% right over here."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "And so this is going to be a probability. 12 20ths is the same thing as, if we multiply both the numerator and the denominator by five, this is a 60% probability, or I could say a 0.6 probability of being delayed when it is snowy. This is, of course, an experimental probability, which is much higher than this. This is less than 10% right over here. This right over here is less than 0.1. I could get a calculator to calculate it exactly. It'll be 9 point something percent or 0.9 something."}, {"video_title": "Conditional probability and independence Probability AP Statistics Khan Academy.mp3", "Sentence": "This is less than 10% right over here. This right over here is less than 0.1. I could get a calculator to calculate it exactly. It'll be 9 point something percent or 0.9 something. But clearly, this, you are much more likely, at least from the experimental data, it seems like you have a much higher proportion of your snowy days are delayed than just general days in general, than just general days. And so based on this data, because the experimental probability of being delayed given snowy is so much higher than the experimental probability of just being delayed, I would make the statement that these are not independent. So for these days, are the events delayed and snowy independent?"}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "That should be choices. Each problem has only one correct answer. What is the probability of randomly guessing the correct answer on both problems? Now, the probability of guessing the correct answer on each problem, these are independent events. So let's write this down. The probability of correct on problem number one is independent. Or let me write it this way."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "Now, the probability of guessing the correct answer on each problem, these are independent events. So let's write this down. The probability of correct on problem number one is independent. Or let me write it this way. Probability of correct on number one and probability of correct on problem two are independent. Independent. Which means that the outcome of one of the events of guessing on the first problem isn't going to affect the probability of guessing correctly on the second problem, independent events."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "Or let me write it this way. Probability of correct on number one and probability of correct on problem two are independent. Independent. Which means that the outcome of one of the events of guessing on the first problem isn't going to affect the probability of guessing correctly on the second problem, independent events. So the probability of guessing on both of them, so that means that the probability of being correct on one and number two is going to be equal to the product of these probabilities. And we're going to see why that is visually in a second. But it's going to be the probability of correct on number one times the probability of being correct on number two."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "Which means that the outcome of one of the events of guessing on the first problem isn't going to affect the probability of guessing correctly on the second problem, independent events. So the probability of guessing on both of them, so that means that the probability of being correct on one and number two is going to be equal to the product of these probabilities. And we're going to see why that is visually in a second. But it's going to be the probability of correct on number one times the probability of being correct on number two. Now, what are each of these probabilities? On number one, there are four choices. There are four possible outcomes."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "But it's going to be the probability of correct on number one times the probability of being correct on number two. Now, what are each of these probabilities? On number one, there are four choices. There are four possible outcomes. And only one of them is going to be correct. Each one only has one correct answer. So the probability of being correct on problem one is 1 fourth, and then the probability of being correct on problem number two has three choices."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "There are four possible outcomes. And only one of them is going to be correct. Each one only has one correct answer. So the probability of being correct on problem one is 1 fourth, and then the probability of being correct on problem number two has three choices. So there's three possible outcomes. And there's only one correct one. So only one of them are correct."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "So the probability of being correct on problem one is 1 fourth, and then the probability of being correct on problem number two has three choices. So there's three possible outcomes. And there's only one correct one. So only one of them are correct. So probability of correct on number two is 1 third. Probability of guessing correct on number one is 1 fourth. The probability of doing on both of them is going to be its product."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "So only one of them are correct. So probability of correct on number two is 1 third. Probability of guessing correct on number one is 1 fourth. The probability of doing on both of them is going to be its product. So it's going to be equal to 1 fourth times 1 third is 1 twelfth. Now, to see kind of visually why this makes sense, let's draw a little chart here. And we did a similar thing when we thought about rolling two separate dice."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "The probability of doing on both of them is going to be its product. So it's going to be equal to 1 fourth times 1 third is 1 twelfth. Now, to see kind of visually why this makes sense, let's draw a little chart here. And we did a similar thing when we thought about rolling two separate dice. So let's think about problem number one has four choices, only one of which is correct. So let's write incorrect choice one, incorrect choice two, incorrect choice three, and then it has the correct choice over there. So those are the four choices."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "And we did a similar thing when we thought about rolling two separate dice. So let's think about problem number one has four choices, only one of which is correct. So let's write incorrect choice one, incorrect choice two, incorrect choice three, and then it has the correct choice over there. So those are the four choices. They're not going to necessarily be in that order on the exam, but we can just list them in this order. Now, problem number two has three choices, only one of which is correct. So problem number two has incorrect choice one, incorrect choice two, and then let's say the third choice is correct."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "So those are the four choices. They're not going to necessarily be in that order on the exam, but we can just list them in this order. Now, problem number two has three choices, only one of which is correct. So problem number two has incorrect choice one, incorrect choice two, and then let's say the third choice is correct. It's not necessarily in that order, but we know it has two incorrect and one correct choices. Now, what are all of the different possible outcomes? We can draw a grid here, all of these possible outcomes."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "So problem number two has incorrect choice one, incorrect choice two, and then let's say the third choice is correct. It's not necessarily in that order, but we know it has two incorrect and one correct choices. Now, what are all of the different possible outcomes? We can draw a grid here, all of these possible outcomes. Let's draw all of the outcomes. Each of these cells or each of these boxes in a grid are a possible outcome. You're just guessing."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "We can draw a grid here, all of these possible outcomes. Let's draw all of the outcomes. Each of these cells or each of these boxes in a grid are a possible outcome. You're just guessing. You're randomly choosing one of these four. You're randomly choosing one of these four. So you might get incorrect choice one and incorrect choice one, incorrect choice in problem number one, and then incorrect choice in problem number two."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "You're just guessing. You're randomly choosing one of these four. You're randomly choosing one of these four. So you might get incorrect choice one and incorrect choice one, incorrect choice in problem number one, and then incorrect choice in problem number two. That would be that cell right there. Maybe you get problem number one correct, but you get incorrect choice number two in problem number two. So these would represent all of the possible outcomes when you guess on each problem."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "So you might get incorrect choice one and incorrect choice one, incorrect choice in problem number one, and then incorrect choice in problem number two. That would be that cell right there. Maybe you get problem number one correct, but you get incorrect choice number two in problem number two. So these would represent all of the possible outcomes when you guess on each problem. And which of these outcomes represent getting correct on both? Well, getting correct on both is only this one. Correct on choice one and correct on choice on problem number two."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "So these would represent all of the possible outcomes when you guess on each problem. And which of these outcomes represent getting correct on both? Well, getting correct on both is only this one. Correct on choice one and correct on choice on problem number two. And so that's one of the possible outcomes. And how many total outcomes are there? There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12."}, {"video_title": "Test taking probability and independent events Precalculus Khan Academy.mp3", "Sentence": "Correct on choice one and correct on choice on problem number two. And so that's one of the possible outcomes. And how many total outcomes are there? There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Out of 12 possible outcomes. Or, since these are independent events, you can multiply. You see that there are 12 outcomes because there's 12 possible outcomes."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So right over here, we have a fairly simple least squares regression. We're trying to fit four points. And in previous videos, we actually came up with the equation of this least squares regression line. What I'm going to do now is plot the residuals for each of these points. So what is a residual? Well, just as a reminder, your residual for a given point is equal to the actual minus the expected. So how do I make that tangible?"}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "What I'm going to do now is plot the residuals for each of these points. So what is a residual? Well, just as a reminder, your residual for a given point is equal to the actual minus the expected. So how do I make that tangible? Well, what's the residual for this point right over here? For this point here, the actual y, when x equals one, is one, but the expected, when x equals one for this least squares regression line, 2.5 times one minus two, well, that's going to be.5. And so our residual is one minus.5, so we have a positive, we have a positive 0.5 residual."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So how do I make that tangible? Well, what's the residual for this point right over here? For this point here, the actual y, when x equals one, is one, but the expected, when x equals one for this least squares regression line, 2.5 times one minus two, well, that's going to be.5. And so our residual is one minus.5, so we have a positive, we have a positive 0.5 residual. Over for this point, you have zero residual. The actual is the expected. For this point right over here, the actual, when x equals two, for y is two, but the expected is three."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "And so our residual is one minus.5, so we have a positive, we have a positive 0.5 residual. Over for this point, you have zero residual. The actual is the expected. For this point right over here, the actual, when x equals two, for y is two, but the expected is three. So our residual over here, once again, the actual is y equals two when x equals two. The expected, two times 2.5 minus two is three, so this is going to be two minus three, which equals a residual of negative one. And then over here, our residual, our actual, when x equals three, is six."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "For this point right over here, the actual, when x equals two, for y is two, but the expected is three. So our residual over here, once again, the actual is y equals two when x equals two. The expected, two times 2.5 minus two is three, so this is going to be two minus three, which equals a residual of negative one. And then over here, our residual, our actual, when x equals three, is six. Our expected, when x equals three, is 5.5. So six minus 5.5, that is a positive 0.5. So those are the residuals, but how do we plot it?"}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "And then over here, our residual, our actual, when x equals three, is six. Our expected, when x equals three, is 5.5. So six minus 5.5, that is a positive 0.5. So those are the residuals, but how do we plot it? Well, we would set up our axes. Let me do it right over here. One, two, and three."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So those are the residuals, but how do we plot it? Well, we would set up our axes. Let me do it right over here. One, two, and three. And let's see, the maximum residual here is positive 0.5, and then the minimum one here is negative one. So let's see, this could be 0.51, negative 0.5, negative one. So this is negative one, this is positive one here."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "One, two, and three. And let's see, the maximum residual here is positive 0.5, and then the minimum one here is negative one. So let's see, this could be 0.51, negative 0.5, negative one. So this is negative one, this is positive one here. And so when x equals one, what was the residual? Well, the actual was one, expected was 0.5, one minus 0.5 is 0.5. So this right over here, we can plot right over here, the residual is 0.5."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So this is negative one, this is positive one here. And so when x equals one, what was the residual? Well, the actual was one, expected was 0.5, one minus 0.5 is 0.5. So this right over here, we can plot right over here, the residual is 0.5. When x equals two, we actually have two data points. First, I'll do this one. When we have the point two comma three, the residual there is zero."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So this right over here, we can plot right over here, the residual is 0.5. When x equals two, we actually have two data points. First, I'll do this one. When we have the point two comma three, the residual there is zero. So for one of them, the residual is zero. Now for the other one, the residual is negative one. Let me do that in a different color."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "When we have the point two comma three, the residual there is zero. So for one of them, the residual is zero. Now for the other one, the residual is negative one. Let me do that in a different color. For the other one, the residual is negative one, so we would plot it right over here. And then this last point, the residual is positive 0.5. So it is just like that."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "Let me do that in a different color. For the other one, the residual is negative one, so we would plot it right over here. And then this last point, the residual is positive 0.5. So it is just like that. And so this thing that I have just created, where we're just seeing for each x where we have a corresponding point, we plot the point above or below the line based on the residual. This is called a residual plot. Now one question is, why do people even go through the trouble of creating a residual plot like this?"}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So it is just like that. And so this thing that I have just created, where we're just seeing for each x where we have a corresponding point, we plot the point above or below the line based on the residual. This is called a residual plot. Now one question is, why do people even go through the trouble of creating a residual plot like this? The answer is, regardless of whether the regression line is upward sloping or downward sloping, this gives you a sense of how good a fit it is and whether a line is good at explaining the relationship between the variables. The general idea is, if you see the points pretty evenly scattered or randomly scattered above and below this line, you don't really discern any trend here, then a line is probably a good model for the data. But if you do see some type of trend, if the residuals had an upward trend like this, or if they were curving up and then curving down, or they had a downward trend, then you might say, hey, this line isn't a good fit, and maybe we would have to do a nonlinear model."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "Now one question is, why do people even go through the trouble of creating a residual plot like this? The answer is, regardless of whether the regression line is upward sloping or downward sloping, this gives you a sense of how good a fit it is and whether a line is good at explaining the relationship between the variables. The general idea is, if you see the points pretty evenly scattered or randomly scattered above and below this line, you don't really discern any trend here, then a line is probably a good model for the data. But if you do see some type of trend, if the residuals had an upward trend like this, or if they were curving up and then curving down, or they had a downward trend, then you might say, hey, this line isn't a good fit, and maybe we would have to do a nonlinear model. What are some examples of other residual plots? And let's try to analyze them a bit. So right here you have a regression line and its corresponding residual plot."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "But if you do see some type of trend, if the residuals had an upward trend like this, or if they were curving up and then curving down, or they had a downward trend, then you might say, hey, this line isn't a good fit, and maybe we would have to do a nonlinear model. What are some examples of other residual plots? And let's try to analyze them a bit. So right here you have a regression line and its corresponding residual plot. And once again, you see here, the residual is slightly positive. The actual is slightly above the line, and you see it right over there, it's slightly positive. This one's even more positive, you see it there."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So right here you have a regression line and its corresponding residual plot. And once again, you see here, the residual is slightly positive. The actual is slightly above the line, and you see it right over there, it's slightly positive. This one's even more positive, you see it there. But like the example we just looked at, it looks like these residuals are pretty evenly scattered above and below the line. There isn't any discernible trend. And so I would say that a linear model here, and in particular this regression line, is a good model for this data."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "This one's even more positive, you see it there. But like the example we just looked at, it looks like these residuals are pretty evenly scattered above and below the line. There isn't any discernible trend. And so I would say that a linear model here, and in particular this regression line, is a good model for this data. But if we see something like this, a different picture emerges. When I look at just the residual plot, it doesn't look like they're evenly scattered. It looks like there's some type of trend here."}, {"video_title": "Residual plots Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "And so I would say that a linear model here, and in particular this regression line, is a good model for this data. But if we see something like this, a different picture emerges. When I look at just the residual plot, it doesn't look like they're evenly scattered. It looks like there's some type of trend here. I'm going down here, but then I'm going back up. When you see something like this, where on the residual plot you're going below the x-axis and then above, then it might say, hey, a linear model might not be appropriate. Maybe some type of nonlinear model, some type of nonlinear curve might better fit the data, or the relationship between the y and the x is nonlinear."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "We're told that a board game has players roll two three-sided dice. These exist, and actually I looked it up, they do exist and they're actually fascinating. And subtract the numbers showing on the faces. The game only looks at non-negative differences. For example, if a player rolls a one and a three, the difference is two. Let D represent the difference in a given roll. Construct the theoretical probability distribution of D. So pause this video and see if you can have a go at that before we work through it together."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "The game only looks at non-negative differences. For example, if a player rolls a one and a three, the difference is two. Let D represent the difference in a given roll. Construct the theoretical probability distribution of D. So pause this video and see if you can have a go at that before we work through it together. All right, now let's work through it together. So let's just think about all of the scenarios for the two die. So let me draw a little table here."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "Construct the theoretical probability distribution of D. So pause this video and see if you can have a go at that before we work through it together. All right, now let's work through it together. So let's just think about all of the scenarios for the two die. So let me draw a little table here. So let me do it like that. And let me do it like this. And then let me put a little divider right over here."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "So let me draw a little table here. So let me do it like that. And let me do it like this. And then let me put a little divider right over here. And for this top, this is going to be die one, and then this is going to be die two. Die one can take on one, two, or three. And die two could be one, two, or three."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "And then let me put a little divider right over here. And for this top, this is going to be die one, and then this is going to be die two. Die one can take on one, two, or three. And die two could be one, two, or three. And so let me finish making this a bit of a table here. And what we wanna do is look at the difference, but the non-negative difference. So we'll always subtract the lower die from the higher die."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "And die two could be one, two, or three. And so let me finish making this a bit of a table here. And what we wanna do is look at the difference, but the non-negative difference. So we'll always subtract the lower die from the higher die. So what's the difference here? Well, this is going to be zero if I roll a one and a one. Now, what if I roll a two and a one?"}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "So we'll always subtract the lower die from the higher die. So what's the difference here? Well, this is going to be zero if I roll a one and a one. Now, what if I roll a two and a one? Well, here the difference is going to be two minus one, which is one. Here the difference is three minus one, which is two. Now, what about right over here?"}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "Now, what if I roll a two and a one? Well, here the difference is going to be two minus one, which is one. Here the difference is three minus one, which is two. Now, what about right over here? Well, here the higher die is two, the lower one is one right over here. So two minus one is one, two minus two is zero. And now this is going to be the higher roll."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "Now, what about right over here? Well, here the higher die is two, the lower one is one right over here. So two minus one is one, two minus two is zero. And now this is going to be the higher roll. Die one is gonna have the higher roll in this scenario. Three minus two is one. And then right over here, three minus one is two."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "And now this is going to be the higher roll. Die one is gonna have the higher roll in this scenario. Three minus two is one. And then right over here, three minus one is two. Now, if die one rolls a two, die two rolls a three. Die three is higher, three minus two is one. And then three minus three is zero."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "And then right over here, three minus one is two. Now, if die one rolls a two, die two rolls a three. Die three is higher, three minus two is one. And then three minus three is zero. So we've come up with all of the scenarios, and we can see that we're either gonna end up with a zero or one or a two when we look at the positive difference. So there's a scenario of getting a zero, a one, or a two. Those are the different differences that we could actually get."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "And then three minus three is zero. So we've come up with all of the scenarios, and we can see that we're either gonna end up with a zero or one or a two when we look at the positive difference. So there's a scenario of getting a zero, a one, or a two. Those are the different differences that we could actually get. And so let's think about the probability of each of them. What's the probability that the difference is zero? Well, we can see that one, two, three of the nine equally likely outcomes result in a difference of zero."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "Those are the different differences that we could actually get. And so let's think about the probability of each of them. What's the probability that the difference is zero? Well, we can see that one, two, three of the nine equally likely outcomes result in a difference of zero. So it's gonna be three out of nine or 1 3rd. What about a difference of, let me use blue, one? Well, we could see there are one, two, three, four of the nine scenarios have that."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "Well, we can see that one, two, three of the nine equally likely outcomes result in a difference of zero. So it's gonna be three out of nine or 1 3rd. What about a difference of, let me use blue, one? Well, we could see there are one, two, three, four of the nine scenarios have that. So there is a 4 9th probability. And then last but not least, a difference of two. Well, there's two out of the nine scenarios that have that."}, {"video_title": "Theoretical probability distribution example tables Probability & combinatorics.mp3", "Sentence": "Well, we could see there are one, two, three, four of the nine scenarios have that. So there is a 4 9th probability. And then last but not least, a difference of two. Well, there's two out of the nine scenarios that have that. So there is a 2 9th probability right over there. And we're done. We've constructed the theoretical probability distribution of D."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And we want to get a sense of how these students feel about the quality of math instruction at this school. So we construct a survey, and we just need to decide who are we going to get to actually answer this survey. One option is to just go to every member of the population, but let's just say it's a really large school. Let's say we're a college, and there's 10,000 people in the college. We say, well, we can't just talk to everyone. So instead we say, let's sample this population to get an indication of how the entire school feels. So we are going to sample it."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Let's say we're a college, and there's 10,000 people in the college. We say, well, we can't just talk to everyone. So instead we say, let's sample this population to get an indication of how the entire school feels. So we are going to sample it. We're going to sample that population. Now, in order to avoid having bias in our response, in order for it to have the best chance of it being indicative of the entire population, we want our sample to be random. So our sample could either be random, random, or not random, not random."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So we are going to sample it. We're going to sample that population. Now, in order to avoid having bias in our response, in order for it to have the best chance of it being indicative of the entire population, we want our sample to be random. So our sample could either be random, random, or not random, not random. And it might seem at first pretty straightforward to do a random sample, but when you actually get down to it, it's not always as straightforward as you would think. So one type of random sample is just a simple random sample. So simple, simple, random, random sample."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So our sample could either be random, random, or not random, not random. And it might seem at first pretty straightforward to do a random sample, but when you actually get down to it, it's not always as straightforward as you would think. So one type of random sample is just a simple random sample. So simple, simple, random, random sample. And this is saying, all right, let me maybe assign a number to every person in the school. Maybe they already have a student ID number. And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So simple, simple, random, random sample. And this is saying, all right, let me maybe assign a number to every person in the school. Maybe they already have a student ID number. And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students. So let's say there's a sample of 100 students that I'm going to apply the survey to. So that would be a simple random sample. We are just going into this whole population and randomly, let me just draw this."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students. So let's say there's a sample of 100 students that I'm going to apply the survey to. So that would be a simple random sample. We are just going into this whole population and randomly, let me just draw this. So this is the population. We are just randomly picking people out. And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "We are just going into this whole population and randomly, let me just draw this. So this is the population. We are just randomly picking people out. And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students. Now that's pretty good. It's unlikely that you're going to have bias from this sample. But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students. Now that's pretty good. It's unlikely that you're going to have bias from this sample. But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors. And that's a possibility. So even though you're taking a simple random sample and it's truly random, once again, it's some probability that's not indicative of the entire population. And so to mitigate that, there are other techniques at our disposal."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors. And that's a possibility. So even though you're taking a simple random sample and it's truly random, once again, it's some probability that's not indicative of the entire population. And so to mitigate that, there are other techniques at our disposal. One technique is a stratified sample. Stratified. And so this is the idea of taking our entire population and essentially stratifying it."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And so to mitigate that, there are other techniques at our disposal. One technique is a stratified sample. Stratified. And so this is the idea of taking our entire population and essentially stratifying it. So let's say we take that same population, we take that same population, I'll draw it as a square here just for convenience, and we're gonna stratify it by, let's say we're concerned that we get an appropriate sample of freshmen, sophomores, juniors, and seniors. So we'll stratify it by freshmen, sophomores, juniors, and seniors. And then we sample 25 from each of these groups."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And so this is the idea of taking our entire population and essentially stratifying it. So let's say we take that same population, we take that same population, I'll draw it as a square here just for convenience, and we're gonna stratify it by, let's say we're concerned that we get an appropriate sample of freshmen, sophomores, juniors, and seniors. So we'll stratify it by freshmen, sophomores, juniors, and seniors. And then we sample 25 from each of these groups. So these are the stratifications. This is freshmen, sophomore, juniors, and seniors. And instead of just sampling 100 out of the entire pool, we sample 25 from each of these."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And then we sample 25 from each of these groups. So these are the stratifications. This is freshmen, sophomore, juniors, and seniors. And instead of just sampling 100 out of the entire pool, we sample 25 from each of these. So just like that. And so that makes sure that you are getting indicative responses from at least all of the different age groups or levels within your university. Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And instead of just sampling 100 out of the entire pool, we sample 25 from each of these. So just like that. And so that makes sure that you are getting indicative responses from at least all of the different age groups or levels within your university. Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school. And there is some probability, if I do 100 random people, it's very likely that it's close to 50-50, but there's some chance just due to randomness that it's disproportionately male or disproportionately female. And that's even possible in the stratified case. And so what you might say is, well, you know what I'm gonna do?"}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school. And there is some probability, if I do 100 random people, it's very likely that it's close to 50-50, but there's some chance just due to randomness that it's disproportionately male or disproportionately female. And that's even possible in the stratified case. And so what you might say is, well, you know what I'm gonna do? I'm going to, there's a technique called a clustered sample. Let me write this right over here. Clustered, a clustered sample."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And so what you might say is, well, you know what I'm gonna do? I'm going to, there's a technique called a clustered sample. Let me write this right over here. Clustered, a clustered sample. And what we do is we sample groups. Each of those groups, we feel confident, has a good balance of male, females. So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Clustered, a clustered sample. And what we do is we sample groups. Each of those groups, we feel confident, has a good balance of male, females. So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do. We'll, let's just say that we can split our, let's say we can split our population into groups. Maybe these are classrooms. And each of these classrooms have an even distribution of males and females, or pretty close to even distributions."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do. We'll, let's just say that we can split our, let's say we can split our population into groups. Maybe these are classrooms. And each of these classrooms have an even distribution of males and females, or pretty close to even distributions. And so what we do is we sample the actual classrooms. So that's why it's called cluster, or cluster technique, or clustered random sample, because we're going to randomly sample our classrooms, each of which have a close, or maybe an exact balance of males and females. So we know that we're gonna get good representation, but we are still sampling."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And each of these classrooms have an even distribution of males and females, or pretty close to even distributions. And so what we do is we sample the actual classrooms. So that's why it's called cluster, or cluster technique, or clustered random sample, because we're going to randomly sample our classrooms, each of which have a close, or maybe an exact balance of males and females. So we know that we're gonna get good representation, but we are still sampling. We are sampling from the clusters, but then we're gonna survey every single person in each of these clusters, every single person in one of these classrooms. So once again, these are all forms of random surveys, or random samples. You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So we know that we're gonna get good representation, but we are still sampling. We are sampling from the clusters, but then we're gonna survey every single person in each of these clusters, every single person in one of these classrooms. So once again, these are all forms of random surveys, or random samples. You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster. Now, if these are all random samples, what are the non-random things like? Well, one case of non-random, you could have a voluntary, voluntary survey, or voluntary sample. This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster. Now, if these are all random samples, what are the non-random things like? Well, one case of non-random, you could have a voluntary, voluntary survey, or voluntary sample. This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey. And that's likely to introduce bias, because you might have, maybe, the students who really like the math instruction at their school, more likely to fill it out. Maybe the students who really don't like it, more likely to fill it out. Maybe it's just the kids who have more time, more likely to fill it out."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey. And that's likely to introduce bias, because you might have, maybe, the students who really like the math instruction at their school, more likely to fill it out. Maybe the students who really don't like it, more likely to fill it out. Maybe it's just the kids who have more time, more likely to fill it out. So this has a good chance of introducing bias. The students who fill out the survey might be just more skewed one way or the other, because they volunteered for it. Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Maybe it's just the kids who have more time, more likely to fill it out. So this has a good chance of introducing bias. The students who fill out the survey might be just more skewed one way or the other, because they volunteered for it. Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used. And this might say, well, let's just sample the hundred first students who show up in school. And that's just convenient for me, because I didn't have to do these random numbers, or do the stratification, or doing any of this clustering. But you can understand how this also would introduce bias."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used. And this might say, well, let's just sample the hundred first students who show up in school. And that's just convenient for me, because I didn't have to do these random numbers, or do the stratification, or doing any of this clustering. But you can understand how this also would introduce bias. Because the first hundred students who show up at school, maybe those are the most diligent students, maybe they all take an early math class that has a very good instructor, or they're all happy about it. Or it might go the other way. The instructor there isn't the best one, and so it might introduce bias the other way."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "But you can understand how this also would introduce bias. Because the first hundred students who show up at school, maybe those are the most diligent students, maybe they all take an early math class that has a very good instructor, or they're all happy about it. Or it might go the other way. The instructor there isn't the best one, and so it might introduce bias the other way. So if you let people volunteer, or you just say, oh, let me go to the first N students, or you say, hey, let me just talk to all the students who happen to be in front of me right now. They might be in front of you out of convenience, but they might not be a true random sample. Now, there's other reasons why you might introduce bias, and it might not be because of the sampling."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "The instructor there isn't the best one, and so it might introduce bias the other way. So if you let people volunteer, or you just say, oh, let me go to the first N students, or you say, hey, let me just talk to all the students who happen to be in front of me right now. They might be in front of you out of convenience, but they might not be a true random sample. Now, there's other reasons why you might introduce bias, and it might not be because of the sampling. You might introduce bias because of the wording of your survey. You could imagine a survey that says, do you consider yourself lucky to get a math education that very few other people in the world have access to? Well, that might bias you to say, well, yeah, I guess I feel lucky."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Now, there's other reasons why you might introduce bias, and it might not be because of the sampling. You might introduce bias because of the wording of your survey. You could imagine a survey that says, do you consider yourself lucky to get a math education that very few other people in the world have access to? Well, that might bias you to say, well, yeah, I guess I feel lucky. Well, if the wording was, do you like the fact that a disproportionate, more students at your school tend to fail algebra than our surrounding schools? Well, that might bias you negatively. So the wording really, really, really matters in surveys, and there's a lot that would go into this."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Well, that might bias you to say, well, yeah, I guess I feel lucky. Well, if the wording was, do you like the fact that a disproportionate, more students at your school tend to fail algebra than our surrounding schools? Well, that might bias you negatively. So the wording really, really, really matters in surveys, and there's a lot that would go into this. And the other one is just people's, it's called response bias. And once again, this isn't about response bias. And this is just people not wanting to tell the truth or maybe not wanting to respond at all."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So the wording really, really, really matters in surveys, and there's a lot that would go into this. And the other one is just people's, it's called response bias. And once again, this isn't about response bias. And this is just people not wanting to tell the truth or maybe not wanting to respond at all. Maybe they're afraid that somehow their response is gonna show up in front of their math teacher or the administrators, or if they're too negative, it might be taken out on them in some way. And because of that, they might not be truthful. And so they might be overly positive or not fill it out at all."}, {"video_title": "Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And this is just people not wanting to tell the truth or maybe not wanting to respond at all. Maybe they're afraid that somehow their response is gonna show up in front of their math teacher or the administrators, or if they're too negative, it might be taken out on them in some way. And because of that, they might not be truthful. And so they might be overly positive or not fill it out at all. So anyway, this is a very high-level overview of how you could think about sampling. You wanna go random because it lowers the probability of their introducing some bias into it. And then these are some techniques."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And what I'm hoping to do in this video is get a little bit of practice interpreting this. And what I have here are five different statements. And I want you to look at these statements. Pause the video, look at these statements, and think about which of these, based on the information in the box and whiskers plot, which of these are for sure true, which of these are for sure false, and which of these we don't have enough information. It could go either way. All right, so let's work through these. So the first statement is that all of the students are less than 17 years old."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Pause the video, look at these statements, and think about which of these, based on the information in the box and whiskers plot, which of these are for sure true, which of these are for sure false, and which of these we don't have enough information. It could go either way. All right, so let's work through these. So the first statement is that all of the students are less than 17 years old. Well, we see right over here that the maximum age, that's the right end of this right whisker, is 16. So it is the case that all of the students are less than 17 years old. So this is definitely going to be true."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So the first statement is that all of the students are less than 17 years old. Well, we see right over here that the maximum age, that's the right end of this right whisker, is 16. So it is the case that all of the students are less than 17 years old. So this is definitely going to be true. The next statement, at least 75% of the students are 10 years old or older. So when you look at this, this feels right because 10 is the value, that is at the beginning of the second quartile. This is the second quartile right over there."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So this is definitely going to be true. The next statement, at least 75% of the students are 10 years old or older. So when you look at this, this feels right because 10 is the value, that is at the beginning of the second quartile. This is the second quartile right over there. And actually, let me do this in a different color. So this is the second quartile. So 25% of the value of the numbers are in the second, or roughly, sometimes it's not exactly, so approximately, I'll say roughly, 25% are going to be in the second quartile."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "This is the second quartile right over there. And actually, let me do this in a different color. So this is the second quartile. So 25% of the value of the numbers are in the second, or roughly, sometimes it's not exactly, so approximately, I'll say roughly, 25% are going to be in the second quartile. Approximately 25% are going to be in the third quartile. And approximately 25% are going to be in the fourth quartile. So it seems reasonable for saying 10 years old or older that this is going to be true."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So 25% of the value of the numbers are in the second, or roughly, sometimes it's not exactly, so approximately, I'll say roughly, 25% are going to be in the second quartile. Approximately 25% are going to be in the third quartile. And approximately 25% are going to be in the fourth quartile. So it seems reasonable for saying 10 years old or older that this is going to be true. In fact, you could even have a couple of values in the first quartile that are 10. But to make that a little bit more tangible, let's look at some, so I'm feeling good that this is true, but let's look at a few more examples to make this a little bit more concrete. So they don't know, we don't know, based on the information here, exactly how many students are at the party."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So it seems reasonable for saying 10 years old or older that this is going to be true. In fact, you could even have a couple of values in the first quartile that are 10. But to make that a little bit more tangible, let's look at some, so I'm feeling good that this is true, but let's look at a few more examples to make this a little bit more concrete. So they don't know, we don't know, based on the information here, exactly how many students are at the party. We'll have to construct some scenarios. So we could do a scenario. Let's see if we can do, we could do a scenario where, well, let's see, let's see if I can construct something where, let's see, the median is 13."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So they don't know, we don't know, based on the information here, exactly how many students are at the party. We'll have to construct some scenarios. So we could do a scenario. Let's see if we can do, we could do a scenario where, well, let's see, let's see if I can construct something where, let's see, the median is 13. We know that for sure. The median is 13. So if I have an odd number, I would have 13 in the middle, just like that."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's see if we can do, we could do a scenario where, well, let's see, let's see if I can construct something where, let's see, the median is 13. We know that for sure. The median is 13. So if I have an odd number, I would have 13 in the middle, just like that. And maybe I have three on each side. And I'm just making that number up. I'm just trying to see what I can learn about the different types of data sets that could be described by this box and whiskers plot."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So if I have an odd number, I would have 13 in the middle, just like that. And maybe I have three on each side. And I'm just making that number up. I'm just trying to see what I can learn about the different types of data sets that could be described by this box and whiskers plot. So 10 is going to be the middle of the bottom half. So that's 10 right over there. And 15 is going to be the middle of the top half."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "I'm just trying to see what I can learn about the different types of data sets that could be described by this box and whiskers plot. So 10 is going to be the middle of the bottom half. So that's 10 right over there. And 15 is going to be the middle of the top half. That's what this box and whiskers plot is telling us. And they of course tell us what the minimum, the minimum is seven, and they tell us that the maximum is 16. So we know that's seven, and then that is 16."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And 15 is going to be the middle of the top half. That's what this box and whiskers plot is telling us. And they of course tell us what the minimum, the minimum is seven, and they tell us that the maximum is 16. So we know that's seven, and then that is 16. And then this right over here could be anything. It could be 10, it could be 11, it could be 12, it could be 13. It wouldn't change what these medians are."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So we know that's seven, and then that is 16. And then this right over here could be anything. It could be 10, it could be 11, it could be 12, it could be 13. It wouldn't change what these medians are. It wouldn't change this box and whiskers plot. Similarly, this could be 13, it could be 14, it could be 15. And so any of those values wouldn't change it."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It wouldn't change what these medians are. It wouldn't change this box and whiskers plot. Similarly, this could be 13, it could be 14, it could be 15. And so any of those values wouldn't change it. And so 75% are 10 or older. Well, this value, in this case, six out of seven are 10 years old or older. And we could try it out with other scenarios where let's try to minimize the number of 10s given this data set."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And so any of those values wouldn't change it. And so 75% are 10 or older. Well, this value, in this case, six out of seven are 10 years old or older. And we could try it out with other scenarios where let's try to minimize the number of 10s given this data set. Well, we could do something like, let's say that we have eight. So let's see, one, two, three, four, five, six, seven, eight. And so here, we know that the minimum, we know that the minimum is seven, we know that the maximum is 16."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And we could try it out with other scenarios where let's try to minimize the number of 10s given this data set. Well, we could do something like, let's say that we have eight. So let's see, one, two, three, four, five, six, seven, eight. And so here, we know that the minimum, we know that the minimum is seven, we know that the maximum is 16. We know that the mean of these middle two values, we have an even number now. So the median is going to be the mean of these two values. So it's going to be the mean of this and this is going to be 13."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And so here, we know that the minimum, we know that the minimum is seven, we know that the maximum is 16. We know that the mean of these middle two values, we have an even number now. So the median is going to be the mean of these two values. So it's going to be the mean of this and this is going to be 13. And we know that the mean of, we know that the mean of this and this is going to be 10, and that the mean of this and this is going to be 15. So what could we construct? Well, actually, we don't even have to construct to answer this question."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So it's going to be the mean of this and this is going to be 13. And we know that the mean of, we know that the mean of this and this is going to be 10, and that the mean of this and this is going to be 15. So what could we construct? Well, actually, we don't even have to construct to answer this question. We know that this is going to have to be 10 or larger, and then all of these other things are going to be 10 or larger. So this is exactly 75%, exactly 75%, if we assume that this is less than 10, are going to be 10 years old or older. So feeling very good, very good, about this one right over here."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, actually, we don't even have to construct to answer this question. We know that this is going to have to be 10 or larger, and then all of these other things are going to be 10 or larger. So this is exactly 75%, exactly 75%, if we assume that this is less than 10, are going to be 10 years old or older. So feeling very good, very good, about this one right over here. And actually, just to make this concrete, I'll put in some values here. You know, this could be a nine and an 11. This could be a 12 and a 14."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So feeling very good, very good, about this one right over here. And actually, just to make this concrete, I'll put in some values here. You know, this could be a nine and an 11. This could be a 12 and a 14. This could be a 14 and a 16. Or it could be a 15 and a 15. You could think about it in any of those ways."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "This could be a 12 and a 14. This could be a 14 and a 16. Or it could be a 15 and a 15. You could think about it in any of those ways. But feeling very good that this is definitely going to be true based on the information given in this plot. Now they say there's only one seven-year-old at the party. One seven-year-old at the party."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "You could think about it in any of those ways. But feeling very good that this is definitely going to be true based on the information given in this plot. Now they say there's only one seven-year-old at the party. One seven-year-old at the party. Well, this first possibility that we looked at, that was the case. There was only one seven-year-old at the party, and there was one 16-year-old at the party. And actually, that was the next statement."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "One seven-year-old at the party. Well, this first possibility that we looked at, that was the case. There was only one seven-year-old at the party, and there was one 16-year-old at the party. And actually, that was the next statement. There's only one 16-year-old at the party. So both of these seem like we can definitely construct data that's consistent with this box plot, box and whiskers plot, where this is true. But could we construct one where it's not true?"}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And actually, that was the next statement. There's only one 16-year-old at the party. So both of these seem like we can definitely construct data that's consistent with this box plot, box and whiskers plot, where this is true. But could we construct one where it's not true? Well, sure. Let's imagine, let's see, we have our median at 13. Median at 13."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But could we construct one where it's not true? Well, sure. Let's imagine, let's see, we have our median at 13. Median at 13. And then we have, let's see, one, two, three, four, five. This is going to be the 10, the median of this bottom half. This is going to be 15."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Median at 13. And then we have, let's see, one, two, three, four, five. This is going to be the 10, the median of this bottom half. This is going to be 15. This is going to be 7. This is going to be 16. Well, this could also be 7."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "This is going to be 15. This is going to be 7. This is going to be 16. Well, this could also be 7. It doesn't have to be. It could be 7, 8, 9, or 10. This could also be 16."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, this could also be 7. It doesn't have to be. It could be 7, 8, 9, or 10. This could also be 16. It doesn't have to be. It could be 15 as well. But just like that, I've constructed a data set."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "This could also be 16. It doesn't have to be. It could be 15 as well. But just like that, I've constructed a data set. And these could be 10, 11, 12, 13. This could be 10, 11, 12, 13. This could be 13, 14, 15."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But just like that, I've constructed a data set. And these could be 10, 11, 12, 13. This could be 10, 11, 12, 13. This could be 13, 14, 15. This one also could be 13, 14, 15. But the simple thing is, or the basic idea here, I can have a data set where I have multiple 7's and multiple 16's. Or I could have a data set where I only have one 7, or only one 16."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "This could be 13, 14, 15. This one also could be 13, 14, 15. But the simple thing is, or the basic idea here, I can have a data set where I have multiple 7's and multiple 16's. Or I could have a data set where I only have one 7, or only one 16. So both of these statements, we just plain don't know. Now the next statement, exactly half the students are older than 13. Well, if you look at this possibility up here, we saw that 3 out of the 7 are older than 13."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Or I could have a data set where I only have one 7, or only one 16. So both of these statements, we just plain don't know. Now the next statement, exactly half the students are older than 13. Well, if you look at this possibility up here, we saw that 3 out of the 7 are older than 13. So that's not exactly half. 3 7's is not 1 half. But in this one over here, we did see that exactly half are older than 13."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, if you look at this possibility up here, we saw that 3 out of the 7 are older than 13. So that's not exactly half. 3 7's is not 1 half. But in this one over here, we did see that exactly half are older than 13. In fact, if you're saying exactly half, well, in this one, we're saying that exactly half are older than 13. We have an even number right over here. So it is exactly half."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But in this one over here, we did see that exactly half are older than 13. In fact, if you're saying exactly half, well, in this one, we're saying that exactly half are older than 13. We have an even number right over here. So it is exactly half. So it's possible that it's true. It's possible that it's not true based on the information given. We once again do not know."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So it is exactly half. So it's possible that it's true. It's possible that it's not true based on the information given. We once again do not know. Anyway, hopefully you found this interesting. The whole point of me doing this is, when you look at statistics, sometimes it's easy to kind of say, OK, I think it roughly means that. And that's sometimes OK."}, {"video_title": "Interpreting box plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We once again do not know. Anyway, hopefully you found this interesting. The whole point of me doing this is, when you look at statistics, sometimes it's easy to kind of say, OK, I think it roughly means that. And that's sometimes OK. But it's very important to think about what types of actual statements you can make and what you can't make. And it's very important when you're looking at statistics to say, well, you know what? I just don't know that the data actually is not telling me that thing for sure."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And as we begin our journey into the world of statistics, we will be doing a lot of what we can call descriptive statistics. So if we have a bunch of data, and if we want to tell something about all of that data without giving them all of the data, can we somehow describe it with a smaller set of numbers? So that's what we're going to focus on. And then once we build our toolkit on the descriptive statistics, then we can start to make inferences about that data, start to make conclusions, start to make judgments, and we'll start to do a lot of inferential, inferential statistics, make inferences. So with that out of the way, let's think about how we can describe the data. So let's say we have a set of numbers. We can consider this to be data."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then once we build our toolkit on the descriptive statistics, then we can start to make inferences about that data, start to make conclusions, start to make judgments, and we'll start to do a lot of inferential, inferential statistics, make inferences. So with that out of the way, let's think about how we can describe the data. So let's say we have a set of numbers. We can consider this to be data. Maybe we're measuring the heights of our plants in our garden. And let's say we have six plants, and the heights are four inches, three inches, one inch, six inches, and another one's one inch, and then another one is seven inches. And let's say someone just said, in another room, not looking at your plants, just said, well, you know, how tall are your plants?"}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We can consider this to be data. Maybe we're measuring the heights of our plants in our garden. And let's say we have six plants, and the heights are four inches, three inches, one inch, six inches, and another one's one inch, and then another one is seven inches. And let's say someone just said, in another room, not looking at your plants, just said, well, you know, how tall are your plants? And they only want to hear one number. They want to somehow have one number that represents all of these different heights of plants. How would you do that?"}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And let's say someone just said, in another room, not looking at your plants, just said, well, you know, how tall are your plants? And they only want to hear one number. They want to somehow have one number that represents all of these different heights of plants. How would you do that? Well, you'd say, well, how can I find something that maybe I want a typical number, maybe I want some number that somehow represents the middle, maybe I want the most frequent number, maybe I want the number that somehow represents the center of all of these numbers. And if you said any of those things, you would actually have done the same things that the people who first came up with descriptive statistics said. They said, well, how can we do it?"}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "How would you do that? Well, you'd say, well, how can I find something that maybe I want a typical number, maybe I want some number that somehow represents the middle, maybe I want the most frequent number, maybe I want the number that somehow represents the center of all of these numbers. And if you said any of those things, you would actually have done the same things that the people who first came up with descriptive statistics said. They said, well, how can we do it? And we'll start by thinking of the idea of average. Average. And in everyday terminology, average has a very particular meaning."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "They said, well, how can we do it? And we'll start by thinking of the idea of average. Average. And in everyday terminology, average has a very particular meaning. As we'll see, when many people talk about average, they're talking about the arithmetic mean, which we'll see shortly. But in statistics, average means something more general. It really means give me a typical, give me a typical, or give me a middle, give me a middle number."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And in everyday terminology, average has a very particular meaning. As we'll see, when many people talk about average, they're talking about the arithmetic mean, which we'll see shortly. But in statistics, average means something more general. It really means give me a typical, give me a typical, or give me a middle, give me a middle number. Or, and these are ors, and really, it's an attempt to find a measure of central tendency. Central, central tendency. So once again, you have a bunch of numbers."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It really means give me a typical, give me a typical, or give me a middle, give me a middle number. Or, and these are ors, and really, it's an attempt to find a measure of central tendency. Central, central tendency. So once again, you have a bunch of numbers. You're somehow trying to represent these with one number. We'll call it the average. That's somehow typical or a middle or the center somehow of these numbers."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So once again, you have a bunch of numbers. You're somehow trying to represent these with one number. We'll call it the average. That's somehow typical or a middle or the center somehow of these numbers. And as we'll see, there's many types of averages. The first is the one that you're probably most familiar with. It's the one that people talk about, hey, the average on this exam or the average height."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "That's somehow typical or a middle or the center somehow of these numbers. And as we'll see, there's many types of averages. The first is the one that you're probably most familiar with. It's the one that people talk about, hey, the average on this exam or the average height. And that's the arithmetic mean. So let me write it in, I'll write it in yellow. Arith, arithmetic, arithmetic mean."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It's the one that people talk about, hey, the average on this exam or the average height. And that's the arithmetic mean. So let me write it in, I'll write it in yellow. Arith, arithmetic, arithmetic mean. When arithmetic is a noun, we call it arithmetic. When it's an adjective like this, we call it arithmetic. Arithmetic mean."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Arith, arithmetic, arithmetic mean. When arithmetic is a noun, we call it arithmetic. When it's an adjective like this, we call it arithmetic. Arithmetic mean. And this is really just the sum of all the numbers divided by, and this is a human constructed definition that we've found useful, the sum of all of these numbers divided by the number of numbers we have. So given that, what is the arithmetic mean of this data set? Well, let's just compute it."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Arithmetic mean. And this is really just the sum of all the numbers divided by, and this is a human constructed definition that we've found useful, the sum of all of these numbers divided by the number of numbers we have. So given that, what is the arithmetic mean of this data set? Well, let's just compute it. It's going to be four plus three plus one plus six plus one plus seven over the number of data points we have. So we have six data points, so we're gonna divide by six. And we get four plus three is seven, plus one is eight, plus six is 14, plus one is 15, plus seven, 15 plus seven is 22."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, let's just compute it. It's going to be four plus three plus one plus six plus one plus seven over the number of data points we have. So we have six data points, so we're gonna divide by six. And we get four plus three is seven, plus one is eight, plus six is 14, plus one is 15, plus seven, 15 plus seven is 22. We'll do that one more time. You have seven, eight, 14, 15, 22. All of that over six."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And we get four plus three is seven, plus one is eight, plus six is 14, plus one is 15, plus seven, 15 plus seven is 22. We'll do that one more time. You have seven, eight, 14, 15, 22. All of that over six. And we could write this as a mixed number. Six goes into 22 three times with the remainder of four. So it's three and 4 6, which is the same thing as three and 2 3rds."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "All of that over six. And we could write this as a mixed number. Six goes into 22 three times with the remainder of four. So it's three and 4 6, which is the same thing as three and 2 3rds. We could write this as a decimal with 3.6 repeating. So this is also 3.6 repeating. We could write it any one of those ways."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So it's three and 4 6, which is the same thing as three and 2 3rds. We could write this as a decimal with 3.6 repeating. So this is also 3.6 repeating. We could write it any one of those ways. But this is kind of a representative number. This is trying to get at a central tendency. Once again, these are human constructed."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We could write it any one of those ways. But this is kind of a representative number. This is trying to get at a central tendency. Once again, these are human constructed. No one ever, it's not like someone just found some religious document that said, this is the way that the arithmetic mean must be defined. It's not as pure of a computation as say finding the circumference of the circle, which there really is. That was kind of, we studied the universe and that just fell out of our study of the universe."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Once again, these are human constructed. No one ever, it's not like someone just found some religious document that said, this is the way that the arithmetic mean must be defined. It's not as pure of a computation as say finding the circumference of the circle, which there really is. That was kind of, we studied the universe and that just fell out of our study of the universe. It's a human constructed definition that we found useful. Now, there are other ways to measure the average or find a typical or middle value. The other very typical way is the median."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "That was kind of, we studied the universe and that just fell out of our study of the universe. It's a human constructed definition that we found useful. Now, there are other ways to measure the average or find a typical or middle value. The other very typical way is the median. And I will write median, I'm running out of colors. I will write median in pink. So there is the median."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "The other very typical way is the median. And I will write median, I'm running out of colors. I will write median in pink. So there is the median. And the median is literally looking for the middle number. So if you were to order all the numbers in your set and find the middle one, then that is your median. So given that, what's the median of this set of numbers going to be?"}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So there is the median. And the median is literally looking for the middle number. So if you were to order all the numbers in your set and find the middle one, then that is your median. So given that, what's the median of this set of numbers going to be? Let's try to figure it out. Let's try to order it. So we have one, then we have another one, then we have a three, then we have a four, a six and a seven."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So given that, what's the median of this set of numbers going to be? Let's try to figure it out. Let's try to order it. So we have one, then we have another one, then we have a three, then we have a four, a six and a seven. So all I did is I reordered this. And so what's the middle number? Well, you look here, since we have an even number of numbers we have six numbers."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So we have one, then we have another one, then we have a three, then we have a four, a six and a seven. So all I did is I reordered this. And so what's the middle number? Well, you look here, since we have an even number of numbers we have six numbers. There's not one middle number. You actually have two middle numbers here. You have two middle numbers right over here."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, you look here, since we have an even number of numbers we have six numbers. There's not one middle number. You actually have two middle numbers here. You have two middle numbers right over here. You have the three and the four. And in this case, when you have two middle numbers, you actually go halfway between these two numbers. Or essentially, you're taking the arithmetic mean of these two numbers to find the median."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "You have two middle numbers right over here. You have the three and the four. And in this case, when you have two middle numbers, you actually go halfway between these two numbers. Or essentially, you're taking the arithmetic mean of these two numbers to find the median. So the median is going to be halfway in between three and four which is going to be 3.5. So the median in this case is 3.5. So if you have an even number of numbers, the median or the middle two, essentially the arithmetic mean of the middle two are halfway between the middle two."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Or essentially, you're taking the arithmetic mean of these two numbers to find the median. So the median is going to be halfway in between three and four which is going to be 3.5. So the median in this case is 3.5. So if you have an even number of numbers, the median or the middle two, essentially the arithmetic mean of the middle two are halfway between the middle two. If you have an odd number of numbers, it's a little bit easier to compute. And just so that we see that, let me give you another data set. Let's say our data set, and I'll order it for us."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So if you have an even number of numbers, the median or the middle two, essentially the arithmetic mean of the middle two are halfway between the middle two. If you have an odd number of numbers, it's a little bit easier to compute. And just so that we see that, let me give you another data set. Let's say our data set, and I'll order it for us. Let's say our data set was 0,750, I don't know, 10,000, 10,000, and 1,000,000. 1,000,000. Let's say that that is our data set."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's say our data set, and I'll order it for us. Let's say our data set was 0,750, I don't know, 10,000, 10,000, and 1,000,000. 1,000,000. Let's say that that is our data set. Kind of a crazy data set. But in this situation, what is our median? Well, here we have five numbers."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's say that that is our data set. Kind of a crazy data set. But in this situation, what is our median? Well, here we have five numbers. We have an odd number of numbers. So it's easier to pick out a middle. The middle is a number that is greater than two of the numbers and is less than two of the numbers."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, here we have five numbers. We have an odd number of numbers. So it's easier to pick out a middle. The middle is a number that is greater than two of the numbers and is less than two of the numbers. It's exactly in the middle. So in this case, our median is 50. Now, the third measure of central tendency, and this is the one that's probably used least often in life, is the mode."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "The middle is a number that is greater than two of the numbers and is less than two of the numbers. It's exactly in the middle. So in this case, our median is 50. Now, the third measure of central tendency, and this is the one that's probably used least often in life, is the mode. And people often forget about it and it sounds like something very complex. But what we'll see is it's actually a very straightforward idea. And in some ways, it is the most basic idea."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Now, the third measure of central tendency, and this is the one that's probably used least often in life, is the mode. And people often forget about it and it sounds like something very complex. But what we'll see is it's actually a very straightforward idea. And in some ways, it is the most basic idea. So the mode is actually the most common number in a data set, if there is a most common number. If all the numbers are represented equally, if there's no one single most common number, then you have no mode. But given that definition of the mode, what is the single most common number in our original data set, in this data set right over here?"}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And in some ways, it is the most basic idea. So the mode is actually the most common number in a data set, if there is a most common number. If all the numbers are represented equally, if there's no one single most common number, then you have no mode. But given that definition of the mode, what is the single most common number in our original data set, in this data set right over here? Well, we only have one four. We only have one three. But we have two ones."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But given that definition of the mode, what is the single most common number in our original data set, in this data set right over here? Well, we only have one four. We only have one three. But we have two ones. We have one six and one seven. So the number that shows up the most number of times here is our one. So the mode, the most typical number, the most common number here is a one."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But we have two ones. We have one six and one seven. So the number that shows up the most number of times here is our one. So the mode, the most typical number, the most common number here is a one. So you see, these are all different ways of trying to get at a typical or middle or central tendency. But you do it in very, very different ways. And as we study more and more statistics, we'll see that they're good for different things."}, {"video_title": "Statistics intro Mean, median, and mode Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So the mode, the most typical number, the most common number here is a one. So you see, these are all different ways of trying to get at a typical or middle or central tendency. But you do it in very, very different ways. And as we study more and more statistics, we'll see that they're good for different things. This is used very frequently. The median is really good if you have some kind of crazy number out here that could have otherwise skewed the arithmetic mean. The mode could also be useful in situations like that, especially if you do have one number that's showing up a lot more frequently."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "Each dot plot below represents a different set of data. We see that here. Order the dot plots from largest standard deviation top to smallest standard deviation bottom. So pause this video and see if you can do that, or at least if you could rank these from largest standard deviation to smallest standard deviation. All right, now let's work through this together. And I'm doing this on Khan Academy where I can move these around to order them. But let's just remind ourselves what the standard deviation is or how we can perceive it."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "So pause this video and see if you can do that, or at least if you could rank these from largest standard deviation to smallest standard deviation. All right, now let's work through this together. And I'm doing this on Khan Academy where I can move these around to order them. But let's just remind ourselves what the standard deviation is or how we can perceive it. You could view the standard deviation as a measure of the typical distance from each of the data points to the mean. So the largest standard deviation, which we wanna put on top, would be the one where typically our data points are further from the mean and our smallest standard deviation would be the ones where it feels like, on average, our data points are closer to the mean. When all of these examples, our mean looks to be right in the center, right between 50 and 100, so right around 75."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "But let's just remind ourselves what the standard deviation is or how we can perceive it. You could view the standard deviation as a measure of the typical distance from each of the data points to the mean. So the largest standard deviation, which we wanna put on top, would be the one where typically our data points are further from the mean and our smallest standard deviation would be the ones where it feels like, on average, our data points are closer to the mean. When all of these examples, our mean looks to be right in the center, right between 50 and 100, so right around 75. So it's really about how spread apart they are from that. And if you look at this first one, it has these two data points, the one on the left and one on the right, that are pretty far, and then you have these two that are a little bit closer and then these two that are inside. This one right over here, to get from this top one to this middle one, you essentially are taking this data point and making it go further, and taking this data point and making it go further."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "When all of these examples, our mean looks to be right in the center, right between 50 and 100, so right around 75. So it's really about how spread apart they are from that. And if you look at this first one, it has these two data points, the one on the left and one on the right, that are pretty far, and then you have these two that are a little bit closer and then these two that are inside. This one right over here, to get from this top one to this middle one, you essentially are taking this data point and making it go further, and taking this data point and making it go further. And so this one is going to have a higher standard deviation than that one. So let me put it just like that. And I just wanna make it very clear."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "This one right over here, to get from this top one to this middle one, you essentially are taking this data point and making it go further, and taking this data point and making it go further. And so this one is going to have a higher standard deviation than that one. So let me put it just like that. And I just wanna make it very clear. Keep track of what's the difference between these two things. Here you have this data point and this data point that was closer in, and then if you move it further, that's going to make your typical distance from the middle more, which is exactly what happened there. Now what about this one?"}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "And I just wanna make it very clear. Keep track of what's the difference between these two things. Here you have this data point and this data point that was closer in, and then if you move it further, that's going to make your typical distance from the middle more, which is exactly what happened there. Now what about this one? Well, this one is starting here and then taking this point and taking this point and moving it closer. And so that would make our typical distance from the middle, from the mean, shorter. So this would have the smallest standard deviation and this would have the largest."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "Now what about this one? Well, this one is starting here and then taking this point and taking this point and moving it closer. And so that would make our typical distance from the middle, from the mean, shorter. So this would have the smallest standard deviation and this would have the largest. Let's do another example. So same idea, order the dot plots from largest standard deviation on the top to smallest standard deviation on the bottom. Pause this video and see if you can figure that out."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "So this would have the smallest standard deviation and this would have the largest. Let's do another example. So same idea, order the dot plots from largest standard deviation on the top to smallest standard deviation on the bottom. Pause this video and see if you can figure that out. So this is interesting because these all have different means. Just eyeballing it, the mean for this first one is right around here. The mean for the second one is right around here at around 10."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "Pause this video and see if you can figure that out. So this is interesting because these all have different means. Just eyeballing it, the mean for this first one is right around here. The mean for the second one is right around here at around 10. And the mean for the third one, it looks like the same mean as this top one. And so pause this video. How would you order them?"}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "The mean for the second one is right around here at around 10. And the mean for the third one, it looks like the same mean as this top one. And so pause this video. How would you order them? All right, so just eyeballing it, these, this middle one right over here, your typical data point seems furthest from the mean. You definitely have, if the mean is here, you have these, this data point and this data point that are quite far from that mean. And even this data point and this data point are at least as far as any of the data points that we have in the top or the bottom one."}, {"video_title": "Visually assessing standard deviation AP Statistics Khan Academy.mp3", "Sentence": "How would you order them? All right, so just eyeballing it, these, this middle one right over here, your typical data point seems furthest from the mean. You definitely have, if the mean is here, you have these, this data point and this data point that are quite far from that mean. And even this data point and this data point are at least as far as any of the data points that we have in the top or the bottom one. So I would say this has the largest standard deviation. And if I were to compare between these two, if you think about how you would get the difference between these two is if you took this data point and moved it at, and you moved it to the mean, and if you took this data point and you moved it to the mean, you would get this third situation. And so this third situation, you have the fewest data points that are sitting away from the mean relative to this one."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And in particular, I am going to put 8 green cubes. I'm also going to put some spheres in that bag. Let's say I'm going to put 9 spheres, and these are the green spheres. I'm also going to put some yellow cubes in that bag. Some yellow cubes. I'm going to put 5 of those. And I'm also going to put some yellow spheres in this bag."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "I'm also going to put some yellow cubes in that bag. Some yellow cubes. I'm going to put 5 of those. And I'm also going to put some yellow spheres in this bag. Yellow spheres. And let's say I put 7 of those. I'm going to stick them all in this bag, and then I'm going to shake that bag, and then I'm going to pour it out, and I'm going to look at the first object that falls out of that bag."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And I'm also going to put some yellow spheres in this bag. Yellow spheres. And let's say I put 7 of those. I'm going to stick them all in this bag, and then I'm going to shake that bag, and then I'm going to pour it out, and I'm going to look at the first object that falls out of that bag. And what I want to think about in this video is what are the probabilities of getting different types of objects. So for example, what is the probability of getting a cube? A cube of any color."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "I'm going to stick them all in this bag, and then I'm going to shake that bag, and then I'm going to pour it out, and I'm going to look at the first object that falls out of that bag. And what I want to think about in this video is what are the probabilities of getting different types of objects. So for example, what is the probability of getting a cube? A cube of any color. What is the probability of getting a cube? Well, to think about that, we should think about, or this is one way to think about it, what are all of the equally likely possibilities that might pop out of the bag? Well, we have 8 plus 9 is 17, 17 plus 5 is 22, 22 plus 7 is 29."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "A cube of any color. What is the probability of getting a cube? Well, to think about that, we should think about, or this is one way to think about it, what are all of the equally likely possibilities that might pop out of the bag? Well, we have 8 plus 9 is 17, 17 plus 5 is 22, 22 plus 7 is 29. So we have 29 objects. There are 29 objects in the bag. Did I do that right?"}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Well, we have 8 plus 9 is 17, 17 plus 5 is 22, 22 plus 7 is 29. So we have 29 objects. There are 29 objects in the bag. Did I do that right? This is 14, yep, 29 objects. So let's draw all of the possible objects. And I'll represent it as this big area right over here."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Did I do that right? This is 14, yep, 29 objects. So let's draw all of the possible objects. And I'll represent it as this big area right over here. So these are all the possible objects. There are 29 possible objects. So there's 29 equal possibilities for the outcome of my experiment of seeing what pops out of that bag, assuming that it's equally likely for a cube or a sphere to pop out first."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And I'll represent it as this big area right over here. So these are all the possible objects. There are 29 possible objects. So there's 29 equal possibilities for the outcome of my experiment of seeing what pops out of that bag, assuming that it's equally likely for a cube or a sphere to pop out first. And how many of them meet our constraint of being a cube? Well, I have 8 green cubes, and I have 5 yellow cubes. So there are a total of 13 cubes."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So there's 29 equal possibilities for the outcome of my experiment of seeing what pops out of that bag, assuming that it's equally likely for a cube or a sphere to pop out first. And how many of them meet our constraint of being a cube? Well, I have 8 green cubes, and I have 5 yellow cubes. So there are a total of 13 cubes. So let me draw that set of cubes. So there's 13 cubes. I could draw it like this."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So there are a total of 13 cubes. So let me draw that set of cubes. So there's 13 cubes. I could draw it like this. There are 13 cubes. This right here is the set of cubes. This area, and I'm not drawing it exact, I'm approximating, represents the set of all the cubes."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "I could draw it like this. There are 13 cubes. This right here is the set of cubes. This area, and I'm not drawing it exact, I'm approximating, represents the set of all the cubes. So the probability of getting a cube is the number of events that meet our criterion. So there's 13 possible cubes that have an equal likely chance of popping out over all of the possible equally likely events, which are 29. That includes the cubes and the spheres."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "This area, and I'm not drawing it exact, I'm approximating, represents the set of all the cubes. So the probability of getting a cube is the number of events that meet our criterion. So there's 13 possible cubes that have an equal likely chance of popping out over all of the possible equally likely events, which are 29. That includes the cubes and the spheres. Now let's ask a different question. What is the probability of getting a yellow object, either a cube or a sphere? So once again, how many things meet our conditions here?"}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "That includes the cubes and the spheres. Now let's ask a different question. What is the probability of getting a yellow object, either a cube or a sphere? So once again, how many things meet our conditions here? We have 5 plus 7. There's 12 yellow objects in the bag. So we have 29 equally likely possibilities."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So once again, how many things meet our conditions here? We have 5 plus 7. There's 12 yellow objects in the bag. So we have 29 equally likely possibilities. I'll do that in the same color. We have 29 equally likely possibilities, and of those, 12 meet our criterion. So let me draw 12 right over here."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So we have 29 equally likely possibilities. I'll do that in the same color. We have 29 equally likely possibilities, and of those, 12 meet our criterion. So let me draw 12 right over here. I'll do my best attempt. It looks something like the set of yellow objects. There are 12 objects that are yellow."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So let me draw 12 right over here. I'll do my best attempt. It looks something like the set of yellow objects. There are 12 objects that are yellow. So the 12 that meet our conditions are 12 over all the possibilities, 29. So the probability of getting a cube, 13 29ths. Probability of getting a yellow, 12 29ths."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "There are 12 objects that are yellow. So the 12 that meet our conditions are 12 over all the possibilities, 29. So the probability of getting a cube, 13 29ths. Probability of getting a yellow, 12 29ths. Now let's ask something a little bit more interesting. What is the probability of getting a yellow cube? I'll put it in yellow, so we care about the color now."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Probability of getting a yellow, 12 29ths. Now let's ask something a little bit more interesting. What is the probability of getting a yellow cube? I'll put it in yellow, so we care about the color now. So this thing is yellow. What is the probability of getting a yellow cube? Well, there's 29 equally likely possibilities."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "I'll put it in yellow, so we care about the color now. So this thing is yellow. What is the probability of getting a yellow cube? Well, there's 29 equally likely possibilities. And of those 29 equally likely possibilities, 5 of those are yellow cubes, or yellow cubes. 5 of them. So the probability is 5 29ths."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Well, there's 29 equally likely possibilities. And of those 29 equally likely possibilities, 5 of those are yellow cubes, or yellow cubes. 5 of them. So the probability is 5 29ths. And where would we see that on this Venn diagram that I've drawn? This is just a way to visualize the different probabilities. And they become interesting when you start thinking about where sets overlap, or even where they don't overlap."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So the probability is 5 29ths. And where would we see that on this Venn diagram that I've drawn? This is just a way to visualize the different probabilities. And they become interesting when you start thinking about where sets overlap, or even where they don't overlap. So here we're thinking about things that are members of the set yellow. So they're in this set, and they are cubes. So this area right over here, that's the overlap of these two sets."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And they become interesting when you start thinking about where sets overlap, or even where they don't overlap. So here we're thinking about things that are members of the set yellow. So they're in this set, and they are cubes. So this area right over here, that's the overlap of these two sets. So this area right over here, this represents things that are both yellow and cubes, because they are inside both circles. So this right over here, let me write it over here. So there's 5 objects that are both yellow and cubes."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So this area right over here, that's the overlap of these two sets. So this area right over here, this represents things that are both yellow and cubes, because they are inside both circles. So this right over here, let me write it over here. So there's 5 objects that are both yellow and cubes. Now let's ask, and this is probably the most interesting thing to ask, what is the probability of getting something that is yellow or a cube? A cube of any color. Well we still know that the denominator here is going to be 29."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So there's 5 objects that are both yellow and cubes. Now let's ask, and this is probably the most interesting thing to ask, what is the probability of getting something that is yellow or a cube? A cube of any color. Well we still know that the denominator here is going to be 29. These are all of the equally likely possibilities that might jump out of the bag. But what are the possibilities that meet our conditions? Well one way to think about it is, well there's 12 things that would meet the yellow condition."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Well we still know that the denominator here is going to be 29. These are all of the equally likely possibilities that might jump out of the bag. But what are the possibilities that meet our conditions? Well one way to think about it is, well there's 12 things that would meet the yellow condition. So that would be this entire circle right over here. 12 things that meet the yellow condition. So this right over here is 12."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Well one way to think about it is, well there's 12 things that would meet the yellow condition. So that would be this entire circle right over here. 12 things that meet the yellow condition. So this right over here is 12. This is the number of yellow, that is 12. And then to that, we can't just add the number of cubes, because if we add the number of cubes, we've already counted these 5. These 5 are counted as part of this 12."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So this right over here is 12. This is the number of yellow, that is 12. And then to that, we can't just add the number of cubes, because if we add the number of cubes, we've already counted these 5. These 5 are counted as part of this 12. One way to think about it is, there are 7 yellow objects that are not cubes. Those are the spheres. There are 5 yellow objects that are cubes."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "These 5 are counted as part of this 12. One way to think about it is, there are 7 yellow objects that are not cubes. Those are the spheres. There are 5 yellow objects that are cubes. And then there are 8 cubes that are not yellow. That's one way to think about it. So when we counted this 12, the number of yellow, we counted all of this."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "There are 5 yellow objects that are cubes. And then there are 8 cubes that are not yellow. That's one way to think about it. So when we counted this 12, the number of yellow, we counted all of this. So we can't just add the number of cubes to it, because then we would count this middle part again. So then we have to essentially count cubes, the number of cubes, which is 13. So number of cubes."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So when we counted this 12, the number of yellow, we counted all of this. So we can't just add the number of cubes to it, because then we would count this middle part again. So then we have to essentially count cubes, the number of cubes, which is 13. So number of cubes. And we'll have to subtract out this middle section right over here. Let me do this. So subtract out the middle section right over here."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So number of cubes. And we'll have to subtract out this middle section right over here. Let me do this. So subtract out the middle section right over here. So minus 5. So this is the number of yellow cubes. It feels weird to write the word yellow in green."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So subtract out the middle section right over here. So minus 5. So this is the number of yellow cubes. It feels weird to write the word yellow in green. The number of yellow cubes. Or another way to think about it, and you could just do this math right here, 12 plus 13 minus 5 is what? It's 20."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "It feels weird to write the word yellow in green. The number of yellow cubes. Or another way to think about it, and you could just do this math right here, 12 plus 13 minus 5 is what? It's 20. Did I do that right? 12 minus, yep, it's 20. So that's one way you just get this is equal to 20 over 29."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "It's 20. Did I do that right? 12 minus, yep, it's 20. So that's one way you just get this is equal to 20 over 29. But the more interesting thing than even the answer of the probability of getting that is expressing this in terms of the other probabilities that we figured out earlier in the video. So let's think about this a little bit. We can rewrite this fraction right over here."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So that's one way you just get this is equal to 20 over 29. But the more interesting thing than even the answer of the probability of getting that is expressing this in terms of the other probabilities that we figured out earlier in the video. So let's think about this a little bit. We can rewrite this fraction right over here. We can rewrite this as 12 over 29 plus 13 over 29 minus 5 over 29. And this was the number of yellow over the total possibilities. So this right over here was the probability of getting a yellow."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "We can rewrite this fraction right over here. We can rewrite this as 12 over 29 plus 13 over 29 minus 5 over 29. And this was the number of yellow over the total possibilities. So this right over here was the probability of getting a yellow. This right over here was the number of cubes over the total possibilities. So this is plus the probability of getting a cube. And this right over here was the number of yellow cubes over the total possibilities."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So this right over here was the probability of getting a yellow. This right over here was the number of cubes over the total possibilities. So this is plus the probability of getting a cube. And this right over here was the number of yellow cubes over the total possibilities. So this right over here was minus the probability of yellow and a cube. Let me write it that way. Minus the probability of yellow."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And this right over here was the number of yellow cubes over the total possibilities. So this right over here was minus the probability of yellow and a cube. Let me write it that way. Minus the probability of yellow. I'll write yellow in yellow. Yellow and getting a cube. And so what we've just done here, and you could play with the numbers, the numbers I just used as an example right here to make things a little bit concrete, but you can see this is a generalizable thing."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Minus the probability of yellow. I'll write yellow in yellow. Yellow and getting a cube. And so what we've just done here, and you could play with the numbers, the numbers I just used as an example right here to make things a little bit concrete, but you can see this is a generalizable thing. If we have the probability of one condition or another condition, so let me rewrite it. The probability, and I'll just write a little bit more generally here. This gives us an interesting idea."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And so what we've just done here, and you could play with the numbers, the numbers I just used as an example right here to make things a little bit concrete, but you can see this is a generalizable thing. If we have the probability of one condition or another condition, so let me rewrite it. The probability, and I'll just write a little bit more generally here. This gives us an interesting idea. The probability of getting one condition of an object being a member of set A or a member of set B is equal to the probability that it is a member of set A plus the probability that it is a member of set B minus the probability that it is a member of both. And this is a really useful result. I think sometimes it's called the addition rule of probability, but I wanted to show you it's a completely common sense thing."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "This gives us an interesting idea. The probability of getting one condition of an object being a member of set A or a member of set B is equal to the probability that it is a member of set A plus the probability that it is a member of set B minus the probability that it is a member of both. And this is a really useful result. I think sometimes it's called the addition rule of probability, but I wanted to show you it's a completely common sense thing. The reason why you can't just add these two probabilities is because they might have some overlap. There's a probability of getting both. And if you just added both of these, you would be double counting that overlap, which we've already seen earlier in this video."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "I think sometimes it's called the addition rule of probability, but I wanted to show you it's a completely common sense thing. The reason why you can't just add these two probabilities is because they might have some overlap. There's a probability of getting both. And if you just added both of these, you would be double counting that overlap, which we've already seen earlier in this video. So you have to subtract one version of the overlap out so you are not double counting it. And I'll throw one other idea out. Sometimes you have possibilities that have no overlap."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "And if you just added both of these, you would be double counting that overlap, which we've already seen earlier in this video. So you have to subtract one version of the overlap out so you are not double counting it. And I'll throw one other idea out. Sometimes you have possibilities that have no overlap. So let's say this is a set of all possibilities. And let's say this is a set that meets condition A. Let's say that this is the set that meets condition B."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Sometimes you have possibilities that have no overlap. So let's say this is a set of all possibilities. And let's say this is a set that meets condition A. Let's say that this is the set that meets condition B. So in this situation, there is no overlap. Nothing is a member of both set A and B. So in this situation, the probability of A and B is 0."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "Let's say that this is the set that meets condition B. So in this situation, there is no overlap. Nothing is a member of both set A and B. So in this situation, the probability of A and B is 0. There is no overlap. And these type of conditions, or these two events, are called mutually exclusive. So if events are mutually exclusive, that means that they both cannot happen at the same time."}, {"video_title": "Addition rule for probability Probability and Statistics Khan Academy.mp3", "Sentence": "So in this situation, the probability of A and B is 0. There is no overlap. And these type of conditions, or these two events, are called mutually exclusive. So if events are mutually exclusive, that means that they both cannot happen at the same time. There's no event that meets both of these conditions. And if things are mutually exclusive, then you can say the probability of A or B is a probability of A plus B, because this thing is 0. But if things are not mutually exclusive, you would have to subtract out the overlap."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "I have this article right here from WebMD. And the point of this isn't to poke holes at WebMD. I think they have some great articles and they have some great information on their site. But what I want to do here is to think about what a lot of articles you might read or a lot of research you might read are implying and to think about whether they really imply what they claim to be implying. So this is an excerpt of an article. And the title of the article says, eating breakfast may beat teen obesity. So they're already trying to kind of create this cause and effect relationship."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "But what I want to do here is to think about what a lot of articles you might read or a lot of research you might read are implying and to think about whether they really imply what they claim to be implying. So this is an excerpt of an article. And the title of the article says, eating breakfast may beat teen obesity. So they're already trying to kind of create this cause and effect relationship. The title itself says, if you eat breakfast, then you're less likely or you won't be obese. You're not going to be obese. So the title right there already sets up this, that eating breakfast may beat teen obesity."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "So they're already trying to kind of create this cause and effect relationship. The title itself says, if you eat breakfast, then you're less likely or you won't be obese. You're not going to be obese. So the title right there already sets up this, that eating breakfast may beat teen obesity. And then they tell us about the study. In the study, published in Pediatrics, researchers analyzed the dietary and weight patterns of a group of 2,216 adolescents over a five-year period from public schools in Minneapolis, St. Paul, Minnesota. And I won't talk too much about it."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "So the title right there already sets up this, that eating breakfast may beat teen obesity. And then they tell us about the study. In the study, published in Pediatrics, researchers analyzed the dietary and weight patterns of a group of 2,216 adolescents over a five-year period from public schools in Minneapolis, St. Paul, Minnesota. And I won't talk too much about it. This looks like a good sample size. It was over a large period of time. I'll just give the researchers the benefit of the doubt, assume that it was over a broad audience that they were able to control for a lot of variables."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "And I won't talk too much about it. This looks like a good sample size. It was over a large period of time. I'll just give the researchers the benefit of the doubt, assume that it was over a broad audience that they were able to control for a lot of variables. But then they go on to say, the researchers write that teens who ate breakfast regularly had a lower percentage of total calories from saturated fat and ate more fiber and carbohydrates. And to some degree, that first, then those who skipped breakfast, and to some degree this first sentence is obvious. Breakfast tends to be things like cereals, grains, you eat syrup, you eat waffles."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "I'll just give the researchers the benefit of the doubt, assume that it was over a broad audience that they were able to control for a lot of variables. But then they go on to say, the researchers write that teens who ate breakfast regularly had a lower percentage of total calories from saturated fat and ate more fiber and carbohydrates. And to some degree, that first, then those who skipped breakfast, and to some degree this first sentence is obvious. Breakfast tends to be things like cereals, grains, you eat syrup, you eat waffles. That all tends to fall in the category of carbohydrates and sugars. And frankly, that's not even necessarily a good thing. Not obvious to me whether bacon is more or less healthy than downing a bunch of syrup or Froot Loops or whatever else."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Breakfast tends to be things like cereals, grains, you eat syrup, you eat waffles. That all tends to fall in the category of carbohydrates and sugars. And frankly, that's not even necessarily a good thing. Not obvious to me whether bacon is more or less healthy than downing a bunch of syrup or Froot Loops or whatever else. But we'll let that be right here. In addition, regular breakfast eaters seemed more physically active than their breakfast skippers. So over here, they're once again trying to create this other cause and effect relationship."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Not obvious to me whether bacon is more or less healthy than downing a bunch of syrup or Froot Loops or whatever else. But we'll let that be right here. In addition, regular breakfast eaters seemed more physically active than their breakfast skippers. So over here, they're once again trying to create this other cause and effect relationship. Regular breakfast eaters seemed more physically active than their breakfast skippers. So the implication here is that breakfast makes you more active. And then this last sentence right over here, they say, over time, researchers found teens who regularly ate breakfast tended to gain less weight and had a lower body mass index than breakfast skippers."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "So over here, they're once again trying to create this other cause and effect relationship. Regular breakfast eaters seemed more physically active than their breakfast skippers. So the implication here is that breakfast makes you more active. And then this last sentence right over here, they say, over time, researchers found teens who regularly ate breakfast tended to gain less weight and had a lower body mass index than breakfast skippers. So they're telling us that breakfast skipping, this is the implication here, is more likely or it can be a cause of making you overweight or maybe even making you obese. So the entire narrative here, from the title all the way through every paragraph, is look, breakfast prevents obesity. Breakfast makes you active."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "And then this last sentence right over here, they say, over time, researchers found teens who regularly ate breakfast tended to gain less weight and had a lower body mass index than breakfast skippers. So they're telling us that breakfast skipping, this is the implication here, is more likely or it can be a cause of making you overweight or maybe even making you obese. So the entire narrative here, from the title all the way through every paragraph, is look, breakfast prevents obesity. Breakfast makes you active. Breakfast skipping will make you obese. So you just say, boy, I have to eat breakfast. And you should always think about the motivations and the industries around things like breakfast."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Breakfast makes you active. Breakfast skipping will make you obese. So you just say, boy, I have to eat breakfast. And you should always think about the motivations and the industries around things like breakfast. But the more interesting question is, does this research really tell us that eating breakfast can prevent obesity? Does it really tell us that eating breakfast will cause someone to become more active? Does it really tell us that breakfast skipping can make you overweight or make you obese?"}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "And you should always think about the motivations and the industries around things like breakfast. But the more interesting question is, does this research really tell us that eating breakfast can prevent obesity? Does it really tell us that eating breakfast will cause someone to become more active? Does it really tell us that breakfast skipping can make you overweight or make you obese? Or, it is more likely, are they just showing that these two things tend to go together? And this is a really important difference. And let me kind of state slightly technical words here."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Does it really tell us that breakfast skipping can make you overweight or make you obese? Or, it is more likely, are they just showing that these two things tend to go together? And this is a really important difference. And let me kind of state slightly technical words here. And they sound fancy, but they really aren't that fancy. Are they pointing out causality? Are they pointing out causality, which is what it seems like they're implying?"}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "And let me kind of state slightly technical words here. And they sound fancy, but they really aren't that fancy. Are they pointing out causality? Are they pointing out causality, which is what it seems like they're implying? Eating breakfast causes you to not be obese. Breakfast causes you to be active. Breakfast skipping causes you to be obese."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Are they pointing out causality, which is what it seems like they're implying? Eating breakfast causes you to not be obese. Breakfast causes you to be active. Breakfast skipping causes you to be obese. So it looks like they're kind of implying causality. They're implying cause and effect. But really, what the study looked at is correlation."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Breakfast skipping causes you to be obese. So it looks like they're kind of implying causality. They're implying cause and effect. But really, what the study looked at is correlation. So the whole point of this is to understand the difference between causality and correlation, because they're saying very different things. Causality versus correlation. And as I said, causality says A causes B."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "But really, what the study looked at is correlation. So the whole point of this is to understand the difference between causality and correlation, because they're saying very different things. Causality versus correlation. And as I said, causality says A causes B. Well, correlation just says A and B tend to be observed at the same time. Whenever I see B happening, it looks like A is happening at the same time. Whenever A is happening, it looks like it also tends to happen with B."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "And as I said, causality says A causes B. Well, correlation just says A and B tend to be observed at the same time. Whenever I see B happening, it looks like A is happening at the same time. Whenever A is happening, it looks like it also tends to happen with B. And the reason why it's super important to notice the distinction between these is you can come to very, very, very, very different conclusions. So the one thing that this research does do, assuming that it was performed well, is it does show a correlation. So this study does show a correlation."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Whenever A is happening, it looks like it also tends to happen with B. And the reason why it's super important to notice the distinction between these is you can come to very, very, very, very different conclusions. So the one thing that this research does do, assuming that it was performed well, is it does show a correlation. So this study does show a correlation. It does show, if we believe all of their data, that breakfast skipping correlates with obesity, and obesity correlates with breakfast skipping. We're seeing it at the same time. Activity correlates with breakfast, and breakfast correlates with activity."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "So this study does show a correlation. It does show, if we believe all of their data, that breakfast skipping correlates with obesity, and obesity correlates with breakfast skipping. We're seeing it at the same time. Activity correlates with breakfast, and breakfast correlates with activity. All of these correlate. Well, they don't say, and there's no data here that lets me know one way or the other, what is causing what, or maybe you have some underlying cause that is causing both. So for example, they're saying breakfast causes activity, or they're implying breakfast causes activity."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Activity correlates with breakfast, and breakfast correlates with activity. All of these correlate. Well, they don't say, and there's no data here that lets me know one way or the other, what is causing what, or maybe you have some underlying cause that is causing both. So for example, they're saying breakfast causes activity, or they're implying breakfast causes activity. They're not saying it explicitly. But maybe activity causes breakfast. Maybe."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "So for example, they're saying breakfast causes activity, or they're implying breakfast causes activity. They're not saying it explicitly. But maybe activity causes breakfast. Maybe. They didn't write the study that people who are active, maybe they're more likely to be hungry in the morning. Activity causes breakfast. And then you start having a different takeaway."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Maybe. They didn't write the study that people who are active, maybe they're more likely to be hungry in the morning. Activity causes breakfast. And then you start having a different takeaway. Then you don't say, wait, maybe if you're active and you skip breakfast, and I'm not telling you that you should, I have no data one or the other, maybe you'll lose even more weight. Maybe it's even a healthier thing to do. We're not sure."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "And then you start having a different takeaway. Then you don't say, wait, maybe if you're active and you skip breakfast, and I'm not telling you that you should, I have no data one or the other, maybe you'll lose even more weight. Maybe it's even a healthier thing to do. We're not sure. So they're trying to say, look, if you have breakfast, it's going to make you active, which is a very positive outcome. But maybe you can have the positive outcome without breakfast. Who knows?"}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "We're not sure. So they're trying to say, look, if you have breakfast, it's going to make you active, which is a very positive outcome. But maybe you can have the positive outcome without breakfast. Who knows? Likewise, they say breakfast skipping, or they're implying breakfast skipping can cause obesity. But maybe it's the other way around. Maybe people who have high body fat, maybe for whatever reason, they're less likely to get hungry in the morning."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Who knows? Likewise, they say breakfast skipping, or they're implying breakfast skipping can cause obesity. But maybe it's the other way around. Maybe people who have high body fat, maybe for whatever reason, they're less likely to get hungry in the morning. So maybe it goes this way. Maybe there's a causality there. Or even more likely, maybe there's some underlying cause that causes both of these things to happen."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Maybe people who have high body fat, maybe for whatever reason, they're less likely to get hungry in the morning. So maybe it goes this way. Maybe there's a causality there. Or even more likely, maybe there's some underlying cause that causes both of these things to happen. And you could think of a bunch of different examples of that. One could be the physical activity. So physical activity, and these are all just theories."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Or even more likely, maybe there's some underlying cause that causes both of these things to happen. And you could think of a bunch of different examples of that. One could be the physical activity. So physical activity, and these are all just theories. I have no proof for it. But I just want to give you different ways of thinking about the same data, and maybe not just coming to the same conclusion that this article seems like it's trying to lead us to conclude that we should eat breakfast if we don't want to become obese. So maybe if you're physically active, that leads to you being hungry in the morning."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "So physical activity, and these are all just theories. I have no proof for it. But I just want to give you different ways of thinking about the same data, and maybe not just coming to the same conclusion that this article seems like it's trying to lead us to conclude that we should eat breakfast if we don't want to become obese. So maybe if you're physically active, that leads to you being hungry in the morning. So you're more likely to eat breakfast. And obviously, being physically active also makes it so that you've burned calories, you have more muscle, so that you're not obese. So notice, if you view things this way, if you say physical activity is causing both of these, then all of a sudden you lose this connection between breakfast and obesity."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "So maybe if you're physically active, that leads to you being hungry in the morning. So you're more likely to eat breakfast. And obviously, being physically active also makes it so that you've burned calories, you have more muscle, so that you're not obese. So notice, if you view things this way, if you say physical activity is causing both of these, then all of a sudden you lose this connection between breakfast and obesity. Now you can't make the claim that somehow breakfast is the magic formula for someone to not be obese. So let's say that there is an obese person. Let's say this is the reality, that physical activity is causing both of these things."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "So notice, if you view things this way, if you say physical activity is causing both of these, then all of a sudden you lose this connection between breakfast and obesity. Now you can't make the claim that somehow breakfast is the magic formula for someone to not be obese. So let's say that there is an obese person. Let's say this is the reality, that physical activity is causing both of these things. And let's say that there is an obese person. What will you tell them to do? Will you tell them, eat breakfast and you won't become obese anymore?"}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Let's say this is the reality, that physical activity is causing both of these things. And let's say that there is an obese person. What will you tell them to do? Will you tell them, eat breakfast and you won't become obese anymore? Well, that might not work, especially if they're not physically active. I mean, what's going to happen if you have an obese person who's not physically active? And then you tell them to eat breakfast."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Will you tell them, eat breakfast and you won't become obese anymore? Well, that might not work, especially if they're not physically active. I mean, what's going to happen if you have an obese person who's not physically active? And then you tell them to eat breakfast. Maybe that'll make things worse. And based on that, the advice or the implication from the article is the wrong thing. Physical activity maybe is the thing that should be focused on."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "And then you tell them to eat breakfast. Maybe that'll make things worse. And based on that, the advice or the implication from the article is the wrong thing. Physical activity maybe is the thing that should be focused on. Maybe it's something other than physical activity. Maybe you have sleep. Maybe people who sleep late, and they're not getting enough sleep, maybe someone who's not getting enough sleep, maybe that leads to obesity."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Physical activity maybe is the thing that should be focused on. Maybe it's something other than physical activity. Maybe you have sleep. Maybe people who sleep late, and they're not getting enough sleep, maybe someone who's not getting enough sleep, maybe that leads to obesity. And obviously, because they're not getting enough sleep, they wake up as late as possible and they have to run to the next appointment, or they have to run to school in the case of students. And maybe that's why they skip breakfast. So once again, if you find someone's obese, maybe the rule here isn't to force a breakfast down your throat."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "Maybe people who sleep late, and they're not getting enough sleep, maybe someone who's not getting enough sleep, maybe that leads to obesity. And obviously, because they're not getting enough sleep, they wake up as late as possible and they have to run to the next appointment, or they have to run to school in the case of students. And maybe that's why they skip breakfast. So once again, if you find someone's obese, maybe the rule here isn't to force a breakfast down your throat. Maybe it'll become even worse, because maybe it is the lack of sleep that's causing your metabolism to slow down or whatever. So it's very, very important when you're looking at any of these studies to try to say, is this a correlation, or is this causality? If it's correlation, you cannot make the judgment that, hey, eating breakfast is necessarily going to make someone less obese."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "So once again, if you find someone's obese, maybe the rule here isn't to force a breakfast down your throat. Maybe it'll become even worse, because maybe it is the lack of sleep that's causing your metabolism to slow down or whatever. So it's very, very important when you're looking at any of these studies to try to say, is this a correlation, or is this causality? If it's correlation, you cannot make the judgment that, hey, eating breakfast is necessarily going to make someone less obese. All that tells you is that these things move together. A better study would be one that is able to prove causality. And then we could think of other underlying causes that would kind of break down the narrative that this piece is trying to say."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "If it's correlation, you cannot make the judgment that, hey, eating breakfast is necessarily going to make someone less obese. All that tells you is that these things move together. A better study would be one that is able to prove causality. And then we could think of other underlying causes that would kind of break down the narrative that this piece is trying to say. I'm not saying it's wrong. Maybe it's absolutely true that eating breakfast will fight obesity. But I think it's equally or more important to think about what the other causes are, not to just make a blanket statement like that."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "And then we could think of other underlying causes that would kind of break down the narrative that this piece is trying to say. I'm not saying it's wrong. Maybe it's absolutely true that eating breakfast will fight obesity. But I think it's equally or more important to think about what the other causes are, not to just make a blanket statement like that. So for example, maybe poverty causes you to skip breakfast for multiple reasons. Maybe both of your parents are working. There's no one there to give you breakfast."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "But I think it's equally or more important to think about what the other causes are, not to just make a blanket statement like that. So for example, maybe poverty causes you to skip breakfast for multiple reasons. Maybe both of your parents are working. There's no one there to give you breakfast. Maybe there's more stress in the family. Who knows what it might be? And so when you have poverty, maybe you're more likely to skip breakfast."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "There's no one there to give you breakfast. Maybe there's more stress in the family. Who knows what it might be? And so when you have poverty, maybe you're more likely to skip breakfast. And maybe when there's poverty, and maybe you have two, both your parents are working, and the kids have to make their own dinner and whatever else, maybe they also eat less healthy. So eat less healthy at all times of day, and then that leads to obesity. So once again, in this situation, if this is the reality of things, just telling someone to also eat breakfast, regardless of what that breakfast is, even if it's Froot Loops or syrup, that's probably not going to help the situation."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "And so when you have poverty, maybe you're more likely to skip breakfast. And maybe when there's poverty, and maybe you have two, both your parents are working, and the kids have to make their own dinner and whatever else, maybe they also eat less healthy. So eat less healthy at all times of day, and then that leads to obesity. So once again, in this situation, if this is the reality of things, just telling someone to also eat breakfast, regardless of what that breakfast is, even if it's Froot Loops or syrup, that's probably not going to help the situation. Maybe it's just eating unhealthy dinners. Maybe eating unhealthy dinners is the underlying cause. And if you eat an unhealthy dinner, maybe by breakfast time, you're not hungry still, because you binge so much on breakfast, so you skip breakfast."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy.mp3", "Sentence": "So once again, in this situation, if this is the reality of things, just telling someone to also eat breakfast, regardless of what that breakfast is, even if it's Froot Loops or syrup, that's probably not going to help the situation. Maybe it's just eating unhealthy dinners. Maybe eating unhealthy dinners is the underlying cause. And if you eat an unhealthy dinner, maybe by breakfast time, you're not hungry still, because you binge so much on breakfast, so you skip breakfast. And this also leads to obesity. But once again, if this is the actual reality, doing the advice that that article's saying might actually be a bad thing. If you eat an unhealthy dinner and then force yourself to eat a breakfast when you're not hungry, that might make the obesity even worse."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "We're told that Kai goes to a restaurant that advertises a promotion saying, one in five customers get a free dessert. Suppose Kai goes to the restaurant twice in a given week, and each time he has a 1 5th probability of getting a free dessert. Let X represent the number of free desserts Kai gets in his two trips. Construct the theoretical probability distribution of X. All right, so pause this video and see if you can work through this before we do it together. All right, so first let's just think about the possible values that X could take on. This is the number of free desserts he gets, and he visits twice."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "Construct the theoretical probability distribution of X. All right, so pause this video and see if you can work through this before we do it together. All right, so first let's just think about the possible values that X could take on. This is the number of free desserts he gets, and he visits twice. So there's some world in which he doesn't get any free desserts, so that's zero in his two visits. Maybe on one of the visits, he gets a dessert, and the other one he doesn't. And maybe in both of his visits, he actually is able to get a free dessert."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "This is the number of free desserts he gets, and he visits twice. So there's some world in which he doesn't get any free desserts, so that's zero in his two visits. Maybe on one of the visits, he gets a dessert, and the other one he doesn't. And maybe in both of his visits, he actually is able to get a free dessert. So he's going to have some place from zero to two free desserts in a given week. So we just have to figure out the probability of each of these. So let's first of all think about the probability, let me write it over here, the probability that capital X is equal to zero is going to be equal to what?"}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "And maybe in both of his visits, he actually is able to get a free dessert. So he's going to have some place from zero to two free desserts in a given week. So we just have to figure out the probability of each of these. So let's first of all think about the probability, let me write it over here, the probability that capital X is equal to zero is going to be equal to what? Well, that's going to be the probability that he doesn't get a dessert on both days. And it's important to realize that these are independent events. It's not like the restaurant's gonna say, oh, if you didn't get a dessert on one day, you're more likely to get it on the other day, or somehow if you got it on a previous day, you're less likely on another day, that they are independent events."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "So let's first of all think about the probability, let me write it over here, the probability that capital X is equal to zero is going to be equal to what? Well, that's going to be the probability that he doesn't get a dessert on both days. And it's important to realize that these are independent events. It's not like the restaurant's gonna say, oh, if you didn't get a dessert on one day, you're more likely to get it on the other day, or somehow if you got it on a previous day, you're less likely on another day, that they are independent events. So the probability of not getting it on any one day is four out of five. And the probability of not getting it on two of the days, I would just multiply them because they are independent events. So four over five times four over five."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "It's not like the restaurant's gonna say, oh, if you didn't get a dessert on one day, you're more likely to get it on the other day, or somehow if you got it on a previous day, you're less likely on another day, that they are independent events. So the probability of not getting it on any one day is four out of five. And the probability of not getting it on two of the days, I would just multiply them because they are independent events. So four over five times four over five. So the probability that X is equal to zero is going to be 16 25ths, 16 over 25. Now, what about the probability that X is equal to one? What is this going to be?"}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "So four over five times four over five. So the probability that X is equal to zero is going to be 16 25ths, 16 over 25. Now, what about the probability that X is equal to one? What is this going to be? Well, there are two scenarios over here. There's one scenario where let's say on day one, he does not get the dessert, and on day two, he does get the dessert. But then of course, there's the other scenario where on day one, he gets the dessert, and then on day two, he doesn't get the dessert."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "What is this going to be? Well, there are two scenarios over here. There's one scenario where let's say on day one, he does not get the dessert, and on day two, he does get the dessert. But then of course, there's the other scenario where on day one, he gets the dessert, and then on day two, he doesn't get the dessert. These are the two scenarios where he's going to get X equals one. And so if we add these together, let's see, four fifths times one fifth, this is going to be four over 25. And then this is going to be four over 25 again."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "But then of course, there's the other scenario where on day one, he gets the dessert, and then on day two, he doesn't get the dessert. These are the two scenarios where he's going to get X equals one. And so if we add these together, let's see, four fifths times one fifth, this is going to be four over 25. And then this is going to be four over 25 again. And you add these two together, you're going to get eight 25ths. And then last but not least, and actually we could figure out this last one by subtracting 16 and eight from 25, which would actually give us one 25th. But let's just write this out, the probability that X equals two, this is the probability he gets a dessert on both days."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "And then this is going to be four over 25 again. And you add these two together, you're going to get eight 25ths. And then last but not least, and actually we could figure out this last one by subtracting 16 and eight from 25, which would actually give us one 25th. But let's just write this out, the probability that X equals two, this is the probability he gets a dessert on both days. So one fifth chance on day one, and one fifth chance on the second day. So one fifth times one fifth is one 25th. And you can do a reality check here."}, {"video_title": "Theoretical probability distribution example multiplication Probability & combinatorics.mp3", "Sentence": "But let's just write this out, the probability that X equals two, this is the probability he gets a dessert on both days. So one fifth chance on day one, and one fifth chance on the second day. So one fifth times one fifth is one 25th. And you can do a reality check here. These all need to add up to one, and they do indeed add up to one. 16 plus eight plus one is 25. So 25 25ths is what they all add up to."}, {"video_title": "Probability with permutations & combinations example taste testing Probability & combinatorics.mp3", "Sentence": "From 15 distinct varieties, Samara will choose three different olive oils and blend them together. A contestant will taste the blend and try to identify which three of the 15 varieties were used to make it. Assume that a contestant can't taste any difference and is randomly guessing. What is the probability that a contestant correctly guesses which three varieties were used? So pause this video and see if you can think about that. And if you can just come up with the expression, you don't have to compute it, that is probably good enough, at least for our purposes. All right, now let's work through this together."}, {"video_title": "Probability with permutations & combinations example taste testing Probability & combinatorics.mp3", "Sentence": "What is the probability that a contestant correctly guesses which three varieties were used? So pause this video and see if you can think about that. And if you can just come up with the expression, you don't have to compute it, that is probably good enough, at least for our purposes. All right, now let's work through this together. So we know several things here. We have 15 distinct varieties, and we are choosing three of those varieties. And anytime we're talking about probability and combinatorics, it's always interesting to say, does order matter?"}, {"video_title": "Probability with permutations & combinations example taste testing Probability & combinatorics.mp3", "Sentence": "All right, now let's work through this together. So we know several things here. We have 15 distinct varieties, and we are choosing three of those varieties. And anytime we're talking about probability and combinatorics, it's always interesting to say, does order matter? Does it matter what order that Samara is picking those three from the 15? And it doesn't look like it matters. It looks like we just have to think about what three they are."}, {"video_title": "Probability with permutations & combinations example taste testing Probability & combinatorics.mp3", "Sentence": "And anytime we're talking about probability and combinatorics, it's always interesting to say, does order matter? Does it matter what order that Samara is picking those three from the 15? And it doesn't look like it matters. It looks like we just have to think about what three they are. It doesn't matter what order either she picked them in or the order in which the contestant guesses them in. And so if you think about the total number of ways of picking three things from a group of 15, you could write that as 15 choose three. Once again, this is just shorthand notation for how many combinations are there so you can pick three things from a group of 15."}, {"video_title": "Probability with permutations & combinations example taste testing Probability & combinatorics.mp3", "Sentence": "It looks like we just have to think about what three they are. It doesn't matter what order either she picked them in or the order in which the contestant guesses them in. And so if you think about the total number of ways of picking three things from a group of 15, you could write that as 15 choose three. Once again, this is just shorthand notation for how many combinations are there so you can pick three things from a group of 15. So some of you might have been tempted to say, hey, let me think about permutations here. And I have 15 things. And from that, I wanna figure out how many ways can I pick three things that actually has order mattering."}, {"video_title": "Probability with permutations & combinations example taste testing Probability & combinatorics.mp3", "Sentence": "Once again, this is just shorthand notation for how many combinations are there so you can pick three things from a group of 15. So some of you might have been tempted to say, hey, let me think about permutations here. And I have 15 things. And from that, I wanna figure out how many ways can I pick three things that actually has order mattering. But this would be the situation where we're talking about the contestant actually having to maybe guess in the same order in which the varieties were originally blended or something like that. But we're not doing that. We just care about getting the right three varieties."}, {"video_title": "Probability with permutations & combinations example taste testing Probability & combinatorics.mp3", "Sentence": "And from that, I wanna figure out how many ways can I pick three things that actually has order mattering. But this would be the situation where we're talking about the contestant actually having to maybe guess in the same order in which the varieties were originally blended or something like that. But we're not doing that. We just care about getting the right three varieties. So this will tell us the total number of ways that you can pick three out of 15. And so what's the probability that the contestant correctly guesses which three varieties were used? Well, the contestant is going to be guessing one out of the possible number of scenarios here."}, {"video_title": "Probability with permutations & combinations example taste testing Probability & combinatorics.mp3", "Sentence": "We just care about getting the right three varieties. So this will tell us the total number of ways that you can pick three out of 15. And so what's the probability that the contestant correctly guesses which three varieties were used? Well, the contestant is going to be guessing one out of the possible number of scenarios here. So the probability would be one over 15, choose three. And if you wanted to compute this, this would be equal to one over. Now, how many ways can you pick three things from 15?"}, {"video_title": "Probability with permutations & combinations example taste testing Probability & combinatorics.mp3", "Sentence": "Well, the contestant is going to be guessing one out of the possible number of scenarios here. So the probability would be one over 15, choose three. And if you wanted to compute this, this would be equal to one over. Now, how many ways can you pick three things from 15? And of course, there is a formula here, but I always like to reason through it. Well, you could say, all right, if there's three slots, there's 15 different varieties that could have gone into that first slot. And then there's 14 that could go into that second slot."}, {"video_title": "Probability with permutations & combinations example taste testing Probability & combinatorics.mp3", "Sentence": "Now, how many ways can you pick three things from 15? And of course, there is a formula here, but I always like to reason through it. Well, you could say, all right, if there's three slots, there's 15 different varieties that could have gone into that first slot. And then there's 14 that could go into that second slot. And then there's 13 that can go into that third slot. But then we have to remember that it doesn't matter what order we pick them in. So how many ways can you rearrange three things?"}, {"video_title": "Probability with permutations & combinations example taste testing Probability & combinatorics.mp3", "Sentence": "And then there's 14 that could go into that second slot. And then there's 13 that can go into that third slot. But then we have to remember that it doesn't matter what order we pick them in. So how many ways can you rearrange three things? Well, it would be three factorial or three times two times one. So this would be the same thing as three times two times one over 15 times 14 times 13. See, I can simplify this, divide numerator and denominator by two, divide numerator and denominator by three."}, {"video_title": "Probability with permutations & combinations example taste testing Probability & combinatorics.mp3", "Sentence": "So how many ways can you rearrange three things? Well, it would be three factorial or three times two times one. So this would be the same thing as three times two times one over 15 times 14 times 13. See, I can simplify this, divide numerator and denominator by two, divide numerator and denominator by three. This is going to be equal to one over 35 times 13. This is gonna be one over 350 plus 105, which is 455. And we are done."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "So we want to conduct an experiment to test if this pill really can help people lower their blood sugar. So the first thing we need to think about is how do we even measure or test whether people's blood sugar is getting lowered? Well, for our experiment, what's typically done is we measure folks' hemoglobin A1c. You don't have to worry too much about this in the context of statistics, but a hemoglobin A1c test is a way that's typically used to measure your average blood sugar over the last three months. And we have whole videos on Khan Academy explaining how that works. So our hope would be that our pill lowers people's blood sugar, which shows up as a lowered A1c. Now we have terms for this."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "You don't have to worry too much about this in the context of statistics, but a hemoglobin A1c test is a way that's typically used to measure your average blood sugar over the last three months. And we have whole videos on Khan Academy explaining how that works. So our hope would be that our pill lowers people's blood sugar, which shows up as a lowered A1c. Now we have terms for this. The thing that is causing something else to change, we call this the explanatory variable. Explanatory variable. And the thing that might get changed by that explanatory variable, depending on whether you take the pill or not, we call that our response."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "Now we have terms for this. The thing that is causing something else to change, we call this the explanatory variable. Explanatory variable. And the thing that might get changed by that explanatory variable, depending on whether you take the pill or not, we call that our response. Response variable. So now let's actually conduct the experiment. So what we would do is we would go to the population, population of diabetics, and we would wanna take a random sample from that population of diabetics, a reasonably large one, and later in statistics we talk about what a good-sized sample might be."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "And the thing that might get changed by that explanatory variable, depending on whether you take the pill or not, we call that our response. Response variable. So now let's actually conduct the experiment. So what we would do is we would go to the population, population of diabetics, and we would wanna take a random sample from that population of diabetics, a reasonably large one, and later in statistics we talk about what a good-sized sample might be. But let's say that we randomly sample, randomly sample 100 folks. So we randomly sample 100 folks from that population of diabetics. And then you would want to assign these folks randomly to two different groups."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "So what we would do is we would go to the population, population of diabetics, and we would wanna take a random sample from that population of diabetics, a reasonably large one, and later in statistics we talk about what a good-sized sample might be. But let's say that we randomly sample, randomly sample 100 folks. So we randomly sample 100 folks from that population of diabetics. And then you would want to assign these folks randomly to two different groups. One would be your control group, and this would be the group of people who won't take the new medicine. And then you would have your treatment group. These are the groups of folks who will be given the new medicine, the treatment group."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "And then you would want to assign these folks randomly to two different groups. One would be your control group, and this would be the group of people who won't take the new medicine. And then you would have your treatment group. These are the groups of folks who will be given the new medicine, the treatment group. Now in some cases you can just randomly assign these 100 folks between these two groups. And one way to do it is you could give all of them a random number between one and 100, and then the top 50 go into treatment, and the bottom 50 go into the control, or you could use a computer to randomly assign folks. Now sometimes you might wanna be a little bit more sophisticated than that."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "These are the groups of folks who will be given the new medicine, the treatment group. Now in some cases you can just randomly assign these 100 folks between these two groups. And one way to do it is you could give all of them a random number between one and 100, and then the top 50 go into treatment, and the bottom 50 go into the control, or you could use a computer to randomly assign folks. Now sometimes you might wanna be a little bit more sophisticated than that. For example, there might be evidence that someone's sex might somehow influence how they respond to a drug. So what you could do is something called block design, where let's say this group just happens to have 60 females and 40 males. Well in block design, you can randomly assign, but you can do it in a way that you can ensure that both of these groups have the same proportions of male and female."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "Now sometimes you might wanna be a little bit more sophisticated than that. For example, there might be evidence that someone's sex might somehow influence how they respond to a drug. So what you could do is something called block design, where let's say this group just happens to have 60 females and 40 males. Well in block design, you can randomly assign, but you can do it in a way that you can ensure that both of these groups have the same proportions of male and female. So for example, if you have 60 females here, you can ensure that 30 of them end up in the control, and 30 end up in the treatment. But you would assign those 60 females randomly between these two groups, and similarly, you can do block design. Of these 40 males, 20 end up in the control, and 20 end up in the treatment."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "Well in block design, you can randomly assign, but you can do it in a way that you can ensure that both of these groups have the same proportions of male and female. So for example, if you have 60 females here, you can ensure that 30 of them end up in the control, and 30 end up in the treatment. But you would assign those 60 females randomly between these two groups, and similarly, you can do block design. Of these 40 males, 20 end up in the control, and 20 end up in the treatment. So once you have folks in both of these groups, what you would probably want to do is measure their A1c at the beginning. You could view that as a baseline. And then over the course of the experiment, you would give the pill to the treatment group, and in the control group, you might be saying, oh, we just wouldn't do anything."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "Of these 40 males, 20 end up in the control, and 20 end up in the treatment. So once you have folks in both of these groups, what you would probably want to do is measure their A1c at the beginning. You could view that as a baseline. And then over the course of the experiment, you would give the pill to the treatment group, and in the control group, you might be saying, oh, we just wouldn't do anything. But the best practice is actually to give a pill that looks just like the real thing to the control group. This is known as a placebo. And the reason why we do that is there's definitely evidence that when people think they're taking a pill that might help them, that even psychologically, it can have an effect on them, and sometimes it helps them."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "And then over the course of the experiment, you would give the pill to the treatment group, and in the control group, you might be saying, oh, we just wouldn't do anything. But the best practice is actually to give a pill that looks just like the real thing to the control group. This is known as a placebo. And the reason why we do that is there's definitely evidence that when people think they're taking a pill that might help them, that even psychologically, it can have an effect on them, and sometimes it helps them. This is known as the placebo effect. And not only would you give both groups a pill that looks the same, even though this one in the treatment group actually has the medicine in it, you also would not want to tell folks which group they are in. When you don't tell them which group they are in, that's known as a blind experiment."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "And the reason why we do that is there's definitely evidence that when people think they're taking a pill that might help them, that even psychologically, it can have an effect on them, and sometimes it helps them. This is known as the placebo effect. And not only would you give both groups a pill that looks the same, even though this one in the treatment group actually has the medicine in it, you also would not want to tell folks which group they are in. When you don't tell them which group they are in, that's known as a blind experiment. And you probably also don't wanna tell the people who are administering the experiment which group they are administering, and that's called a double blind. So even the doctors or the nurses that are administering the experiment, when they're giving a pill to the control group, they don't know that that pill is the placebo. And you might say, well, why is it important for an experiment to be blind or especially double blind?"}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "When you don't tell them which group they are in, that's known as a blind experiment. And you probably also don't wanna tell the people who are administering the experiment which group they are administering, and that's called a double blind. So even the doctors or the nurses that are administering the experiment, when they're giving a pill to the control group, they don't know that that pill is the placebo. And you might say, well, why is it important for an experiment to be blind or especially double blind? Well, that avoids, one, any type of psychological effect from the point of the patient, or from the, say, the caregivers in this situation. So they don't kind of give it away. They don't tell these folks, hey, you're actually just pretending to take a pill."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "And you might say, well, why is it important for an experiment to be blind or especially double blind? Well, that avoids, one, any type of psychological effect from the point of the patient, or from the, say, the caregivers in this situation. So they don't kind of give it away. They don't tell these folks, hey, you're actually just pretending to take a pill. And so that ensures that we minimize the amount of influence or bias that might happen. You might even have a triple blind experiment where even the folks who are analyzing the eventual data from this experiment don't know whether they're analyzing the data from the control or the treatment. They just compare the two different groups."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "They don't tell these folks, hey, you're actually just pretending to take a pill. And so that ensures that we minimize the amount of influence or bias that might happen. You might even have a triple blind experiment where even the folks who are analyzing the eventual data from this experiment don't know whether they're analyzing the data from the control or the treatment. They just compare the two different groups. But anyway, you do, you, people take the medicine and the placebo over the course of the experiment. Maybe this lasts for three months. And then you would wanna measure their A1c later."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "They just compare the two different groups. But anyway, you do, you, people take the medicine and the placebo over the course of the experiment. Maybe this lasts for three months. And then you would wanna measure their A1c later. And then you would see their change in the A1c. Now, if you saw that there wasn't really a difference in the change in A1c between the control and the treatment group, then you'd say, well, that probably means that my pill didn't work. Now, if you do get a greater reduction in the treatment group and you do the statistical analysis, which we will learn in statistics, and you show that, hey, there's a very low probability this happened purely due to chance, well, then you've got something."}, {"video_title": "Impact of mass on orbital speed Study design AP Statistics Khan Academy.mp3", "Sentence": "And then you would wanna measure their A1c later. And then you would see their change in the A1c. Now, if you saw that there wasn't really a difference in the change in A1c between the control and the treatment group, then you'd say, well, that probably means that my pill didn't work. Now, if you do get a greater reduction in the treatment group and you do the statistical analysis, which we will learn in statistics, and you show that, hey, there's a very low probability this happened purely due to chance, well, then you've got something. You could probably conclude that there is a causal connection between taking the pill and lowering your A1c level. But once again, you cannot be 100% sure. And so this is why it's very important for people to be able to replicate your experiment."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics (2).mp3", "Sentence": "For the final round, each of them spin a wheel to determine what star ingredient must be in their dish. I guess the primary ingredient, and we can see it could be chard, spinach, romaine, lettuce, I'm guessing, cabbage, arugula, or kale. And so then they give us these different types of events, or at least the symbols for these different types of events, and then give us their meaning. So K sub one means the first contestant lands on kale. K sub two means the second contestant lands on kale. K sub one with this superscript C, which we could view as complement. So K sub one complement, the first contestant does not land on kale."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics (2).mp3", "Sentence": "So K sub one means the first contestant lands on kale. K sub two means the second contestant lands on kale. K sub one with this superscript C, which we could view as complement. So K sub one complement, the first contestant does not land on kale. So it's the complement of this one right over here. And then K sub two complement would be that the second contestant does not land on kale. So the not of K sub two right over here."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics (2).mp3", "Sentence": "So K sub one complement, the first contestant does not land on kale. So it's the complement of this one right over here. And then K sub two complement would be that the second contestant does not land on kale. So the not of K sub two right over here. Using the general multiplication rule, express symbolically the probability that neither contestant lands on kale. So pause this video and see if you can have a go at this. All right."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics (2).mp3", "Sentence": "So the not of K sub two right over here. Using the general multiplication rule, express symbolically the probability that neither contestant lands on kale. So pause this video and see if you can have a go at this. All right. So the general multiplication rule is just saying this notion that the probability of two events, A and B, is going to be equal to the probability of, let's say A given B, times the probability of B. Now, if they're independent events, if the probability of A occurring does not depend in any way on whether B occurred or not, then this would simplify to this probability of A given B would just become the probability of A. And so if you have two independent events, you would just multiply their probabilities."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics (2).mp3", "Sentence": "All right. So the general multiplication rule is just saying this notion that the probability of two events, A and B, is going to be equal to the probability of, let's say A given B, times the probability of B. Now, if they're independent events, if the probability of A occurring does not depend in any way on whether B occurred or not, then this would simplify to this probability of A given B would just become the probability of A. And so if you have two independent events, you would just multiply their probabilities. So that's just all they're talking about, the general multiplication rule. But let me express what they're actually asking us to express, the probability that neither contestant lands on kale. So that means that this is going to happen."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics (2).mp3", "Sentence": "And so if you have two independent events, you would just multiply their probabilities. So that's just all they're talking about, the general multiplication rule. But let me express what they're actually asking us to express, the probability that neither contestant lands on kale. So that means that this is going to happen. The first contestant does not land on kale, and this is going to happen. The second contestant does not land on kale. So I could write it this way."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics (2).mp3", "Sentence": "So that means that this is going to happen. The first contestant does not land on kale, and this is going to happen. The second contestant does not land on kale. So I could write it this way. The probability that K sub one complement and K sub two complement, and I could write it this way. This is going to be equal to, we know that these are independent events because if the first contestant gets kale or whatever they get, it doesn't get taken out of the running for the second contestant. The second contestant still has an equal probability of getting or not getting kale, regardless of what happened for the first contestant."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics (2).mp3", "Sentence": "So I could write it this way. The probability that K sub one complement and K sub two complement, and I could write it this way. This is going to be equal to, we know that these are independent events because if the first contestant gets kale or whatever they get, it doesn't get taken out of the running for the second contestant. The second contestant still has an equal probability of getting or not getting kale, regardless of what happened for the first contestant. So that means we're just in the situation where we multiply these probabilities. So that's going to be the probability of K sub one complement times the probability of K sub two complement. All right, now let's do part two."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics (2).mp3", "Sentence": "The second contestant still has an equal probability of getting or not getting kale, regardless of what happened for the first contestant. So that means we're just in the situation where we multiply these probabilities. So that's going to be the probability of K sub one complement times the probability of K sub two complement. All right, now let's do part two. Interpret what each part of this probability statement represents. So I encourage you, like always, pause this video and try to figure that out. All right, so first let's think about what is going on here."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics (2).mp3", "Sentence": "All right, now let's do part two. Interpret what each part of this probability statement represents. So I encourage you, like always, pause this video and try to figure that out. All right, so first let's think about what is going on here. So this is saying the probability that this is K sub one complement. So the first contestant does not land on kale. So first contestant does not get kale and, all right, and in caps, and second contestant does get kale."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics (2).mp3", "Sentence": "All right, so first let's think about what is going on here. So this is saying the probability that this is K sub one complement. So the first contestant does not land on kale. So first contestant does not get kale and, all right, and in caps, and second contestant does get kale. And second does get kale. So that's what this left hand is saying. And now they say that that is going to be equal to, so this part right over here, probability that the first contestant does not get kale, probability that first does not get kale times, right over here, and the second part right over here is the probability that the second contestant gets kale given that the first contestant does not get kale."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics (2).mp3", "Sentence": "So first contestant does not get kale and, all right, and in caps, and second contestant does get kale. And second does get kale. So that's what this left hand is saying. And now they say that that is going to be equal to, so this part right over here, probability that the first contestant does not get kale, probability that first does not get kale times, right over here, and the second part right over here is the probability that the second contestant gets kale given that the first contestant does not get kale. So probability that the second gets kale given, that's what this vertical line right over here means. It means given, shorthand for given. Given, I wrote it up there too."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics (2).mp3", "Sentence": "And now they say that that is going to be equal to, so this part right over here, probability that the first contestant does not get kale, probability that first does not get kale times, right over here, and the second part right over here is the probability that the second contestant gets kale given that the first contestant does not get kale. So probability that the second gets kale given, that's what this vertical line right over here means. It means given, shorthand for given. Given, I wrote it up there too. Given that first does not get kale. And we're done. I just explained what is going on here."}, {"video_title": "Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "And as you can see, X can take on only a finite number of values, zero, one, two, three, or four, and so because there's a finite number of values here, we would call this a discrete random variable. And you can see that this is a valid probability distribution because the combined probability is one. .1 plus 0.15 plus 0.4 plus 0.25 plus 0.1 is one, and none of these are negative probabilities, which wouldn't have made sense. But what we care about in this video is the notion of an expected value of a discrete random variable, which you would just note this way. And one way to think about it is, once we calculate the expected value of this variable, of this random variable, that in a given week, that would give you a sense of the expected number of workouts. This is also sometimes referred to as the mean of a random variable. This right over here is the Greek letter mu, which is often used to denote the mean, so this is the mean of the random variable X."}, {"video_title": "Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "But what we care about in this video is the notion of an expected value of a discrete random variable, which you would just note this way. And one way to think about it is, once we calculate the expected value of this variable, of this random variable, that in a given week, that would give you a sense of the expected number of workouts. This is also sometimes referred to as the mean of a random variable. This right over here is the Greek letter mu, which is often used to denote the mean, so this is the mean of the random variable X. But how do we actually compute it? To compute this, we essentially just take the weighted sum of the various outcomes, and we weight them by the probabilities. So for example, this is going to be, the first outcome here is zero, and we'll weight it by its probability of 0.1."}, {"video_title": "Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "This right over here is the Greek letter mu, which is often used to denote the mean, so this is the mean of the random variable X. But how do we actually compute it? To compute this, we essentially just take the weighted sum of the various outcomes, and we weight them by the probabilities. So for example, this is going to be, the first outcome here is zero, and we'll weight it by its probability of 0.1. So it's zero times 0.1, plus the next outcome is one, and it would be weighted by its probability of 0.15, so plus one times 0.15, plus the next outcome is two, it has a probability of 0.4, plus two times 0.4, plus the outcome three has a probability of 0.25, plus three times 0.25, and then last but not least, we have the outcome four workouts in a week that has a probability of 0.1, plus four times 0.1. Well, we can simplify this a little bit. Zero times anything is just a zero, so one times 0.15 is 0.15, two times 0.4 is 0.8, three times 0.25 is 0.75, and then four times 0.1 is 0.4."}, {"video_title": "Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "So for example, this is going to be, the first outcome here is zero, and we'll weight it by its probability of 0.1. So it's zero times 0.1, plus the next outcome is one, and it would be weighted by its probability of 0.15, so plus one times 0.15, plus the next outcome is two, it has a probability of 0.4, plus two times 0.4, plus the outcome three has a probability of 0.25, plus three times 0.25, and then last but not least, we have the outcome four workouts in a week that has a probability of 0.1, plus four times 0.1. Well, we can simplify this a little bit. Zero times anything is just a zero, so one times 0.15 is 0.15, two times 0.4 is 0.8, three times 0.25 is 0.75, and then four times 0.1 is 0.4. And so we just have to add up these numbers. So we get 0.15 plus 0.8 plus 0.75 plus 0.4, and let's say 0.4, 0.75, 0.8, let's add them all together, and so let's see, five plus five is 10, and then this is two plus eight is 10, plus seven is 17, plus four is 21. So we get all of this is going to be equal to 2.1."}, {"video_title": "Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "Zero times anything is just a zero, so one times 0.15 is 0.15, two times 0.4 is 0.8, three times 0.25 is 0.75, and then four times 0.1 is 0.4. And so we just have to add up these numbers. So we get 0.15 plus 0.8 plus 0.75 plus 0.4, and let's say 0.4, 0.75, 0.8, let's add them all together, and so let's see, five plus five is 10, and then this is two plus eight is 10, plus seven is 17, plus four is 21. So we get all of this is going to be equal to 2.1. So one way to think about it is, the expected value of x, the expected number of workouts for me in a week, given this probability distribution, is 2.1. Now you might be saying, wait, hold on a second. All of the outcomes here are whole numbers."}, {"video_title": "Mean (expected value) of a discrete random variable AP Statistics Khan Academy.mp3", "Sentence": "So we get all of this is going to be equal to 2.1. So one way to think about it is, the expected value of x, the expected number of workouts for me in a week, given this probability distribution, is 2.1. Now you might be saying, wait, hold on a second. All of the outcomes here are whole numbers. How can you have 2.1 workouts in a week? What is 0.1 of a workout? Well, this isn't saying that in a given week, you would expect me to work out exactly 2.1 times, but this is valuable because you could say, well, in 10 weeks, you would expect me to do roughly 21 workouts."}, {"video_title": "Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "A club of nine people wants to choose a board of three officers. A president, a vice president, and a secretary. How many ways are there to choose the board from the nine people? Now, we're going to assume that one person can't hold more than one office. That if I'm picked for president, that I'm no longer a valid person for vice president or secretary. So let's just think about the three different positions. So you have the president, you have the vice president, VP, and then you have the secretary."}, {"video_title": "Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Now, we're going to assume that one person can't hold more than one office. That if I'm picked for president, that I'm no longer a valid person for vice president or secretary. So let's just think about the three different positions. So you have the president, you have the vice president, VP, and then you have the secretary. Now, let's say that we go for the president first. It actually doesn't matter. Let's say we're picking the president slot first and we haven't appointed any other slots yet."}, {"video_title": "Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So you have the president, you have the vice president, VP, and then you have the secretary. Now, let's say that we go for the president first. It actually doesn't matter. Let's say we're picking the president slot first and we haven't appointed any other slots yet. How many possibilities are there for president? Well, the club has nine people, so there's nine possibilities for president. Now, we're going to pick one of those nine."}, {"video_title": "Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Let's say we're picking the president slot first and we haven't appointed any other slots yet. How many possibilities are there for president? Well, the club has nine people, so there's nine possibilities for president. Now, we're going to pick one of those nine. We're going to kind of take them out of the running for the other two offices, right? Because someone's going to be president. So one of the nine is going to be president."}, {"video_title": "Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Now, we're going to pick one of those nine. We're going to kind of take them out of the running for the other two offices, right? Because someone's going to be president. So one of the nine is going to be president. There's nine possibilities, but one of the nine is going to be president. So you take that person aside. He or she is now the president."}, {"video_title": "Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So one of the nine is going to be president. There's nine possibilities, but one of the nine is going to be president. So you take that person aside. He or she is now the president. How many people are left to be vice president? Well, now there's only eight possible candidates for vice president. Eight possibilities."}, {"video_title": "Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "He or she is now the president. How many people are left to be vice president? Well, now there's only eight possible candidates for vice president. Eight possibilities. Now, he or she also goes aside. Now, how many people are left for secretary? Well, now there's only seven possibilities for secretary."}, {"video_title": "Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Eight possibilities. Now, he or she also goes aside. Now, how many people are left for secretary? Well, now there's only seven possibilities for secretary. So if you want to think about all of the different ways there are to choose a board from the nine people, there's the nine for president times the eight for vice president times the seven for secretary. You didn't have to do it this way. You could have picked secretary first and there would have been nine choices."}, {"video_title": "Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Well, now there's only seven possibilities for secretary. So if you want to think about all of the different ways there are to choose a board from the nine people, there's the nine for president times the eight for vice president times the seven for secretary. You didn't have to do it this way. You could have picked secretary first and there would have been nine choices. And then you could have picked vice president. There would have still been eight choices. Then you could have picked president last and there would have only been seven choices."}, {"video_title": "Example Ways to pick officers Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "You could have picked secretary first and there would have been nine choices. And then you could have picked vice president. There would have still been eight choices. Then you could have picked president last and there would have only been seven choices. But either way, you would have gotten nine times eight times seven. And that is, let's see, nine times eight is 72. 72 times seven is 14."}, {"video_title": "Estimating the line of best fit exercise Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Find the line of best fit, or mark that there is no linear correlation. So let's see, we have a bunch of data points. And we want to find a line that at least shows the trend in the data. And this one seems a little difficult, because if we ignore these three points down here, maybe we could do a line that looks something like this. It seems like it kind of approximates this trend, although it doesn't seem like a great trend. And if we ignore these two points right over here, we could do something like, maybe something like that. But we can't just ignore points like that, so I would say that there's actually no good line of best fit here."}, {"video_title": "Estimating the line of best fit exercise Regression Probability and Statistics Khan Academy.mp3", "Sentence": "And this one seems a little difficult, because if we ignore these three points down here, maybe we could do a line that looks something like this. It seems like it kind of approximates this trend, although it doesn't seem like a great trend. And if we ignore these two points right over here, we could do something like, maybe something like that. But we can't just ignore points like that, so I would say that there's actually no good line of best fit here. So let me check my answer. Let's try a couple more of these. Find the line of best fit."}, {"video_title": "Estimating the line of best fit exercise Regression Probability and Statistics Khan Academy.mp3", "Sentence": "But we can't just ignore points like that, so I would say that there's actually no good line of best fit here. So let me check my answer. Let's try a couple more of these. Find the line of best fit. Well, this feels very similar. It really feels like there's no, I mean, I could do that, but then I'm ignoring these two points. I could do something like that, then I'd be ignoring these points."}, {"video_title": "Estimating the line of best fit exercise Regression Probability and Statistics Khan Academy.mp3", "Sentence": "Find the line of best fit. Well, this feels very similar. It really feels like there's no, I mean, I could do that, but then I'm ignoring these two points. I could do something like that, then I'd be ignoring these points. So I'd also say no good best fit line exists. So let's try one more. So here, it looks like there's very clearly this trend, and I could try to fit it a little bit better than it's fit right now."}, {"video_title": "Estimating the line of best fit exercise Regression Probability and Statistics Khan Academy.mp3", "Sentence": "I could do something like that, then I'd be ignoring these points. So I'd also say no good best fit line exists. So let's try one more. So here, it looks like there's very clearly this trend, and I could try to fit it a little bit better than it's fit right now. So it feels like something like that fits this trend line quite well. I could maybe drop this down a little bit, something like that. Let's check my answer."}, {"video_title": "Estimating the line of best fit exercise Regression Probability and Statistics Khan Academy.mp3", "Sentence": "So here, it looks like there's very clearly this trend, and I could try to fit it a little bit better than it's fit right now. So it feels like something like that fits this trend line quite well. I could maybe drop this down a little bit, something like that. Let's check my answer. A good best fit line exists. Let me check my answer. Got it right."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Now, for each of these scenarios now, so we have 20 scenarios, five times four, we have 20 scenarios where we've seated seat one and seat two. How many people could we now seat in seat three for each of those 20 scenarios? Well, three people haven't sat down yet, so there's three possibilities there. So now, there's five times four times three scenarios for seating the first three people. How many people are left for seat four? Well, two people haven't sat down yes, so there's two possibilities. So now there's five times four times three times two scenarios of seating the first four seats, and for each of those, how many possibilities are there for the fifth seat?"}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So now, there's five times four times three scenarios for seating the first three people. How many people are left for seat four? Well, two people haven't sat down yes, so there's two possibilities. So now there's five times four times three times two scenarios of seating the first four seats, and for each of those, how many possibilities are there for the fifth seat? Well, for each of those scenarios, we only have one person who hasn't sat down left, so there's one possibility. And so the number of permutations, the number of, let me write this down, the number of permutations, permutations of seating these five people in five chairs is five factorial. Five factorial, which is equal to five times four times three times two times one, which of course is equal to, let's see, 20 times six, which is equal to 120."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So now there's five times four times three times two scenarios of seating the first four seats, and for each of those, how many possibilities are there for the fifth seat? Well, for each of those scenarios, we only have one person who hasn't sat down left, so there's one possibility. And so the number of permutations, the number of, let me write this down, the number of permutations, permutations of seating these five people in five chairs is five factorial. Five factorial, which is equal to five times four times three times two times one, which of course is equal to, let's see, 20 times six, which is equal to 120. And we've already covered this in a previous video. But now let's do something maybe more interesting, or maybe you might find it less interesting. Let's say that we still have, let's still say we have these five people, but we don't have as many chairs, so not everyone is going to be able to sit down."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Five factorial, which is equal to five times four times three times two times one, which of course is equal to, let's see, 20 times six, which is equal to 120. And we've already covered this in a previous video. But now let's do something maybe more interesting, or maybe you might find it less interesting. Let's say that we still have, let's still say we have these five people, but we don't have as many chairs, so not everyone is going to be able to sit down. So let's say that we only have three chairs. So we have chair one, we have chair two, and we have chair three. So how many ways can you have five people where only three of them are going to sit down in these three chairs?"}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Let's say that we still have, let's still say we have these five people, but we don't have as many chairs, so not everyone is going to be able to sit down. So let's say that we only have three chairs. So we have chair one, we have chair two, and we have chair three. So how many ways can you have five people where only three of them are going to sit down in these three chairs? And we care which chair they sit in, and I encourage you to pause the video and think about it. So I am assuming you have had your go at it. So let's use the same logic."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So how many ways can you have five people where only three of them are going to sit down in these three chairs? And we care which chair they sit in, and I encourage you to pause the video and think about it. So I am assuming you have had your go at it. So let's use the same logic. So how many, if we seat them in order, we might as well, how many different people, if we haven't sat anyone yet, how many different people could sit in seat one? Well, we could have, if no one sat down, we have five different people. Well, five different people could sit in seat one."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So let's use the same logic. So how many, if we seat them in order, we might as well, how many different people, if we haven't sat anyone yet, how many different people could sit in seat one? Well, we could have, if no one sat down, we have five different people. Well, five different people could sit in seat one. Well, for each of these scenarios where one person has already sat in seat one, how many people could sit in seat two? Well, in each of these scenarios, if one person has sat down, there's four people left who haven't been seated, so four people could sit in seat two. So we have five times four scenarios where we have seated seats one and seat two."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Well, five different people could sit in seat one. Well, for each of these scenarios where one person has already sat in seat one, how many people could sit in seat two? Well, in each of these scenarios, if one person has sat down, there's four people left who haven't been seated, so four people could sit in seat two. So we have five times four scenarios where we have seated seats one and seat two. Now, for each of those 20 scenarios, how many people could sit in seat three? Well, we haven't sat, we haven't, we haven't seaten or sat three of the people yet, so for each of these 20, we could put three different people in seat three. So that gives us five times four times three scenarios."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So we have five times four scenarios where we have seated seats one and seat two. Now, for each of those 20 scenarios, how many people could sit in seat three? Well, we haven't sat, we haven't, we haven't seaten or sat three of the people yet, so for each of these 20, we could put three different people in seat three. So that gives us five times four times three scenarios. So this is equal to five times four times three scenarios, which is equal to, this is equal to 60. So there's 60 permutations of sitting five people in three chairs. Now, this, and this is my brain, you know, whenever I start to think in terms of permutations, I actually think in these ways."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So that gives us five times four times three scenarios. So this is equal to five times four times three scenarios, which is equal to, this is equal to 60. So there's 60 permutations of sitting five people in three chairs. Now, this, and this is my brain, you know, whenever I start to think in terms of permutations, I actually think in these ways. I just literally draw it out because especially, you know, I don't like formulas. I like to actually conceptualize and visualize what I'm doing. But you might say, hey, you know, when we just did five different people in five different chairs and we cared which seat they sit in, we had this five factorial."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Now, this, and this is my brain, you know, whenever I start to think in terms of permutations, I actually think in these ways. I just literally draw it out because especially, you know, I don't like formulas. I like to actually conceptualize and visualize what I'm doing. But you might say, hey, you know, when we just did five different people in five different chairs and we cared which seat they sit in, we had this five factorial. Well, you know, factorial is kind of a neat little operation. How can I relate factorial to what we did just now? Well, it looks like we kind of did factorial, but then we stopped."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "But you might say, hey, you know, when we just did five different people in five different chairs and we cared which seat they sit in, we had this five factorial. Well, you know, factorial is kind of a neat little operation. How can I relate factorial to what we did just now? Well, it looks like we kind of did factorial, but then we stopped. We stopped at, we didn't go times two times one. So one way to think about what we just did is we just did five times four times three times two times one. But of course, we actually didn't do the two times one."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Well, it looks like we kind of did factorial, but then we stopped. We stopped at, we didn't go times two times one. So one way to think about what we just did is we just did five times four times three times two times one. But of course, we actually didn't do the two times one. So you could take that and you could divide by two times one. And if you did that, then this two times one would cancel with that two times one and you'd be left with five times four times three. And the whole reason I'm writing this way is that now I could write it in terms of factorial."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "But of course, we actually didn't do the two times one. So you could take that and you could divide by two times one. And if you did that, then this two times one would cancel with that two times one and you'd be left with five times four times three. And the whole reason I'm writing this way is that now I could write it in terms of factorial. I could write this as five factorial over two factorial. But then you might have the question, well, where did this two come from? You know, I have three seats."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And the whole reason I'm writing this way is that now I could write it in terms of factorial. I could write this as five factorial over two factorial. But then you might have the question, well, where did this two come from? You know, I have three seats. Where did this two come from? Well, think about it. I multiplied five times four times three."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "You know, I have three seats. Where did this two come from? Well, think about it. I multiplied five times four times three. I kept going until I had that many seats and then I didn't do the remainder. So the number of, so the things that I left out, the things that I left out, that was essentially the number of people minus the number of chairs. So I was trying to put five things in three places."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "I multiplied five times four times three. I kept going until I had that many seats and then I didn't do the remainder. So the number of, so the things that I left out, the things that I left out, that was essentially the number of people minus the number of chairs. So I was trying to put five things in three places. So five minus three, that gave me two left over. So I could write it like this. I could write it as five, let me use those same colors."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So I was trying to put five things in three places. So five minus three, that gave me two left over. So I could write it like this. I could write it as five, let me use those same colors. I could write it as five factorial over, over five minus three, which of course is two. Five minus three factorial. And so another way of thinking about it, if we wanted to generalize, is if you're trying to put, if you're trying to figure out the number of permutations, and there's a bunch of notations for writing this."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "I could write it as five, let me use those same colors. I could write it as five factorial over, over five minus three, which of course is two. Five minus three factorial. And so another way of thinking about it, if we wanted to generalize, is if you're trying to put, if you're trying to figure out the number of permutations, and there's a bunch of notations for writing this. If you're trying to figure out the number of permutations where you could put n people in r seats, or the number of permutations, you could put n people in r seats, and there's other notations as well. Well this is just going to be n factorial over n minus r factorial. Here n was five, r was three, and five minus three is two."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And so another way of thinking about it, if we wanted to generalize, is if you're trying to put, if you're trying to figure out the number of permutations, and there's a bunch of notations for writing this. If you're trying to figure out the number of permutations where you could put n people in r seats, or the number of permutations, you could put n people in r seats, and there's other notations as well. Well this is just going to be n factorial over n minus r factorial. Here n was five, r was three, and five minus three is two. Now, you'll see this in a probability or statistics class, and people might memorize this thing, it seems like this kind of daunting thing. I'll just tell you right now, the whole reason why I just showed this to you is so that you could connect it with what you might see in your textbook, or what you might see in a class, or when you see this type of formula, you see that it's not coming out of, it's not some type of voodoo magic, but I will tell you that for me, personally, I never use this formula. I always reason it through, because if you just memorize the formula, you're always going to wait, does this formula apply there?"}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Here n was five, r was three, and five minus three is two. Now, you'll see this in a probability or statistics class, and people might memorize this thing, it seems like this kind of daunting thing. I'll just tell you right now, the whole reason why I just showed this to you is so that you could connect it with what you might see in your textbook, or what you might see in a class, or when you see this type of formula, you see that it's not coming out of, it's not some type of voodoo magic, but I will tell you that for me, personally, I never use this formula. I always reason it through, because if you just memorize the formula, you're always going to wait, does this formula apply there? What's n, what's r? But if you reason it through, it comes out of straight logic. You don't have to memorize anything, you don't feel like you're just memorizing without understanding, you're just using your deductive reasoning, your logic."}, {"video_title": "Permutation formula Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "I always reason it through, because if you just memorize the formula, you're always going to wait, does this formula apply there? What's n, what's r? But if you reason it through, it comes out of straight logic. You don't have to memorize anything, you don't feel like you're just memorizing without understanding, you're just using your deductive reasoning, your logic. And that's especially valuable, because as we'll see, not every scenario is going to fit so cleanly into what we did. There might be some tweaks on this, where maybe only person B likes sitting in one of the chairs, or who knows what it might be, and then your formula is going to be useless. So I like reasoning through it like this, but I just showed you this so that you could connect it to a formula that you might see in a lecture, or in a class."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "And what I'm going to ask you is, which of these intervals, interval A, B, or C, which one contains the median of the scores, and which one, or give an estimate of which one contains the mean of the scores? Pause this video and see if you can figure that out. So let's just start with the median. Remember, the median you could view as the middle number, or if you have an even number of data points, it would be the average of the middle two. Here we have an odd number of data points, so it would be the middle number. So what would be the middle number if you were to order them from least to greatest? Well, it would be the one that has 15 on either side, so it would be the 16th data point."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "Remember, the median you could view as the middle number, or if you have an even number of data points, it would be the average of the middle two. Here we have an odd number of data points, so it would be the middle number. So what would be the middle number if you were to order them from least to greatest? Well, it would be the one that has 15 on either side, so it would be the 16th data point. 16th data point. And so we could just think about which interval here contains the 16th data point. You could view it for the 16th from the highest, or the 16th from the lowest."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "Well, it would be the one that has 15 on either side, so it would be the 16th data point. 16th data point. And so we could just think about which interval here contains the 16th data point. You could view it for the 16th from the highest, or the 16th from the lowest. It is the middle one. All right, so let's start from the highest. So this interval C contains the 13 highest data points, and then interval B goes from the 14th highest all the way to the 18th highest."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "You could view it for the 16th from the highest, or the 16th from the lowest. It is the middle one. All right, so let's start from the highest. So this interval C contains the 13 highest data points, and then interval B goes from the 14th highest all the way to the 18th highest. So this B contains the median. It contains the 16th highest data point, or if you started from the left, it would also be the 16th lowest data point. So that's where the median is, the median."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "So this interval C contains the 13 highest data points, and then interval B goes from the 14th highest all the way to the 18th highest. So this B contains the median. It contains the 16th highest data point, or if you started from the left, it would also be the 16th lowest data point. So that's where the median is, the median. Now what about an estimate for the mean? Well, you have calculated the mean in the past, but when you're looking at a distribution like this, when you're looking at a histogram, one way to think about the mean is it would be the balancing point. If you imagine that this histogram was made out of some material of, let's say, uniform density, where would you put a fulcrum in order to balance it?"}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "So that's where the median is, the median. Now what about an estimate for the mean? Well, you have calculated the mean in the past, but when you're looking at a distribution like this, when you're looking at a histogram, one way to think about the mean is it would be the balancing point. If you imagine that this histogram was made out of some material of, let's say, uniform density, where would you put a fulcrum in order to balance it? If you put the fulcrum right over here, it feels like you would have, it feels like you would tip over to the left because this is a left-skewed distribution. You have this long tail to the left. If you really wanted to balance it out, it seems like you would have to move your fulcrum in the direction of that left skew, in the direction of the tail."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "If you imagine that this histogram was made out of some material of, let's say, uniform density, where would you put a fulcrum in order to balance it? If you put the fulcrum right over here, it feels like you would have, it feels like you would tip over to the left because this is a left-skewed distribution. You have this long tail to the left. If you really wanted to balance it out, it seems like you would have to move your fulcrum in the direction of that left skew, in the direction of the tail. And so I would estimate to balance it out, it would actually be closer to that, which would be interval A. Interval A would contain the mean. The intention of this type of exercise isn't for you to try to calculate every data point. In fact, they don't give you all the information here and add them all up and then divide by 31."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "If you really wanted to balance it out, it seems like you would have to move your fulcrum in the direction of that left skew, in the direction of the tail. And so I would estimate to balance it out, it would actually be closer to that, which would be interval A. Interval A would contain the mean. The intention of this type of exercise isn't for you to try to calculate every data point. In fact, they don't give you all the information here and add them all up and then divide by 31. It's really to estimate and to also get the intuition that when you have a left-skewed distribution like this, you will often see a situation where your mean is to the left of the median. If you have a right-skewed distribution, it would be the other way around. And as we will see, when you see a symmetric distribution, the mean and the median will be awfully close to each other or when you have a roughly symmetric distribution."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "In fact, they don't give you all the information here and add them all up and then divide by 31. It's really to estimate and to also get the intuition that when you have a left-skewed distribution like this, you will often see a situation where your mean is to the left of the median. If you have a right-skewed distribution, it would be the other way around. And as we will see, when you see a symmetric distribution, the mean and the median will be awfully close to each other or when you have a roughly symmetric distribution. If you have a perfectly symmetric distribution, they might be exactly in the same place. So let's do another example. So here it says we have the ages of 14 coworkers and what I want you to do is say roughly where is the mean and roughly where is the median?"}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "And as we will see, when you see a symmetric distribution, the mean and the median will be awfully close to each other or when you have a roughly symmetric distribution. If you have a perfectly symmetric distribution, they might be exactly in the same place. So let's do another example. So here it says we have the ages of 14 coworkers and what I want you to do is say roughly where is the mean and roughly where is the median? Is it roughly at A, is it roughly at B, or is it roughly at C? Pause this video and try to figure it out. So let's first start off with the median."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "So here it says we have the ages of 14 coworkers and what I want you to do is say roughly where is the mean and roughly where is the median? Is it roughly at A, is it roughly at B, or is it roughly at C? Pause this video and try to figure it out. So let's first start off with the median. We have 14 data points. So this would be the average of the middle two data points. It would be the average of the seventh and eighth data point."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "So let's first start off with the median. We have 14 data points. So this would be the average of the middle two data points. It would be the average of the seventh and eighth data point. Well, you could say one, two, three, four, five, six, seven, and then the eighth one is here. So the seventh data point is a 30. The eighth one is in the 31 bucket."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "It would be the average of the seventh and eighth data point. Well, you could say one, two, three, four, five, six, seven, and then the eighth one is here. So the seventh data point is a 30. The eighth one is in the 31 bucket. So the average of the two would get you to B. Another way that you could think about it is you can just eyeball it and see you have just as many data points below B as you do have above B and so that also gives you a good indication that B would be where the median is. So that is where the median is."}, {"video_title": "Estimating mean and median in data displays AP Statistics Khan Academy.mp3", "Sentence": "The eighth one is in the 31 bucket. So the average of the two would get you to B. Another way that you could think about it is you can just eyeball it and see you have just as many data points below B as you do have above B and so that also gives you a good indication that B would be where the median is. So that is where the median is. Now what about the mean? Well, this is a perfectly symmetric distribution. If I wanted to balance it, I would put the fulcrum right in the middle."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's say that you have a cherry pie store, and you've noticed that there is variability in the number of cherries on each pie that you sell. Some pies might have over 100 cherries, while other pies might have fewer than 50 cherries. So what you're curious about is what is the distribution? How many of the different types of pies do you have? How many pies do you have that have a lot of cherries? How many pies do you have that have very few cherries? How many pies are in between?"}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "How many of the different types of pies do you have? How many pies do you have that have a lot of cherries? How many pies do you have that have very few cherries? How many pies are in between? And so to do that, you set up a histogram. What you do is you take each pie in your store. Let's see if I can draw a pie of some kind."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "How many pies are in between? And so to do that, you set up a histogram. What you do is you take each pie in your store. Let's see if I can draw a pie of some kind. It's a cherry pie. I don't know if this is an adequate drawing of a pie. But you take each of the pies in your store, and you count the number of cherries on it."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's see if I can draw a pie of some kind. It's a cherry pie. I don't know if this is an adequate drawing of a pie. But you take each of the pies in your store, and you count the number of cherries on it. So this pie right over here is one, two, three, four, five, six, seven, eight, nine, ten. Let's see, you keep counting, and let's say it has 32 cherries. And you do that for every pie."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "But you take each of the pies in your store, and you count the number of cherries on it. So this pie right over here is one, two, three, four, five, six, seven, eight, nine, ten. Let's see, you keep counting, and let's say it has 32 cherries. And you do that for every pie. And then you created buckets, because you don't want to create just a graph of how many have exactly 32. You just want to get a general sense of things. So you create buckets of 30."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And you do that for every pie. And then you created buckets, because you don't want to create just a graph of how many have exactly 32. You just want to get a general sense of things. So you create buckets of 30. You say, how many pies have between zero and 29 cherries? How many pies have between 30 and 59, including 30 and 59? How many pies have at least 60 and at most 89 cherries?"}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So you create buckets of 30. You say, how many pies have between zero and 29 cherries? How many pies have between 30 and 59, including 30 and 59? How many pies have at least 60 and at most 89 cherries? How many pies have at least 90 and at most 119? And then how many pies have at least 120 and at most 149? And you know that you don't have any pies that have more than 149 cherries."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "How many pies have at least 60 and at most 89 cherries? How many pies have at least 90 and at most 119? And then how many pies have at least 120 and at most 149? And you know that you don't have any pies that have more than 149 cherries. So this should account for everything. And then you count them. So for example, you say, okay, five pies have 30 to 59 cherries."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And you know that you don't have any pies that have more than 149 cherries. So this should account for everything. And then you count them. So for example, you say, okay, five pies have 30 to 59 cherries. And so we create a histogram, or you create a histogram, and you make this magenta bar go up to five. So that's how you would construct this histogram. That's what this pies at different cherry levels histogram is telling us."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So for example, you say, okay, five pies have 30 to 59 cherries. And so we create a histogram, or you create a histogram, and you make this magenta bar go up to five. So that's how you would construct this histogram. That's what this pies at different cherry levels histogram is telling us. So now that we know how to construct it, let's see if we can interpret it based on the information given in the histogram. So the first question is, based on just this information, can you figure out the total number of pies in your store, assuming that they're all accounted for by this histogram? And I encourage you to pause the video and try to figure it out on your own."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "That's what this pies at different cherry levels histogram is telling us. So now that we know how to construct it, let's see if we can interpret it based on the information given in the histogram. So the first question is, based on just this information, can you figure out the total number of pies in your store, assuming that they're all accounted for by this histogram? And I encourage you to pause the video and try to figure it out on your own. Well, what's the total number of pies? Well, let's see. There's five pies that have more than, or 30 or more, have at least 30 cherries, but no more than 59."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And I encourage you to pause the video and try to figure it out on your own. Well, what's the total number of pies? Well, let's see. There's five pies that have more than, or 30 or more, have at least 30 cherries, but no more than 59. You have eight pies in this blue bucket. You have four pies in this green bucket. And then you have, what is this, five, no, this is three pies that have at least 120, but no more than 149 cherries."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There's five pies that have more than, or 30 or more, have at least 30 cherries, but no more than 59. You have eight pies in this blue bucket. You have four pies in this green bucket. And then you have, what is this, five, no, this is three pies that have at least 120, but no more than 149 cherries. And this accounts for all of the pies. So the total number of pies you have at this store are five plus eight plus four plus three, which is what? Five plus eight is 13, plus four is 17, plus three is 20."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then you have, what is this, five, no, this is three pies that have at least 120, but no more than 149 cherries. And this accounts for all of the pies. So the total number of pies you have at this store are five plus eight plus four plus three, which is what? Five plus eight is 13, plus four is 17, plus three is 20. So there are 20 pies in this store. But then you can ask more nuanced questions. What if you wanted to know the number of pies with more than 60 cherries?"}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Five plus eight is 13, plus four is 17, plus three is 20. So there are 20 pies in this store. But then you can ask more nuanced questions. What if you wanted to know the number of pies with more than 60 cherries? The number of pies with more than 60. So number of pies with, I'll say, let's say 60 or more. 60 or more, 60 or more cherries."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "What if you wanted to know the number of pies with more than 60 cherries? The number of pies with more than 60. So number of pies with, I'll say, let's say 60 or more. 60 or more, 60 or more cherries. So let's think about it. Well, this magenta bar doesn't apply because these all have less than 60, but all of these other bars are counting pies that have 60 or more cherries. This is 60 to 89, this is 90 to 119, this is 120 to 149."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "60 or more, 60 or more cherries. So let's think about it. Well, this magenta bar doesn't apply because these all have less than 60, but all of these other bars are counting pies that have 60 or more cherries. This is 60 to 89, this is 90 to 119, this is 120 to 149. So it's going to be these eight cherries that are, sorry, these eight pies that are in this bucket plus these four pies, plus these three pies. So it is going to be essentially everything but this first bucket. Everything but all the pies except for these five pies have 60 or more cherries."}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "This is 60 to 89, this is 90 to 119, this is 120 to 149. So it's going to be these eight cherries that are, sorry, these eight pies that are in this bucket plus these four pies, plus these three pies. So it is going to be essentially everything but this first bucket. Everything but all the pies except for these five pies have 60 or more cherries. So it should be five less than 20, and so let's see, eight plus four is 12, plus three is 15, which is five less than 20. So using this histogram, we can answer a really interesting question. We can say, well, wait, how many more pies do we have that have 60 to 89 cherries than 120 to 149 cherries?"}, {"video_title": "How to interpret a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Everything but all the pies except for these five pies have 60 or more cherries. So it should be five less than 20, and so let's see, eight plus four is 12, plus three is 15, which is five less than 20. So using this histogram, we can answer a really interesting question. We can say, well, wait, how many more pies do we have that have 60 to 89 cherries than 120 to 149 cherries? Well, we say, well, we have eight pies that have 60 to 89 cherries, three that have 120 to 149. So we have five more pies in the 60 to 89 category than we do in the 120 to 149 category. So a lot of questions that we can start to answer, and hopefully this gives you a sense of how you can interpret histograms."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "And you don't want to cut open every watermelon in your watermelon farm or patch or whatever it might be called, because you want to sell most of them. You just want to sample a few watermelons and then take samples of those watermelons to figure out how dense the seeds are, and hope that you can calculate statistics on those samples that are decent estimates of the parameters for the population. So let's start doing that. So let's say that you take these little cubic inch chunks out of a random sample of your watermelons, and then you count the number of seeds in them. And you have eight samples like this. So in one of them, you found four seeds. In the next, you found three, five, seven, two, nine, 11, and seven."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So let's say that you take these little cubic inch chunks out of a random sample of your watermelons, and then you count the number of seeds in them. And you have eight samples like this. So in one of them, you found four seeds. In the next, you found three, five, seven, two, nine, 11, and seven. So this is a sample just to make sure we're visualizing it right. If this is a population of all of the chunks, these little cubic, I guess if we view this as a cubic inch, the cubic inch chunks in my entire watermelon farm, I'm sampling a very small sample of them. So I'm sampling a very small sample."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "In the next, you found three, five, seven, two, nine, 11, and seven. So this is a sample just to make sure we're visualizing it right. If this is a population of all of the chunks, these little cubic, I guess if we view this as a cubic inch, the cubic inch chunks in my entire watermelon farm, I'm sampling a very small sample of them. So I'm sampling a very small sample. Maybe I could have had a million over here. A million chunks of watermelon could have been produced from my farm, but I'm only sampling. So capital N would be 1 million, lowercase n is equal to 8."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So I'm sampling a very small sample. Maybe I could have had a million over here. A million chunks of watermelon could have been produced from my farm, but I'm only sampling. So capital N would be 1 million, lowercase n is equal to 8. And once again, you might want to have more samples, but this will make our math easy. Now, let's think about what statistics we can measure. Well, the first one that we often do is a measure of central tendency, and that's the arithmetic mean."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So capital N would be 1 million, lowercase n is equal to 8. And once again, you might want to have more samples, but this will make our math easy. Now, let's think about what statistics we can measure. Well, the first one that we often do is a measure of central tendency, and that's the arithmetic mean. But here, we were trying to estimate the population mean by coming up with the sample mean. So what is the sample mean going to be? Well, all we have to do is add up these points, add up these measurements, and then divide by the number of measurements we have."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Well, the first one that we often do is a measure of central tendency, and that's the arithmetic mean. But here, we were trying to estimate the population mean by coming up with the sample mean. So what is the sample mean going to be? Well, all we have to do is add up these points, add up these measurements, and then divide by the number of measurements we have. So let's get our calculator out for that. Actually, maybe I don't need my calculator. Let's see."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Well, all we have to do is add up these points, add up these measurements, and then divide by the number of measurements we have. So let's get our calculator out for that. Actually, maybe I don't need my calculator. Let's see. So 4 plus 3 is 7. 7 plus 5 is 12. 12 plus 7 is 19."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Let's see. So 4 plus 3 is 7. 7 plus 5 is 12. 12 plus 7 is 19. 19 plus 2 is 21. Plus 9 is 30. Plus 11 is 41."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "12 plus 7 is 19. 19 plus 2 is 21. Plus 9 is 30. Plus 11 is 41. Plus 7 is 48. So I'm going to get 48 over 8 data points. So this worked out quite well."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Plus 11 is 41. Plus 7 is 48. So I'm going to get 48 over 8 data points. So this worked out quite well. 48 divided by 8 is equal to 6. So our sample mean is 6. It's our estimate of what the population mean might be."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So this worked out quite well. 48 divided by 8 is equal to 6. So our sample mean is 6. It's our estimate of what the population mean might be. But we also want to think about how much in our population, we want to estimate how much in our population, how much spread is there? How much do our measurements vary from this mean? So there we say, well, we can try to estimate the population variance by calculating the sample variance."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "It's our estimate of what the population mean might be. But we also want to think about how much in our population, we want to estimate how much in our population, how much spread is there? How much do our measurements vary from this mean? So there we say, well, we can try to estimate the population variance by calculating the sample variance. And we're going to calculate the unbiased sample variance. Hopefully we're fairly convinced at this point why we divide by n minus 1. So we're going to calculate the unbiased sample variance."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So there we say, well, we can try to estimate the population variance by calculating the sample variance. And we're going to calculate the unbiased sample variance. Hopefully we're fairly convinced at this point why we divide by n minus 1. So we're going to calculate the unbiased sample variance. And if we do that, what do we get? Well, it's just going to be, I'll do this in a different color, it's going to be 4 minus 6 squared plus 3 minus 6 squared plus 5 minus 6 squared plus 7 minus 6 squared plus 2 minus 6 squared plus 9 minus 6 squared plus 11 minus 6 squared plus 7 minus 6 squared, all of that divided by not by 8. Remember, we want the unbiased sample variance."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So we're going to calculate the unbiased sample variance. And if we do that, what do we get? Well, it's just going to be, I'll do this in a different color, it's going to be 4 minus 6 squared plus 3 minus 6 squared plus 5 minus 6 squared plus 7 minus 6 squared plus 2 minus 6 squared plus 9 minus 6 squared plus 11 minus 6 squared plus 7 minus 6 squared, all of that divided by not by 8. Remember, we want the unbiased sample variance. We're going to divide it by 8 minus 1. So we're going to divide by 7. And so this is going to be equal to, let me give myself a little bit more real estate, the unbiased sample variance."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Remember, we want the unbiased sample variance. We're going to divide it by 8 minus 1. So we're going to divide by 7. And so this is going to be equal to, let me give myself a little bit more real estate, the unbiased sample variance. And I could even denote it by this to make it clear that we're dividing by lowercase n minus 1, is going to be equal to, let's see, 4 minus 6 is negative 2. That squared is positive 4. So I did that one."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "And so this is going to be equal to, let me give myself a little bit more real estate, the unbiased sample variance. And I could even denote it by this to make it clear that we're dividing by lowercase n minus 1, is going to be equal to, let's see, 4 minus 6 is negative 2. That squared is positive 4. So I did that one. 3 minus 6 is negative 3. That squared is going to be 9. 5 minus 6 squared is 1 squared, which is 1."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So I did that one. 3 minus 6 is negative 3. That squared is going to be 9. 5 minus 6 squared is 1 squared, which is 1. 7 minus 6 is, once again, 1 squared, which is 1. 2 minus 6, negative 4 squared. Negative 4 squared is 16."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "5 minus 6 squared is 1 squared, which is 1. 7 minus 6 is, once again, 1 squared, which is 1. 2 minus 6, negative 4 squared. Negative 4 squared is 16. 9 minus 6 squared, well, that's going to be 9. 11 minus 6 squared, that is 25. And then finally, 7 minus 6 squared, that's another 1."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Negative 4 squared is 16. 9 minus 6 squared, well, that's going to be 9. 11 minus 6 squared, that is 25. And then finally, 7 minus 6 squared, that's another 1. And we're going to divide it by 7. Now let's see if we can add this up in our heads. 4 plus 9 is 13."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "And then finally, 7 minus 6 squared, that's another 1. And we're going to divide it by 7. Now let's see if we can add this up in our heads. 4 plus 9 is 13. Plus 1 is 14, 15, 31, 40, 65, 66. So this is going to be equal to 66 over 7. And we could either divide."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "4 plus 9 is 13. Plus 1 is 14, 15, 31, 40, 65, 66. So this is going to be equal to 66 over 7. And we could either divide. That's 9 and 3 sevenths. We could write that as 9 and 3 sevenths. Or if we want to write that as a decimal, I can just take 66."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "And we could either divide. That's 9 and 3 sevenths. We could write that as 9 and 3 sevenths. Or if we want to write that as a decimal, I can just take 66. 66 divided by 7 gives us 9 point, I'll just round it. So it's approximately 9.43. So this is approximately 9.43."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Or if we want to write that as a decimal, I can just take 66. 66 divided by 7 gives us 9 point, I'll just round it. So it's approximately 9.43. So this is approximately 9.43. Now, that gave us our unbiased sample variance. Well, how could we calculate a sample standard deviation? We want to somehow get at an estimate of what the population standard deviation might be."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So this is approximately 9.43. Now, that gave us our unbiased sample variance. Well, how could we calculate a sample standard deviation? We want to somehow get at an estimate of what the population standard deviation might be. Well, the logic should, I guess, is reasonable to say, well, this is our unbiased sample variance. It's our best estimate of what the true population variance is. When we think about population parameters to get the population standard deviation, we just take the square root of the population variance."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "We want to somehow get at an estimate of what the population standard deviation might be. Well, the logic should, I guess, is reasonable to say, well, this is our unbiased sample variance. It's our best estimate of what the true population variance is. When we think about population parameters to get the population standard deviation, we just take the square root of the population variance. So if we want to get an estimate of the sample standard deviation, why don't we just take the square root of the unbiased sample variance? So that's what we'll do. So we'll define it that way."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "When we think about population parameters to get the population standard deviation, we just take the square root of the population variance. So if we want to get an estimate of the sample standard deviation, why don't we just take the square root of the unbiased sample variance? So that's what we'll do. So we'll define it that way. We'll call the sample standard deviation, we're going to define it, to be equal to the square root of the unbiased sample variance. So it's going to be the square root of this quantity. And we could take our calculator out."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So we'll define it that way. We'll call the sample standard deviation, we're going to define it, to be equal to the square root of the unbiased sample variance. So it's going to be the square root of this quantity. And we could take our calculator out. It's going to be the square root of what I just typed in. I could do second answer. It'll be the last entry here."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "And we could take our calculator out. It's going to be the square root of what I just typed in. I could do second answer. It'll be the last entry here. So the square root of that is, and I'll just round, it's approximately equal to 3.07. Approximately equal to 3.07. Now, I'm going to tell you something very counterintuitive."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "It'll be the last entry here. So the square root of that is, and I'll just round, it's approximately equal to 3.07. Approximately equal to 3.07. Now, I'm going to tell you something very counterintuitive. Or at least initially it's counterintuitive, but hopefully you'll appreciate this over time. This we've already talked about in some depth. People have even created simulations to show that this is an unbiased estimate of population variance when we divide it by n minus 1."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Now, I'm going to tell you something very counterintuitive. Or at least initially it's counterintuitive, but hopefully you'll appreciate this over time. This we've already talked about in some depth. People have even created simulations to show that this is an unbiased estimate of population variance when we divide it by n minus 1. And that's a good starting point if we're going to take the square root of anything. But it actually turns out that because the square root, the square root function is nonlinear, that this sample standard deviation, and this is how it tends to be defined, sample standard deviation, that this sample standard deviation, which is the square root of our sample variance. So from i equals 1 to n of our unbiased sample variance."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "People have even created simulations to show that this is an unbiased estimate of population variance when we divide it by n minus 1. And that's a good starting point if we're going to take the square root of anything. But it actually turns out that because the square root, the square root function is nonlinear, that this sample standard deviation, and this is how it tends to be defined, sample standard deviation, that this sample standard deviation, which is the square root of our sample variance. So from i equals 1 to n of our unbiased sample variance. So we divide it by n minus 1. This is how we literally divide our sample standard deviation. Because the square root function is nonlinear, this is nonlinear, it turns out that this is not an unbiased estimate of the true population standard deviation."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "So from i equals 1 to n of our unbiased sample variance. So we divide it by n minus 1. This is how we literally divide our sample standard deviation. Because the square root function is nonlinear, this is nonlinear, it turns out that this is not an unbiased estimate of the true population standard deviation. And I encourage people to make simulations of that if they're interested. But then you might say, OK, well, we went through great pains to divide by n minus 1 here in order to get an unbiased estimate of the population variance. Why don't we go through similar pains and somehow figure out a formula for an unbiased estimate of the population standard deviation?"}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Because the square root function is nonlinear, this is nonlinear, it turns out that this is not an unbiased estimate of the true population standard deviation. And I encourage people to make simulations of that if they're interested. But then you might say, OK, well, we went through great pains to divide by n minus 1 here in order to get an unbiased estimate of the population variance. Why don't we go through similar pains and somehow figure out a formula for an unbiased estimate of the population standard deviation? And the reason why that's difficult is to unbiased the sample variance, we just had to divide by n minus 1 instead of n. And that worked for any probability distribution for our population. It turns out to do the same thing for the standard deviation. It's not that easy."}, {"video_title": "Sample standard deviation and bias Probability and Statistics Khan Academy.mp3", "Sentence": "Why don't we go through similar pains and somehow figure out a formula for an unbiased estimate of the population standard deviation? And the reason why that's difficult is to unbiased the sample variance, we just had to divide by n minus 1 instead of n. And that worked for any probability distribution for our population. It turns out to do the same thing for the standard deviation. It's not that easy. It's actually dependent on how that population is actually distributed. So in statistics, we just define the sample standard deviation. And the one that we typically use is based on the square root of the unbiased sample variance."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "You can never have too much practice dealing with the normal distribution, because it's really one of those super important building blocks for the rest of statistics, and really a lot of your life. So what I've done here is I've taken some sample problems. This is from ck12.org's open source flexbook, their AP Statistics flexbook. And I've taken the problems from their normal distribution chapter, so you could go to their site and actually look up these same problems. So this first problem, which of the following data sets is most likely to be normally distributed? For the other choices, explain why you believe they would not follow a normal distribution. So let's see, choice A."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "And I've taken the problems from their normal distribution chapter, so you could go to their site and actually look up these same problems. So this first problem, which of the following data sets is most likely to be normally distributed? For the other choices, explain why you believe they would not follow a normal distribution. So let's see, choice A. So this is really, my beliefs come into play. So this is unusual in the math context. It's more of a, what do I think?"}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So let's see, choice A. So this is really, my beliefs come into play. So this is unusual in the math context. It's more of a, what do I think? It's kind of an essay question. So let's see what they have here. A, the hand span, measured from the tip of the thumb to the tip of the extended fifth finger."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "It's more of a, what do I think? It's kind of an essay question. So let's see what they have here. A, the hand span, measured from the tip of the thumb to the tip of the extended fifth finger. So I think they're talking about, let me see if I can draw a hand. So if that's the index finger, and then you've got the middle finger, and then you've got your ring finger, and then you've got your pinky, and the hand will look something like that. I think they're talking about this distance, from the tip of the thumb to the tip of the extended fifth finger, which is a fancy way of saying the pinky, I think."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "A, the hand span, measured from the tip of the thumb to the tip of the extended fifth finger. So I think they're talking about, let me see if I can draw a hand. So if that's the index finger, and then you've got the middle finger, and then you've got your ring finger, and then you've got your pinky, and the hand will look something like that. I think they're talking about this distance, from the tip of the thumb to the tip of the extended fifth finger, which is a fancy way of saying the pinky, I think. They're talking about that distance right there. And they're saying, if I were to measure it of a random sample of high school seniors, what would it look like? Well, you know, how far this is, this is a combination of genetics and environmental factors."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "I think they're talking about this distance, from the tip of the thumb to the tip of the extended fifth finger, which is a fancy way of saying the pinky, I think. They're talking about that distance right there. And they're saying, if I were to measure it of a random sample of high school seniors, what would it look like? Well, you know, how far this is, this is a combination of genetics and environmental factors. Maybe how much milk you drank, or how much you hung from your pinky from a bar while you were growing up. So I would think that it is a sum of a huge number of random processes. So I would guess that it is roughly normally distributed."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "Well, you know, how far this is, this is a combination of genetics and environmental factors. Maybe how much milk you drank, or how much you hung from your pinky from a bar while you were growing up. So I would think that it is a sum of a huge number of random processes. So I would guess that it is roughly normally distributed. If I look at my own hand, and my hand I don't think has grown much since I was a high school senior. It looks like, I don't know, it looks like roughly 9 inches or so. I play guitar, maybe that helped me stretch my hand."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So I would guess that it is roughly normally distributed. If I look at my own hand, and my hand I don't think has grown much since I was a high school senior. It looks like, I don't know, it looks like roughly 9 inches or so. I play guitar, maybe that helped me stretch my hand. But it's really an essay question, so I just have to say what I feel. So I would guess that the distribution would look something like this. I don't know, I've never done this, but maybe it has a mean of 8 inches or 9 inches, and it's distributed something like this."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "I play guitar, maybe that helped me stretch my hand. But it's really an essay question, so I just have to say what I feel. So I would guess that the distribution would look something like this. I don't know, I've never done this, but maybe it has a mean of 8 inches or 9 inches, and it's distributed something like this. It's distributed something like this. So maybe it probably does look like a normal distribution. But it probably won't be a perfect, in fact, I can guarantee you it won't be a perfect normal distribution."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "I don't know, I've never done this, but maybe it has a mean of 8 inches or 9 inches, and it's distributed something like this. It's distributed something like this. So maybe it probably does look like a normal distribution. But it probably won't be a perfect, in fact, I can guarantee you it won't be a perfect normal distribution. Because one, no one can have negative length of that span. This distance can never be negative. So they're going to have a, I guess, you can have no hand, so that would maybe be counted as 0."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "But it probably won't be a perfect, in fact, I can guarantee you it won't be a perfect normal distribution. Because one, no one can have negative length of that span. This distance can never be negative. So they're going to have a, I guess, you can have no hand, so that would maybe be counted as 0. But the distribution wouldn't go into the negative domain, so it wouldn't be a perfect normal distribution on the left-hand side. It would really just end here at 0. And even on the right-hand side, there are some physically impossible hand lengths."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So they're going to have a, I guess, you can have no hand, so that would maybe be counted as 0. But the distribution wouldn't go into the negative domain, so it wouldn't be a perfect normal distribution on the left-hand side. It would really just end here at 0. And even on the right-hand side, there are some physically impossible hand lengths. No one can have a hand that's larger than the height of Earth's atmosphere or an astronomical unit. You would start touching the sun. There's some point at which it is physically impossible to get to."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "And even on the right-hand side, there are some physically impossible hand lengths. No one can have a hand that's larger than the height of Earth's atmosphere or an astronomical unit. You would start touching the sun. There's some point at which it is physically impossible to get to. And in a true normal distribution, if I were to flip a bunch of coins, there's some very, very small probability that I could get a million heads in a row. It's almost 0, but there's some probability. But in the case of hand spans, there's no way out here the probability of a human being who happens to be a high school senior having a, I don't know, one mile length hand span, that's 0."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "There's some point at which it is physically impossible to get to. And in a true normal distribution, if I were to flip a bunch of coins, there's some very, very small probability that I could get a million heads in a row. It's almost 0, but there's some probability. But in the case of hand spans, there's no way out here the probability of a human being who happens to be a high school senior having a, I don't know, one mile length hand span, that's 0. So it's not going to be a perfect normal distribution at the outliers or as we get further and further away from the mean. But I think it'll be a pretty good, in our everyday world, as good as we're going to get approximation, the normal distribution is going to be a pretty good approximation for the distribution that we see. I guess one thing that, you know, this is high school seniors, when I did this, it was kind of from my point of view as a guy."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "But in the case of hand spans, there's no way out here the probability of a human being who happens to be a high school senior having a, I don't know, one mile length hand span, that's 0. So it's not going to be a perfect normal distribution at the outliers or as we get further and further away from the mean. But I think it'll be a pretty good, in our everyday world, as good as we're going to get approximation, the normal distribution is going to be a pretty good approximation for the distribution that we see. I guess one thing that, you know, this is high school seniors, when I did this, it was kind of from my point of view as a guy. And I would argue that high school seniors, guys probably have larger hands than women. So it's possible that you actually have a bimodal distribution. So instead of having it like this, it's possible that the distribution looks like this."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "I guess one thing that, you know, this is high school seniors, when I did this, it was kind of from my point of view as a guy. And I would argue that high school seniors, guys probably have larger hands than women. So it's possible that you actually have a bimodal distribution. So instead of having it like this, it's possible that the distribution looks like this. That you have one peak for guys, maybe at 8 inches, and then maybe another peak for women at, I don't know, 7 inches, and then the distribution falls off like that. So it's also possible it could be bimodal. But in general, a normal distribution is going to be a pretty good approximation for part A of this problem."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So instead of having it like this, it's possible that the distribution looks like this. That you have one peak for guys, maybe at 8 inches, and then maybe another peak for women at, I don't know, 7 inches, and then the distribution falls off like that. So it's also possible it could be bimodal. But in general, a normal distribution is going to be a pretty good approximation for part A of this problem. Let's see what part B, what they're asking us to describe. The annual salaries of all employees of a large shipping company. So if we're talking about annual salaries, you know, we have minimum wage laws, whatnot."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "But in general, a normal distribution is going to be a pretty good approximation for part A of this problem. Let's see what part B, what they're asking us to describe. The annual salaries of all employees of a large shipping company. So if we're talking about annual salaries, you know, we have minimum wage laws, whatnot. So I would guess that any corporation, if we're talking about full-time workers at least, there's going to be some minimum salary that people have. And probably a lot of people will have that minimum salary because it'll be probably the most labor-intensive jobs. Most people are down there at the low end of the pay scale."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So if we're talking about annual salaries, you know, we have minimum wage laws, whatnot. So I would guess that any corporation, if we're talking about full-time workers at least, there's going to be some minimum salary that people have. And probably a lot of people will have that minimum salary because it'll be probably the most labor-intensive jobs. Most people are down there at the low end of the pay scale. And then you have your different middle-level managers and whatnot. And then you probably have this big gap, and then you probably have your true executives, maybe your CEO or whatnot, with, you know, if this mean right here is maybe $40,000 a year, and this is probably $80,000 where some of the mid-level managers lie. But this out here, this will probably be, you know, actually if you were to draw a real, the way I've scaled it right now, this would be $80,000, this would be about $200,000, which is actually a reasonable salary for a CEO."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "Most people are down there at the low end of the pay scale. And then you have your different middle-level managers and whatnot. And then you probably have this big gap, and then you probably have your true executives, maybe your CEO or whatnot, with, you know, if this mean right here is maybe $40,000 a year, and this is probably $80,000 where some of the mid-level managers lie. But this out here, this will probably be, you know, actually if you were to draw a real, the way I've scaled it right now, this would be $80,000, this would be about $200,000, which is actually a reasonable salary for a CEO. But the reality is that this actually might get pushed way out from there. It might look something like that. It might be way off the charts."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "But this out here, this will probably be, you know, actually if you were to draw a real, the way I've scaled it right now, this would be $80,000, this would be about $200,000, which is actually a reasonable salary for a CEO. But the reality is that this actually might get pushed way out from there. It might look something like that. It might be way off the charts. You know, let's say the CEO made $5 million in a year because he cashed in a bunch of options or something. So it would be way over here, and maybe it's the CEO and a couple of other people, the CFO or the founders. So my guess is it definitely wouldn't be a normal distribution, and it definitely would have a second, it would be bimodal."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "It might be way off the charts. You know, let's say the CEO made $5 million in a year because he cashed in a bunch of options or something. So it would be way over here, and maybe it's the CEO and a couple of other people, the CFO or the founders. So my guess is it definitely wouldn't be a normal distribution, and it definitely would have a second, it would be bimodal. You would have another peak over here for senior management up at the, unless we're, well, they're not saying, you know, if we're maybe in Europe, this would probably be closer to the left. But it won't be a perfect normal distribution, and you're not going to have any values below a certain threshold, below that kind of minimum wage level. So I would call this, when you have a tail that goes more to the right than to the left, call this a right-skewed distribution."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So my guess is it definitely wouldn't be a normal distribution, and it definitely would have a second, it would be bimodal. You would have another peak over here for senior management up at the, unless we're, well, they're not saying, you know, if we're maybe in Europe, this would probably be closer to the left. But it won't be a perfect normal distribution, and you're not going to have any values below a certain threshold, below that kind of minimum wage level. So I would call this, when you have a tail that goes more to the right than to the left, call this a right-skewed distribution. And since it has two humps right here, one there and one there, we could also say it's bimodal. I mean, it depends on what kind of company this is, but that would be my guess of a lot of large shipping companies' salaries. Let's look at choice C, or problem part C. The annual salaries of a random sample of 50 CEOs of major companies, 25 women and 25 men."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So I would call this, when you have a tail that goes more to the right than to the left, call this a right-skewed distribution. And since it has two humps right here, one there and one there, we could also say it's bimodal. I mean, it depends on what kind of company this is, but that would be my guess of a lot of large shipping companies' salaries. Let's look at choice C, or problem part C. The annual salaries of a random sample of 50 CEOs of major companies, 25 women and 25 men. The fact that they wrote this here, I think they maybe are implying that maybe men and women, you know, the gender gap has not been closed fully, and there is some discrepancy. So if it was just purely 50 CEOs of major companies, I would say it's probably close to a normal distribution. It's probably something like, well, you know, once again, there's going to be some level below which no CEO is willing to work for, although I've heard of some cases where they work for free, but they're really getting paid in other ways."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "Let's look at choice C, or problem part C. The annual salaries of a random sample of 50 CEOs of major companies, 25 women and 25 men. The fact that they wrote this here, I think they maybe are implying that maybe men and women, you know, the gender gap has not been closed fully, and there is some discrepancy. So if it was just purely 50 CEOs of major companies, I would say it's probably close to a normal distribution. It's probably something like, well, you know, once again, there's going to be some level below which no CEO is willing to work for, although I've heard of some cases where they work for free, but they're really getting paid in other ways. If you include all of those things, there's probably some base salary that all CEOs make at least that much, and then it goes up to some value, you know, the highest probability value, and then it probably has a long tail to the right. And this is if there were no gender gap. So this would just be a purely right-skewed distribution, where you have a long tail to the right."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "It's probably something like, well, you know, once again, there's going to be some level below which no CEO is willing to work for, although I've heard of some cases where they work for free, but they're really getting paid in other ways. If you include all of those things, there's probably some base salary that all CEOs make at least that much, and then it goes up to some value, you know, the highest probability value, and then it probably has a long tail to the right. And this is if there were no gender gap. So this would just be a purely right-skewed distribution, where you have a long tail to the right. Now if you assume that there's some gender gap, then you might have two humps here, which would be a bimodal distribution. So if you assume there's some gender gap, this is part c right here, then maybe there's one hump for women, and if you assume that women are less than men, then another hump for men, and there are 25 of each, so there wouldn't be necessarily more men than women, and then it would skew all the way off to the right. And in fact, I think there would probably be a chance that you have this other notion here, where you have these super CEOs or mega CEOs who make millions, while most CEOs probably just make, I'll put it in quotation marks, a few hundred thousand dollars, while there's a small subset that are way off many standard deviations to the right."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So this would just be a purely right-skewed distribution, where you have a long tail to the right. Now if you assume that there's some gender gap, then you might have two humps here, which would be a bimodal distribution. So if you assume there's some gender gap, this is part c right here, then maybe there's one hump for women, and if you assume that women are less than men, then another hump for men, and there are 25 of each, so there wouldn't be necessarily more men than women, and then it would skew all the way off to the right. And in fact, I think there would probably be a chance that you have this other notion here, where you have these super CEOs or mega CEOs who make millions, while most CEOs probably just make, I'll put it in quotation marks, a few hundred thousand dollars, while there's a small subset that are way off many standard deviations to the right. So it could even be a trimodal distribution here. So that's choice c. And so far, choice a looks like the best candidate for a pure, or the closest to being a normal distribution. Let's see what choice d is."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "And in fact, I think there would probably be a chance that you have this other notion here, where you have these super CEOs or mega CEOs who make millions, while most CEOs probably just make, I'll put it in quotation marks, a few hundred thousand dollars, while there's a small subset that are way off many standard deviations to the right. So it could even be a trimodal distribution here. So that's choice c. And so far, choice a looks like the best candidate for a pure, or the closest to being a normal distribution. Let's see what choice d is. The dates of 100 pennies taken from a cash drawer in a convenience store. 100 pennies. So that's actually an interesting experiment."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "Let's see what choice d is. The dates of 100 pennies taken from a cash drawer in a convenience store. 100 pennies. So that's actually an interesting experiment. But I would guess, and once again, this is really a question where I get to express my feelings about these things. As long as your answer is reasonable, I would say that it is right. Most pennies are newer pennies, because they go out of commission, they get traded out, they get worn out as they age, they get lost, or they get pressed at the little tourist place into those little souvenir things."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So that's actually an interesting experiment. But I would guess, and once again, this is really a question where I get to express my feelings about these things. As long as your answer is reasonable, I would say that it is right. Most pennies are newer pennies, because they go out of commission, they get traded out, they get worn out as they age, they get lost, or they get pressed at the little tourist place into those little souvenir things. I'm not even sure if that's legal, if you can do that to money legally. So my guess is that if you were to plot it, you would have a ton of pennies that are within the last few years. So the dates of 100 pennies, not their age."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "Most pennies are newer pennies, because they go out of commission, they get traded out, they get worn out as they age, they get lost, or they get pressed at the little tourist place into those little souvenir things. I'm not even sure if that's legal, if you can do that to money legally. So my guess is that if you were to plot it, you would have a ton of pennies that are within the last few years. So the dates of 100 pennies, not their age. So if we're sitting here in 2000, so if this is 2010, I would guess that right now you're not going to find any 2010 pennies, but you're probably going to find a ton of 2009 pennies, and then it probably just goes down from there. And of course, you're not going to find pennies that are older than, say, the United States, or before they even started printing pennies. So obviously this tail isn't going to go to the left forever, but my guess is you're going to have a left skewed distribution."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So the dates of 100 pennies, not their age. So if we're sitting here in 2000, so if this is 2010, I would guess that right now you're not going to find any 2010 pennies, but you're probably going to find a ton of 2009 pennies, and then it probably just goes down from there. And of course, you're not going to find pennies that are older than, say, the United States, or before they even started printing pennies. So obviously this tail isn't going to go to the left forever, but my guess is you're going to have a left skewed distribution. Where you have the bulk of the distribution on the right, but the tail goes off to the left. That's why it's called a left skewed distribution. Sometimes this is called a negatively skewed distribution, and similarly this right skewed distribution could also, or this right skewed distribution, sometimes it's called positively skewed."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "So obviously this tail isn't going to go to the left forever, but my guess is you're going to have a left skewed distribution. Where you have the bulk of the distribution on the right, but the tail goes off to the left. That's why it's called a left skewed distribution. Sometimes this is called a negatively skewed distribution, and similarly this right skewed distribution could also, or this right skewed distribution, sometimes it's called positively skewed. And if you have only one hump, you don't have a multimodal distribution like this, in a left skewed distribution, your mean is going to be to the left of your median. So in this case, maybe your median might be someplace over here. But since you have this long tail to the left, your mean might be someplace over here."}, {"video_title": "ck12.org normal distribution problems Qualitative sense of normal distributions Khan Academy.mp3", "Sentence": "Sometimes this is called a negatively skewed distribution, and similarly this right skewed distribution could also, or this right skewed distribution, sometimes it's called positively skewed. And if you have only one hump, you don't have a multimodal distribution like this, in a left skewed distribution, your mean is going to be to the left of your median. So in this case, maybe your median might be someplace over here. But since you have this long tail to the left, your mean might be someplace over here. And likewise, in this distribution, your median, your middle value, might be someplace like this. But because it's right skewed, and for the most part it only has one big hump, this hump won't change things too much because it's small, your mean is going to be to the right of it. So that's another reason why it's called a right skewed or positively skewed distribution."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "To win a particular lottery game, a player chooses four numbers from 1 to 60. Each number can only be chosen once. If all four numbers match the four winning numbers, regardless of order, the player wins. What is the probability that the winning numbers are 3, 15, 46, and 49? So the way to think about this problem, they say that we're going to choose four numbers from 60. So one way to think about it is how many different outcomes are there if we choose four numbers out of 60? Now, this is equivalent to saying how many combinations are there if we have 60 items?"}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "What is the probability that the winning numbers are 3, 15, 46, and 49? So the way to think about this problem, they say that we're going to choose four numbers from 60. So one way to think about it is how many different outcomes are there if we choose four numbers out of 60? Now, this is equivalent to saying how many combinations are there if we have 60 items? In this case, we have 60 numbers, and we are going to choose four. And we don't care about the order. That's why we're dealing with combinations, not permutations."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Now, this is equivalent to saying how many combinations are there if we have 60 items? In this case, we have 60 numbers, and we are going to choose four. And we don't care about the order. That's why we're dealing with combinations, not permutations. We don't care about the order. How many different groups of four can we pick out of 60? We don't care what order we pick them in."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "That's why we're dealing with combinations, not permutations. We don't care about the order. How many different groups of four can we pick out of 60? We don't care what order we pick them in. We've seen in previous videos that there is a formula here, but it's important to understand the reasoning behind the formula. I'll write the formula here, but then we'll think about what it's actually saying. This is 60 factorial over 60 minus 4 factorial divided also by 4 factorial, or in the denominator, multiplied by 4 factorial."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "We don't care what order we pick them in. We've seen in previous videos that there is a formula here, but it's important to understand the reasoning behind the formula. I'll write the formula here, but then we'll think about what it's actually saying. This is 60 factorial over 60 minus 4 factorial divided also by 4 factorial, or in the denominator, multiplied by 4 factorial. This is the formula right here. What this is really saying, this part right here, 60 factorial divided by 60 minus 4 factorial, that's just 60 times 59 times 58 times 57. That's what this expression right here is."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "This is 60 factorial over 60 minus 4 factorial divided also by 4 factorial, or in the denominator, multiplied by 4 factorial. This is the formula right here. What this is really saying, this part right here, 60 factorial divided by 60 minus 4 factorial, that's just 60 times 59 times 58 times 57. That's what this expression right here is. And if you think about it, the first number you pick, there's one of 60 numbers, that number is kind of out of the game, then you can pick from one of 59, then from one of 58, then of one of 57. So if you cared about order, this is the number of permutations you could pick four items out of 60 without replacing them. Now, this is when you cared about order, but you're kind of over-counting because it's counting different permutations that are essentially the same combination, the same set of four numbers, and that's why we're dividing by 4 factorial here."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "That's what this expression right here is. And if you think about it, the first number you pick, there's one of 60 numbers, that number is kind of out of the game, then you can pick from one of 59, then from one of 58, then of one of 57. So if you cared about order, this is the number of permutations you could pick four items out of 60 without replacing them. Now, this is when you cared about order, but you're kind of over-counting because it's counting different permutations that are essentially the same combination, the same set of four numbers, and that's why we're dividing by 4 factorial here. Because 4 factorial is essentially the number of ways that four numbers can be arranged in four places. The first number can be in one of four slots, the second in one of three, then two, then one. That's why you're dividing by 4 factorial."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Now, this is when you cared about order, but you're kind of over-counting because it's counting different permutations that are essentially the same combination, the same set of four numbers, and that's why we're dividing by 4 factorial here. Because 4 factorial is essentially the number of ways that four numbers can be arranged in four places. The first number can be in one of four slots, the second in one of three, then two, then one. That's why you're dividing by 4 factorial. But anyway, let's just evaluate this. This will tell us how many possible outcomes are there for the lottery game. So this is equal to, we already said the blue part is equivalent to 60 times 59 times 58 times 57."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "That's why you're dividing by 4 factorial. But anyway, let's just evaluate this. This will tell us how many possible outcomes are there for the lottery game. So this is equal to, we already said the blue part is equivalent to 60 times 59 times 58 times 57. So that's literally 60 factorial divided by essentially 56 factorial. And then you have your 4 factorial over here, which is 4 times 3 times 2 times 1. And we could simplify it a little bit just before we break out the calculator."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So this is equal to, we already said the blue part is equivalent to 60 times 59 times 58 times 57. So that's literally 60 factorial divided by essentially 56 factorial. And then you have your 4 factorial over here, which is 4 times 3 times 2 times 1. And we could simplify it a little bit just before we break out the calculator. 60 divided by 4 is 15. And then let's see, 15 divided by 3 is 5. And let's see, we have a 58 divided by 2."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And we could simplify it a little bit just before we break out the calculator. 60 divided by 4 is 15. And then let's see, 15 divided by 3 is 5. And let's see, we have a 58 divided by 2. 58 divided by 2 is 29. So our answer is going to be 5 times 59 times 29 times 57. This isn't going to be our answer."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And let's see, we have a 58 divided by 2. 58 divided by 2 is 29. So our answer is going to be 5 times 59 times 29 times 57. This isn't going to be our answer. This is going to be the number of combinations we can get if we choose four numbers out of 60 and we don't care about order. So let's take the calculator out now. So we have 5 times 59 times 29 times 57 is equal to 487,635."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "This isn't going to be our answer. This is going to be the number of combinations we can get if we choose four numbers out of 60 and we don't care about order. So let's take the calculator out now. So we have 5 times 59 times 29 times 57 is equal to 487,635. So let me write that down. That is 487,635 combinations. If you're picking four numbers, you're choosing four numbers out of 60 or 60 choose 4."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So we have 5 times 59 times 29 times 57 is equal to 487,635. So let me write that down. That is 487,635 combinations. If you're picking four numbers, you're choosing four numbers out of 60 or 60 choose 4. Now, the question they say is, what is the probability that the winning numbers are 3, 15, 46, and 49? Well, this is just one particular of the combinations. This is just one of the 487,635 possible outcomes."}, {"video_title": "Example Lottery probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "If you're picking four numbers, you're choosing four numbers out of 60 or 60 choose 4. Now, the question they say is, what is the probability that the winning numbers are 3, 15, 46, and 49? Well, this is just one particular of the combinations. This is just one of the 487,635 possible outcomes. So the probability of 3, 15, 46, 49 winning is just equal to, well, this is just one of the outcomes out of 487,635. So that right there is your probability of winning. This is one outcome out of all of the potential outcomes or combinations when you take 60 and you choose 4 from that."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So let me write that down. Number of standard deviations, I'll write it like this. Number of standard deviations from our population mean for a particular, particular data point. Now let's make that a little bit concrete. Let's say that you're some type of marine biologist and you've discovered a new species of winged turtles. And there's a total of seven winged turtles. The entire population of these winged turtles is seven."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Now let's make that a little bit concrete. Let's say that you're some type of marine biologist and you've discovered a new species of winged turtles. And there's a total of seven winged turtles. The entire population of these winged turtles is seven. And so you go and you're actually able to measure all the winged turtles. So, and you care about their length. And you also wanna care about how are those lengths distributed?"}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "The entire population of these winged turtles is seven. And so you go and you're actually able to measure all the winged turtles. So, and you care about their length. And you also wanna care about how are those lengths distributed? Lengths of winged turtles. All right. And let's say, and this is all in centimeters."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And you also wanna care about how are those lengths distributed? Lengths of winged turtles. All right. And let's say, and this is all in centimeters. These are very small turtles. So you discover, and these are all adults, so there's a two-centimeter one, there's another two-centimeter one, there's a three-centimeter one, there's another two-centimeter one, there's a five-centimeter one, a one-centimeter one, and a six-centimeter one. So we have seven data points."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And let's say, and this is all in centimeters. These are very small turtles. So you discover, and these are all adults, so there's a two-centimeter one, there's another two-centimeter one, there's a three-centimeter one, there's another two-centimeter one, there's a five-centimeter one, a one-centimeter one, and a six-centimeter one. So we have seven data points. And from this, and you, I encourage you at any point, if you want, pause this video and see if you wanna calculate, what does the population mean here? We're assuming that this is the population of all the winged turtles. Well, the mean in this situation is going to be equal to, you could add up all of these numbers and divide by seven, and you would then get three."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So we have seven data points. And from this, and you, I encourage you at any point, if you want, pause this video and see if you wanna calculate, what does the population mean here? We're assuming that this is the population of all the winged turtles. Well, the mean in this situation is going to be equal to, you could add up all of these numbers and divide by seven, and you would then get three. And then using these data points and the mean, you can calculate the population standard deviation. And once again, as review, I always encourage you to pause this video and see if you can do it on your own, but I've calculated that ahead of time. The population standard deviation in this situation is approximately, I'll round to the hundredths place, 1.69."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Well, the mean in this situation is going to be equal to, you could add up all of these numbers and divide by seven, and you would then get three. And then using these data points and the mean, you can calculate the population standard deviation. And once again, as review, I always encourage you to pause this video and see if you can do it on your own, but I've calculated that ahead of time. The population standard deviation in this situation is approximately, I'll round to the hundredths place, 1.69. So with this information, you should be able to calculate the z-score for each of these data points. Pause this video and see if you can do that. So let me make a new column here."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "The population standard deviation in this situation is approximately, I'll round to the hundredths place, 1.69. So with this information, you should be able to calculate the z-score for each of these data points. Pause this video and see if you can do that. So let me make a new column here. So here I'm gonna put our z-score. And if you just look at the definition, what you're going to do for each of these data points, let's say each data point is x, you're going to subtract from that the mean, and then you're going to divide that by the standard deviation. The numerator ID over here is gonna tell you how far you are above or below the mean, but you wanna know how many standard deviations you are from the mean."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So let me make a new column here. So here I'm gonna put our z-score. And if you just look at the definition, what you're going to do for each of these data points, let's say each data point is x, you're going to subtract from that the mean, and then you're going to divide that by the standard deviation. The numerator ID over here is gonna tell you how far you are above or below the mean, but you wanna know how many standard deviations you are from the mean. So then you'll divide by the population standard deviation. So for example, this first data point right over here, if I wanna calculate its z-score, I will take two. From that, I will subtract three, and then I will divide by 1.69."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "The numerator ID over here is gonna tell you how far you are above or below the mean, but you wanna know how many standard deviations you are from the mean. So then you'll divide by the population standard deviation. So for example, this first data point right over here, if I wanna calculate its z-score, I will take two. From that, I will subtract three, and then I will divide by 1.69. I will divide by 1.69. And if you got a calculator out, this is going to be negative one divided by 1.69. And if you use a calculator, you would get this is going to be approximately negative 0.59."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "From that, I will subtract three, and then I will divide by 1.69. I will divide by 1.69. And if you got a calculator out, this is going to be negative one divided by 1.69. And if you use a calculator, you would get this is going to be approximately negative 0.59. And the z-score for this data point is going to be the same. That is also going to be negative 0.59. One way to interpret this is, this is a little bit more than half a standard deviation below the mean."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And if you use a calculator, you would get this is going to be approximately negative 0.59. And the z-score for this data point is going to be the same. That is also going to be negative 0.59. One way to interpret this is, this is a little bit more than half a standard deviation below the mean. And we could do a similar calculation for data points that are above the mean. Let's say this data point right over here what is its z-score? Pause this video and see if you can figure that out."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "One way to interpret this is, this is a little bit more than half a standard deviation below the mean. And we could do a similar calculation for data points that are above the mean. Let's say this data point right over here what is its z-score? Pause this video and see if you can figure that out. Well, it's going to be six minus our mean, so minus three. All of that over the standard deviation. All of that over 1.69."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Pause this video and see if you can figure that out. Well, it's going to be six minus our mean, so minus three. All of that over the standard deviation. All of that over 1.69. And this, if you have a calculator, and I calculated it ahead of time, this is going to be approximately 1.77. So more than one, but less than two standard deviations above the mean. I encourage you to pause this video and now try to figure out the z-scores for these other data points."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "All of that over 1.69. And this, if you have a calculator, and I calculated it ahead of time, this is going to be approximately 1.77. So more than one, but less than two standard deviations above the mean. I encourage you to pause this video and now try to figure out the z-scores for these other data points. Now an obvious question that some of you might be asking is why? Why do we care how many standard deviations above or below the mean a data point is? In your future statistical life, z-scores are going to be a really useful way to think about how usual or how unusual a certain data point is."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "I encourage you to pause this video and now try to figure out the z-scores for these other data points. Now an obvious question that some of you might be asking is why? Why do we care how many standard deviations above or below the mean a data point is? In your future statistical life, z-scores are going to be a really useful way to think about how usual or how unusual a certain data point is. And that's going to be really valuable once we start making inferences based on our data. So I will leave you there. Just keep in mind, it's a very useful idea, but at the heart of it, a fairly simple one."}, {"video_title": "Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "In your future statistical life, z-scores are going to be a really useful way to think about how usual or how unusual a certain data point is. And that's going to be really valuable once we start making inferences based on our data. So I will leave you there. Just keep in mind, it's a very useful idea, but at the heart of it, a fairly simple one. If you know the mean, you know the standard deviation. Take your data point, subtract the mean from the data point, and then divide by your standard deviation. That gives you your z-score."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "The line fitted to model the data has a slope of 15. So the line that they're talking about is right here. So this is the scatter plot. This shows that some student who spent some time between half an hour and an hour studying got a little bit less than a 45 on the test. The student here who got a little bit higher than a 60 spent a little under two hours studying. This student over here who looks like they got like a 94 or a 95 spent over four hours studying. And so then they fit a line to it and this line has a slope of 15."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "This shows that some student who spent some time between half an hour and an hour studying got a little bit less than a 45 on the test. The student here who got a little bit higher than a 60 spent a little under two hours studying. This student over here who looks like they got like a 94 or a 95 spent over four hours studying. And so then they fit a line to it and this line has a slope of 15. And before I even read these choices, what's the best interpretation of this slope? Well, if you think this line is indicative of the trend and it does look like that from the scatter plot, that implies that roughly every extra hour that you study is going to improve your score by 15. You could say on average according to this regression."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "And so then they fit a line to it and this line has a slope of 15. And before I even read these choices, what's the best interpretation of this slope? Well, if you think this line is indicative of the trend and it does look like that from the scatter plot, that implies that roughly every extra hour that you study is going to improve your score by 15. You could say on average according to this regression. So if we start over here and we were to increase by one hour our score should improve by 15. And it does indeed look like that. We're going from, we're going in the horizontal direction, we're going one hour, and then in the vertical direction we're going from 45 to 60."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "You could say on average according to this regression. So if we start over here and we were to increase by one hour our score should improve by 15. And it does indeed look like that. We're going from, we're going in the horizontal direction, we're going one hour, and then in the vertical direction we're going from 45 to 60. So that's how I would interpret it. Every hour, based on this regression, you could, it's not unreasonable to expect 15 points improvement, or at least that's what we're seeing, that's what we're seeing from the regression of the data. So let's look at which of these choices actually describe something like that."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "We're going from, we're going in the horizontal direction, we're going one hour, and then in the vertical direction we're going from 45 to 60. So that's how I would interpret it. Every hour, based on this regression, you could, it's not unreasonable to expect 15 points improvement, or at least that's what we're seeing, that's what we're seeing from the regression of the data. So let's look at which of these choices actually describe something like that. The model predicts that the student who scored zero studied for an average of 15 hours. No, it definitely doesn't say that. The model predicts that students who didn't study at all will have an average score of 15 points."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "So let's look at which of these choices actually describe something like that. The model predicts that the student who scored zero studied for an average of 15 hours. No, it definitely doesn't say that. The model predicts that students who didn't study at all will have an average score of 15 points. No, we didn't see that. Students, if you take this, if you believe this model, someone who doesn't study at all would get close to, would get between 35 and 40 points, so like a 37 or a 38. So don't like that choice."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "The model predicts that students who didn't study at all will have an average score of 15 points. No, we didn't see that. Students, if you take this, if you believe this model, someone who doesn't study at all would get close to, would get between 35 and 40 points, so like a 37 or a 38. So don't like that choice. The model predicts that the score will increase 15 points for each additional hour of study time. Yes, that is exactly what we were thinking about when we were looking at the model. That's what a slope of 15 tells you."}, {"video_title": "Interpreting slope of regression line AP Statistics Khan Academy.mp3", "Sentence": "So don't like that choice. The model predicts that the score will increase 15 points for each additional hour of study time. Yes, that is exactly what we were thinking about when we were looking at the model. That's what a slope of 15 tells you. You increase studying time by an hour, it increases score by 15 points. The model predicts that the study time will increase 15 hours for each additional point scored. Well, no, and first of all, hours is the thing that we've used the independent variable and the points being the dependent variable, and this is phrasing it the other way, and you definitely wouldn't expect to do an extra 15 hours for each point."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "What proportion of student heights are lower than Darnell's height? So let's think about what they are asking. So they're saying that heights are normally distributed. So it would have a shape that looks something like that. That's my hand-drawn version of it. There's a mean of 150 centimeters. So right over here, that would be 150 centimeters."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "So it would have a shape that looks something like that. That's my hand-drawn version of it. There's a mean of 150 centimeters. So right over here, that would be 150 centimeters. They tell us that there's a standard deviation of 20 centimeters, and Darnell has a height of 161.4 centimeters. So Darnell is above the mean. So let's say he is right over here, and I'm not drawing it exactly, but you get the idea."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "So right over here, that would be 150 centimeters. They tell us that there's a standard deviation of 20 centimeters, and Darnell has a height of 161.4 centimeters. So Darnell is above the mean. So let's say he is right over here, and I'm not drawing it exactly, but you get the idea. That is 161.4 centimeters. And we want to figure out what proportion of students' heights are lower than Darnell's height. So we want to figure out what is the area under the normal curve right over here."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "So let's say he is right over here, and I'm not drawing it exactly, but you get the idea. That is 161.4 centimeters. And we want to figure out what proportion of students' heights are lower than Darnell's height. So we want to figure out what is the area under the normal curve right over here. That will give us the proportion that are lower than Darnell's height. And I'll give you a hint on how to do this. We need to think about how many standard deviations above the mean is Darnell."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "So we want to figure out what is the area under the normal curve right over here. That will give us the proportion that are lower than Darnell's height. And I'll give you a hint on how to do this. We need to think about how many standard deviations above the mean is Darnell. And we can do that because they tell us what the standard deviation is, and we know the difference between Darnell's height and the mean height. And then once we know how many standard deviations he is above the mean, that's our z-score, we can look at a z-table that tell us what proportion is less than that amount in a normal distribution. So let's do that."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "We need to think about how many standard deviations above the mean is Darnell. And we can do that because they tell us what the standard deviation is, and we know the difference between Darnell's height and the mean height. And then once we know how many standard deviations he is above the mean, that's our z-score, we can look at a z-table that tell us what proportion is less than that amount in a normal distribution. So let's do that. So I have my TI-84 emulator right over here. And let's see, Darnell is 161.4 centimeters, 161.4. Now the mean is 150, minus 150, is equal to, we could have done that in our head, 11.4 centimeters."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "So let's do that. So I have my TI-84 emulator right over here. And let's see, Darnell is 161.4 centimeters, 161.4. Now the mean is 150, minus 150, is equal to, we could have done that in our head, 11.4 centimeters. Now how many standard deviations is that above the mean? Well they tell us that a standard deviation in this case for this distribution is 20 centimeters. So we'll take 11.4 divided by 20."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "Now the mean is 150, minus 150, is equal to, we could have done that in our head, 11.4 centimeters. Now how many standard deviations is that above the mean? Well they tell us that a standard deviation in this case for this distribution is 20 centimeters. So we'll take 11.4 divided by 20. So we will just take our previous answer. So this just means our previous answer divided by 20 centimeters. And that gets us 0.57."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "So we'll take 11.4 divided by 20. So we will just take our previous answer. So this just means our previous answer divided by 20 centimeters. And that gets us 0.57. So we can say that this is 0.57 standard deviations 0.57 standard deviations, deviations above the mean. Now why is that useful? Well you could take this z-score right over here and look at a z-table to figure out what proportion is less than 0.57 standard deviations above the mean."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "And that gets us 0.57. So we can say that this is 0.57 standard deviations 0.57 standard deviations, deviations above the mean. Now why is that useful? Well you could take this z-score right over here and look at a z-table to figure out what proportion is less than 0.57 standard deviations above the mean. So let's get a z-table over here. So what we're going to do is we're gonna look up this z-score on this table. And the way that you do it, this first column, each row tells us our z-score up until the tenths place."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "Well you could take this z-score right over here and look at a z-table to figure out what proportion is less than 0.57 standard deviations above the mean. So let's get a z-table over here. So what we're going to do is we're gonna look up this z-score on this table. And the way that you do it, this first column, each row tells us our z-score up until the tenths place. And then each of these columns after that tell us which hundredths we're in. So 0.57, the tenths place is right over here. So we're going to be in this row."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "And the way that you do it, this first column, each row tells us our z-score up until the tenths place. And then each of these columns after that tell us which hundredths we're in. So 0.57, the tenths place is right over here. So we're going to be in this row. And then our hundredths place is this seven. So we'll look right over here. So 0.57, this tells us the proportion that is lower than 0.57 standard deviations above the mean."}, {"video_title": "Standard normal table for proportion below AP Statistics Khan Academy.mp3", "Sentence": "So we're going to be in this row. And then our hundredths place is this seven. So we'll look right over here. So 0.57, this tells us the proportion that is lower than 0.57 standard deviations above the mean. And so it is 0.7157, or another way to think about it is, if the heights are truly normally distributed, 71.57% of the students would have a height less than Darnell's. But the answer to this question, what proportion of students' heights are lower than Darnell's height? Well, that would be 0.7157."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "What proportion of laptop prices are between $624 and $768? So let's think about what they are asking. So we have a normal distribution for the prices. So it would look something like this. And this is just my hand-drawn sketch of a normal distribution. So it would look something like this. It should be symmetric."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So it would look something like this. And this is just my hand-drawn sketch of a normal distribution. So it would look something like this. It should be symmetric. So I'm making it as symmetric as I can hand-draw it. And we have the mean right in the center. So the mean would be right there."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "It should be symmetric. So I'm making it as symmetric as I can hand-draw it. And we have the mean right in the center. So the mean would be right there. And that is $750. They also tell us that we have a standard deviation of $60. So that means one standard deviation above the mean would be roughly right over here."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So the mean would be right there. And that is $750. They also tell us that we have a standard deviation of $60. So that means one standard deviation above the mean would be roughly right over here. And that'd be 750 plus 60. So that would be $810. One standard deviation below the mean would put us right about there."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So that means one standard deviation above the mean would be roughly right over here. And that'd be 750 plus 60. So that would be $810. One standard deviation below the mean would put us right about there. And that would be 750 minus $60, which would be $690. And then they tell us what proportion of laptop prices are between $624 and $768? So the lower bound, $624, that's going to actually be more than another standard deviation less."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "One standard deviation below the mean would put us right about there. And that would be 750 minus $60, which would be $690. And then they tell us what proportion of laptop prices are between $624 and $768? So the lower bound, $624, that's going to actually be more than another standard deviation less. So that's going to be right around here. So that is $624. And $768 would put us right at about there."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So the lower bound, $624, that's going to actually be more than another standard deviation less. So that's going to be right around here. So that is $624. And $768 would put us right at about there. And once again, this is just a hand-drawn sketch. But that is 768. And so what proportion are between those two values?"}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "And $768 would put us right at about there. And once again, this is just a hand-drawn sketch. But that is 768. And so what proportion are between those two values? We want to find essentially the area under this distribution between these two values. The way we are going to approach it, we're going to figure out the z-score for 768. It's going to be positive because it's above the mean."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "And so what proportion are between those two values? We want to find essentially the area under this distribution between these two values. The way we are going to approach it, we're going to figure out the z-score for 768. It's going to be positive because it's above the mean. And then we're going to use a z-table to figure out what proportion is below 768. So essentially we're going to figure out this entire area. We're even gonna figure out the stuff that's below 624."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "It's going to be positive because it's above the mean. And then we're going to use a z-table to figure out what proportion is below 768. So essentially we're going to figure out this entire area. We're even gonna figure out the stuff that's below 624. That's what that z-table will give us. And then we'll figure out the z-score for 624. That will be negative two point something."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "We're even gonna figure out the stuff that's below 624. That's what that z-table will give us. And then we'll figure out the z-score for 624. That will be negative two point something. And we will use a z-table again to figure out the proportion that is less than that. And so then we can subtract this red area from the proportion that is less than 768 to get this area in between. So let's do that."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "That will be negative two point something. And we will use a z-table again to figure out the proportion that is less than that. And so then we can subtract this red area from the proportion that is less than 768 to get this area in between. So let's do that. Let's figure out first the z-score for 768. And then we'll do it for 624. The z-score for 768, I'll write it like that, is going to be 768 minus 750 over the standard deviation, over 60."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So let's do that. Let's figure out first the z-score for 768. And then we'll do it for 624. The z-score for 768, I'll write it like that, is going to be 768 minus 750 over the standard deviation, over 60. So this is going to be equal to 18 over 60, which is the same thing as six over, let's see if we divide the numerator and denominator by three, 6 20ths, and this is the same thing as 0.30. So that is the z-score for this upper bound. Let's figure out what proportion is less than that."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "The z-score for 768, I'll write it like that, is going to be 768 minus 750 over the standard deviation, over 60. So this is going to be equal to 18 over 60, which is the same thing as six over, let's see if we divide the numerator and denominator by three, 6 20ths, and this is the same thing as 0.30. So that is the z-score for this upper bound. Let's figure out what proportion is less than that. For that, we take out a z-table. Get our z-table. And let's see, we want to get 0.30."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "Let's figure out what proportion is less than that. For that, we take out a z-table. Get our z-table. And let's see, we want to get 0.30. So this is zero. And so this is 0.30, this first column, and we've done this in other videos, this goes up until the 10th place for our z-score, and then if we want to get to our 100th place, that's what these other columns give us. But we're at 0.30, so we're going to be in this row, and our 100th place is right over here, it's a zero."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "And let's see, we want to get 0.30. So this is zero. And so this is 0.30, this first column, and we've done this in other videos, this goes up until the 10th place for our z-score, and then if we want to get to our 100th place, that's what these other columns give us. But we're at 0.30, so we're going to be in this row, and our 100th place is right over here, it's a zero. So this is the proportion that is less than $768. So 0.6179. So 0.6179."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "But we're at 0.30, so we're going to be in this row, and our 100th place is right over here, it's a zero. So this is the proportion that is less than $768. So 0.6179. So 0.6179. So now let's do the same exercise, but do it for the proportion that's below $624. The z-score for 624 is going to be equal to 624 minus the mean of 750, all of that over 60. And so what is that going to be?"}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So 0.6179. So now let's do the same exercise, but do it for the proportion that's below $624. The z-score for 624 is going to be equal to 624 minus the mean of 750, all of that over 60. And so what is that going to be? I'll get my calculator out for this one, don't want to make a careless error. 624 minus 750 is equal to, 624 minus 750 is equal to, and then divide by 60 is equal to negative 2.1. So that lower bound is 2.1 standard deviations below the mean, or you could say it has a z-score of negative 2.1, is equal to negative 2.1."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "And so what is that going to be? I'll get my calculator out for this one, don't want to make a careless error. 624 minus 750 is equal to, 624 minus 750 is equal to, and then divide by 60 is equal to negative 2.1. So that lower bound is 2.1 standard deviations below the mean, or you could say it has a z-score of negative 2.1, is equal to negative 2.1. And so to figure out the proportion that is less than that, this red area right over here, we go back to our z-table. And we'd actually go to the first part of the z-table. So same idea, but this starts at negative, a z-score of negative 3.4, 3.4 standard deviations below the mean."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So that lower bound is 2.1 standard deviations below the mean, or you could say it has a z-score of negative 2.1, is equal to negative 2.1. And so to figure out the proportion that is less than that, this red area right over here, we go back to our z-table. And we'd actually go to the first part of the z-table. So same idea, but this starts at negative, a z-score of negative 3.4, 3.4 standard deviations below the mean. But just like we saw before, this is our zero hundredths, one hundredth, two hundredth, so on and so forth. And we want to go to negative 2.1, we could say negative 2.10, just to be precise. So this is going to get us, let's see, negative 2.1, there we go."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So same idea, but this starts at negative, a z-score of negative 3.4, 3.4 standard deviations below the mean. But just like we saw before, this is our zero hundredths, one hundredth, two hundredth, so on and so forth. And we want to go to negative 2.1, we could say negative 2.10, just to be precise. So this is going to get us, let's see, negative 2.1, there we go. And so we are negative 2.1, and it's negative 2.10, so we have zero hundredths. So we're gonna be right here on our table. So we see the proportion that is less than 624 is.0179, or 0.0179."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So this is going to get us, let's see, negative 2.1, there we go. And so we are negative 2.1, and it's negative 2.10, so we have zero hundredths. So we're gonna be right here on our table. So we see the proportion that is less than 624 is.0179, or 0.0179. So 0.0179. So 0.0179. And so if we want to figure out the proportion that's in between the two, we just subtract this red area from this entire area, the entire proportion that's less than 768, to get what's in between."}, {"video_title": "Standard normal table for proportion between values AP Statistics Khan Academy.mp3", "Sentence": "So we see the proportion that is less than 624 is.0179, or 0.0179. So 0.0179. So 0.0179. And so if we want to figure out the proportion that's in between the two, we just subtract this red area from this entire area, the entire proportion that's less than 768, to get what's in between. 0.6179, once again, I know I keep repeating it, that's this entire area right over here, and we're gonna subtract out what we have in red. So minus 0.0179, so we're gonna subtract this out, to get 0.6. So if we want to give our answer to four decimal places, it would be 0.6000, or another way to think about it is exactly 60% is between 624 and 768."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So you need to shake the hand exactly once of every other person in the room so that you all meet. So my question to you is, if each of these people need to shake the hand of every other person exactly once, how many handshakes are going to occur? The number of handshakes that are going to occur. So like always, pause the video and see if you can make sense of this. All right, I'm assuming you've had a go at it. So one way to think about it is, okay, if you say there's a handshake, well, a handshake has two people are party to a handshake. We're not talking about some new three-person handshake or four-person handshake."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So like always, pause the video and see if you can make sense of this. All right, I'm assuming you've had a go at it. So one way to think about it is, okay, if you say there's a handshake, well, a handshake has two people are party to a handshake. We're not talking about some new three-person handshake or four-person handshake. We're just talking about the traditional two people shake their right hands. And so there's one person, and there's another person that's party to it. And so you say, okay, there's four possibilities of one party."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "We're not talking about some new three-person handshake or four-person handshake. We're just talking about the traditional two people shake their right hands. And so there's one person, and there's another person that's party to it. And so you say, okay, there's four possibilities of one party. And if we assume people aren't shaking their own hands, which we are assuming, or they're always going to shake someone else's hand, then for the other party, there's only three other, for each of these four possibilities, who's this party, there's three possibilities, who's the other party. And so you might say that there's four times three handshakes. Since there's four times three, I guess you could say possible handshakes."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And so you say, okay, there's four possibilities of one party. And if we assume people aren't shaking their own hands, which we are assuming, or they're always going to shake someone else's hand, then for the other party, there's only three other, for each of these four possibilities, who's this party, there's three possibilities, who's the other party. And so you might say that there's four times three handshakes. Since there's four times three, I guess you could say possible handshakes. And what I'd like you to do is think a little bit about whether this is right, whether there would actually be 12 handshakes. Well, you might have thought about it, and you might say, well, you know, this four times three, this would count, this is actually counting the permutations. This is counting how many ways can you permute four people into two buckets, the two buckets of handshakers, where you care about which bucket they are in, whether they're handshaker number one or handshaker number two."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Since there's four times three, I guess you could say possible handshakes. And what I'd like you to do is think a little bit about whether this is right, whether there would actually be 12 handshakes. Well, you might have thought about it, and you might say, well, you know, this four times three, this would count, this is actually counting the permutations. This is counting how many ways can you permute four people into two buckets, the two buckets of handshakers, where you care about which bucket they are in, whether they're handshaker number one or handshaker number two. This would count, this would count A being the number one handshaker and B being the number two handshaker as being different than B being the number one handshaker and A being the number two handshaker. But we don't want both of these things to occur. We don't want A to shake B's hand where A is facing north and B is facing south, and then another time, they need to shake hands again where now B is facing north and A is facing south."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "This is counting how many ways can you permute four people into two buckets, the two buckets of handshakers, where you care about which bucket they are in, whether they're handshaker number one or handshaker number two. This would count, this would count A being the number one handshaker and B being the number two handshaker as being different than B being the number one handshaker and A being the number two handshaker. But we don't want both of these things to occur. We don't want A to shake B's hand where A is facing north and B is facing south, and then another time, they need to shake hands again where now B is facing north and A is facing south. We only have to do it once. These are actually the same thing, so no reason for both of these to occur. So we are going to be double counting."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "We don't want A to shake B's hand where A is facing north and B is facing south, and then another time, they need to shake hands again where now B is facing north and A is facing south. We only have to do it once. These are actually the same thing, so no reason for both of these to occur. So we are going to be double counting. So what we really want to do is think about combinations. One way to think about it is you have four people. In a world of four people or a pool of four people, how many ways can you choose two?"}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So we are going to be double counting. So what we really want to do is think about combinations. One way to think about it is you have four people. In a world of four people or a pool of four people, how many ways can you choose two? How many ways to choose two? Because that's what we're doing. Each handshake is just really a selection of two of these people."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "In a world of four people or a pool of four people, how many ways can you choose two? How many ways to choose two? Because that's what we're doing. Each handshake is just really a selection of two of these people. And so we want to say how many ways can we select two people so that we have a different, each combination, each of these ways to select two people should have a different combination of people in it. If two of them had the same, if A, B, and B, A, these are the same combination. And so this is really a combinations problem."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Each handshake is just really a selection of two of these people. And so we want to say how many ways can we select two people so that we have a different, each combination, each of these ways to select two people should have a different combination of people in it. If two of them had the same, if A, B, and B, A, these are the same combination. And so this is really a combinations problem. This is really equivalent to saying how many ways are there to choose two people from a pool of four, or four choose two, or four choose two. And so this is going to be, well, how many ways are there to permute four people into three spots, which is going to be four times three, which we just figured out right over there, which is 12. Actually, why don't I do it in that green color so you see where that came from."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And so this is really a combinations problem. This is really equivalent to saying how many ways are there to choose two people from a pool of four, or four choose two, or four choose two. And so this is going to be, well, how many ways are there to permute four people into three spots, which is going to be four times three, which we just figured out right over there, which is 12. Actually, why don't I do it in that green color so you see where that came from. So four times three, and then you're going to divide that by the number of ways you can arrange two people. Well, you can arrange two people in two different ways. One's on the left, one's on the right, or the other one's on the left, and the other one's on the right."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Actually, why don't I do it in that green color so you see where that came from. So four times three, and then you're going to divide that by the number of ways you can arrange two people. Well, you can arrange two people in two different ways. One's on the left, one's on the right, or the other one's on the left, and the other one's on the right. Or you could also view that as two factorial, which is also equal to two. So we could write this down as two. So this is the number of ways to arrange two people."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "One's on the left, one's on the right, or the other one's on the left, and the other one's on the right. Or you could also view that as two factorial, which is also equal to two. So we could write this down as two. So this is the number of ways to arrange two people. To arrange two people. Two people, two people. And this up here, let me just do this in a new color."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So this is the number of ways to arrange two people. To arrange two people. Two people, two people. And this up here, let me just do this in a new color. This up here, that's the permutations. That's the way, number of permutations if you take two people from a pool of four. So here you would care about order."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And this up here, let me just do this in a new color. This up here, that's the permutations. That's the way, number of permutations if you take two people from a pool of four. So here you would care about order. And so, one way to think about it, this two is correcting for this double counting here. And if you wanted to apply the formula, you could. I just kind of reasoned through it again."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So here you would care about order. And so, one way to think about it, this two is correcting for this double counting here. And if you wanted to apply the formula, you could. I just kind of reasoned through it again. I mean, you could literally say, okay, four times three is 12, we're double counting, because there's two ways to arrange two people, so you just divided it by two. So you just divide by two. And then you are going to be left with six."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "I just kind of reasoned through it again. I mean, you could literally say, okay, four times three is 12, we're double counting, because there's two ways to arrange two people, so you just divided it by two. So you just divide by two. And then you are going to be left with six. You could think of it in terms of this. Or you could just apply the formula. You could just say, hey look, four choose two, or the number of combinations of selecting two from a group of four."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And then you are going to be left with six. You could think of it in terms of this. Or you could just apply the formula. You could just say, hey look, four choose two, or the number of combinations of selecting two from a group of four. This is going to be four factorial over two factorial times four minus two. Four minus two factorial. And I'm going to make this color different just so you can keep track of how I'm at least applying this."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "You could just say, hey look, four choose two, or the number of combinations of selecting two from a group of four. This is going to be four factorial over two factorial times four minus two. Four minus two factorial. And I'm going to make this color different just so you can keep track of how I'm at least applying this. And so what is this going to be? This is going to be four times three times two times one over two times one times this right over here is two times one. So that will cancel with that."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And I'm going to make this color different just so you can keep track of how I'm at least applying this. And so what is this going to be? This is going to be four times three times two times one over two times one times this right over here is two times one. So that will cancel with that. Four divided by two is two. Two times three divided by one is equal to six. And to just really hit the point home, let's actually draw it out."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So that will cancel with that. Four divided by two is two. Two times three divided by one is equal to six. And to just really hit the point home, let's actually draw it out. Let's draw it out. So A could shake B's hand. A could shake C's hand."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And to just really hit the point home, let's actually draw it out. Let's draw it out. So A could shake B's hand. A could shake C's hand. A could shake D's hand. Or B could shake, and let me just do what we calculated first, the 12 and B could shake A's hand. B could shake C's hand."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "A could shake C's hand. A could shake D's hand. Or B could shake, and let me just do what we calculated first, the 12 and B could shake A's hand. B could shake C's hand. B could shake D's hand. And C could shake A's hand. C could shake B's hand."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "B could shake C's hand. B could shake D's hand. And C could shake A's hand. C could shake B's hand. C could shake D's hand. We could say D could shake A's hand, D could shake B's hand, D could shake C's hand. And this is 12 right over here."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "C could shake B's hand. C could shake D's hand. We could say D could shake A's hand, D could shake B's hand, D could shake C's hand. And this is 12 right over here. And this is a permutations. If D shaking C's hand was actually different than C shaking D's hand, then we would count 12. But we just wanted to say, well, how many ways, they just have to meet each other once."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And this is 12 right over here. And this is a permutations. If D shaking C's hand was actually different than C shaking D's hand, then we would count 12. But we just wanted to say, well, how many ways, they just have to meet each other once. And so we're double counting. So, so A, B is the same thing as B, A. A, C is the same thing as C, A."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "But we just wanted to say, well, how many ways, they just have to meet each other once. And so we're double counting. So, so A, B is the same thing as B, A. A, C is the same thing as C, A. A, D is the same thing as D, A. B, C is the same thing as C, B. B, D is the same thing as D, B."}, {"video_title": "Handshaking combinations Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "A, C is the same thing as C, A. A, D is the same thing as D, A. B, C is the same thing as C, B. B, D is the same thing as D, B. C, D is the same thing as D, C. And so we'd be left with, if we correct for the double counting, we're left with one, two, three, four, five, six combinations. Six possible ways of choosing two from a pool of four. Especially when you don't care about the order in which you choose them."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "What is the maximum average wait time for restaurants where Amelia likes to use the drive-through? Round to the nearest whole second. Like always, if you feel like you can tackle this, pause this video and try to do so. I'm assuming you paused it. Now let's work through this together. So let's think about what's going on. They're telling us that the distribution of average wait times is approximately normal."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "I'm assuming you paused it. Now let's work through this together. So let's think about what's going on. They're telling us that the distribution of average wait times is approximately normal. So let's get a visualization of a normal distribution. And they tell us several things about this normal distribution. They tell us that the mean is 185 seconds."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "They're telling us that the distribution of average wait times is approximately normal. So let's get a visualization of a normal distribution. And they tell us several things about this normal distribution. They tell us that the mean is 185 seconds. So that's 185 there. The standard deviation is 11 seconds. So for example, this is going to be 11 more than the mean."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "They tell us that the mean is 185 seconds. So that's 185 there. The standard deviation is 11 seconds. So for example, this is going to be 11 more than the mean. So this would be 196 seconds. This would be another 11. Each of these dotted lines are one standard deviation more."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So for example, this is going to be 11 more than the mean. So this would be 196 seconds. This would be another 11. Each of these dotted lines are one standard deviation more. So this would be 207. This would be 11 seconds less than the mean. So this would be 174, and so on and so forth."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Each of these dotted lines are one standard deviation more. So this would be 207. This would be 11 seconds less than the mean. So this would be 174, and so on and so forth. And we wanna find the maximum average wait time for restaurants where Amelia likes to use the drive-through. Well, what are those restaurants? That's where the average wait time is in the bottom 10% for that town."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So this would be 174, and so on and so forth. And we wanna find the maximum average wait time for restaurants where Amelia likes to use the drive-through. Well, what are those restaurants? That's where the average wait time is in the bottom 10% for that town. So how do we think about it? Well, there's going to be some value. Let me mark it off right over here in this red color."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "That's where the average wait time is in the bottom 10% for that town. So how do we think about it? Well, there's going to be some value. Let me mark it off right over here in this red color. So we're gonna have some threshold value right over here where this is anything that level or lower is going to be in the bottom 10%. Well, another way to think about it is this is the largest wait time for which you are still in the bottom 10%. And so this area right over here is going to be 10% of the total, or it's going to be 0.10."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Let me mark it off right over here in this red color. So we're gonna have some threshold value right over here where this is anything that level or lower is going to be in the bottom 10%. Well, another way to think about it is this is the largest wait time for which you are still in the bottom 10%. And so this area right over here is going to be 10% of the total, or it's going to be 0.10. So the way we can tackle this is we can get up a z-table and figure out what z-score gives us a proportion of only 0.10 being less than that z-score. And then using that z-score, we can figure out this value, the actual wait time. So let's get our z-table out."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And so this area right over here is going to be 10% of the total, or it's going to be 0.10. So the way we can tackle this is we can get up a z-table and figure out what z-score gives us a proportion of only 0.10 being less than that z-score. And then using that z-score, we can figure out this value, the actual wait time. So let's get our z-table out. And since we know that this is below the mean, the mean would be the 50th percentile, we know we're gonna have a negative z-score. So I'm gonna take out the part of the table that has the negative z-scores on it. And remember, we're looking for 10%, but we don't wanna go beyond 10%."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So let's get our z-table out. And since we know that this is below the mean, the mean would be the 50th percentile, we know we're gonna have a negative z-score. So I'm gonna take out the part of the table that has the negative z-scores on it. And remember, we're looking for 10%, but we don't wanna go beyond 10%. We wanna be sure that that value is within the 10th percentile, that any higher will be out of the 10th percentile. So let's see, when we have these really negative z's, so far it only doesn't even get to the first percentile yet. So let's scroll down a little bit."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And remember, we're looking for 10%, but we don't wanna go beyond 10%. We wanna be sure that that value is within the 10th percentile, that any higher will be out of the 10th percentile. So let's see, when we have these really negative z's, so far it only doesn't even get to the first percentile yet. So let's scroll down a little bit. And let's remember as we do so that this is zero in the hundredths place, one, two, three, four, five, six, seven, eight, nine. So let's remember those columns. So let's see, if we are at a z-score of negative 1.28, remember, this is, the hundredths is zero, one, two, three, four, five, six, seven, eight."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So let's scroll down a little bit. And let's remember as we do so that this is zero in the hundredths place, one, two, three, four, five, six, seven, eight, nine. So let's remember those columns. So let's see, if we are at a z-score of negative 1.28, remember, this is, the hundredths is zero, one, two, three, four, five, six, seven, eight. So this right over here is a z-score of negative 1.28. And that's a little bit crossing the 10th percentile. But if we get a little bit more negative than that, we are in the 10th percentile."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So let's see, if we are at a z-score of negative 1.28, remember, this is, the hundredths is zero, one, two, three, four, five, six, seven, eight. So this right over here is a z-score of negative 1.28. And that's a little bit crossing the 10th percentile. But if we get a little bit more negative than that, we are in the 10th percentile. So this is negative 1.29. And this does seem to be the highest z-score for which we are within the 10th percentile. So negative 1.29 is our z-score."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "But if we get a little bit more negative than that, we are in the 10th percentile. So this is negative 1.29. And this does seem to be the highest z-score for which we are within the 10th percentile. So negative 1.29 is our z-score. So this is z equals negative 1.29. And if we wanna figure out the actual value for that, we would start with the mean, which is 185. And then we would say, well, we wanna go 1.29 standard deviations below the mean."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So negative 1.29 is our z-score. So this is z equals negative 1.29. And if we wanna figure out the actual value for that, we would start with the mean, which is 185. And then we would say, well, we wanna go 1.29 standard deviations below the mean. The negative says we're going below the mean. So we could say minus 1.29 times the standard deviation. And they tell us up here the standard deviation is 11 seconds."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And then we would say, well, we wanna go 1.29 standard deviations below the mean. The negative says we're going below the mean. So we could say minus 1.29 times the standard deviation. And they tell us up here the standard deviation is 11 seconds. So it's going to be 1.29 times 11. And this is going to be equal to 1.29 times 11 is equal to 14.19. And then I'll make that negative and then add that to 185."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And they tell us up here the standard deviation is 11 seconds. So it's going to be 1.29 times 11. And this is going to be equal to 1.29 times 11 is equal to 14.19. And then I'll make that negative and then add that to 185. Plus 185 is equal to 170.81. 170.81. Now they say round to the nearest whole second."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "And then I'll make that negative and then add that to 185. Plus 185 is equal to 170.81. 170.81. Now they say round to the nearest whole second. There's a couple of ways to think about it. If you really wanna ensure that you're not gonna cross the 10th percentile, you might wanna round to the nearest second that is below this threshold. So you might say that this is approximately 170 seconds."}, {"video_title": "Threshold for low percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Now they say round to the nearest whole second. There's a couple of ways to think about it. If you really wanna ensure that you're not gonna cross the 10th percentile, you might wanna round to the nearest second that is below this threshold. So you might say that this is approximately 170 seconds. If you were to just round normally, this would go to 171. But just by doing that, you might have crossed the threshold. But in all likelihood, for this application where someone is concerned about wait time at drive-thru restaurants, that difference in a second between 170 and 171 is not going to be mission critical, so to speak."}, {"video_title": "Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So let's just review factorial a little bit. So if I were to say n factorial, that of course is going to be n times n minus, sorry, times n minus one times n minus two, n minus two, and I would just keep going down until I go to times one. So I would keep decrementing n until I get to one, and then I would multiply all of those things together. So for example, in all of this is review, if I were to say three factorial, that's going to be three times two times one. If I were to say two factorial, that's going to be two times one. One factorial, by that logic, well, I just keep decrementing until I get to one, but I don't even have to decrement here. I'm already at one, so I just multiply it one."}, {"video_title": "Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So for example, in all of this is review, if I were to say three factorial, that's going to be three times two times one. If I were to say two factorial, that's going to be two times one. One factorial, by that logic, well, I just keep decrementing until I get to one, but I don't even have to decrement here. I'm already at one, so I just multiply it one. Now what about zero factorial? This is interesting. Zero factorial."}, {"video_title": "Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "I'm already at one, so I just multiply it one. Now what about zero factorial? This is interesting. Zero factorial. So one logical thing to say, well, hey, maybe zero factorial is zero. I'm just starting with itself. It's already below one."}, {"video_title": "Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Zero factorial. So one logical thing to say, well, hey, maybe zero factorial is zero. I'm just starting with itself. It's already below one. Maybe it is zero. Now what we will see is that this is not the case, that mathematicians have decided. And remember, this is what's interesting."}, {"video_title": "Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "It's already below one. Maybe it is zero. Now what we will see is that this is not the case, that mathematicians have decided. And remember, this is what's interesting. The factorial operation, this is something that humans have invented, that they think is just an interesting thing. It's a useful notation. So they can define what it does."}, {"video_title": "Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And remember, this is what's interesting. The factorial operation, this is something that humans have invented, that they think is just an interesting thing. It's a useful notation. So they can define what it does. And mathematicians have found it far more useful to define zero factorial as something else. To define zero factorial as, and there's a little bit of a drum roll here, they believe zero factorial should be one. And I know, based on the reasoning, the conceptual reasoning of this, this doesn't make any sense."}, {"video_title": "Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "So they can define what it does. And mathematicians have found it far more useful to define zero factorial as something else. To define zero factorial as, and there's a little bit of a drum roll here, they believe zero factorial should be one. And I know, based on the reasoning, the conceptual reasoning of this, this doesn't make any sense. But since we've already been exposed a little bit to permutations, I'll show you why this is a useful concept, especially in the world of permutations and combinations, which is, frankly, where factorial shows up the most. I would say that most of the cases that I've ever seen, factorial in anything, has been in the situations of permutations and combinations. And in a few other things, but mainly permutations and combinations."}, {"video_title": "Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And I know, based on the reasoning, the conceptual reasoning of this, this doesn't make any sense. But since we've already been exposed a little bit to permutations, I'll show you why this is a useful concept, especially in the world of permutations and combinations, which is, frankly, where factorial shows up the most. I would say that most of the cases that I've ever seen, factorial in anything, has been in the situations of permutations and combinations. And in a few other things, but mainly permutations and combinations. So let's review a little bit. We've said that, hey, you know, if we have n things and we want to figure out the number of ways to permute them into k spaces, it's going to be, it's going to be n factorial over n minus, over n minus k factorial. Now we've also said that if we had, if we had n things that we want to permute into n places, well this really should just be n factorial."}, {"video_title": "Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And in a few other things, but mainly permutations and combinations. So let's review a little bit. We've said that, hey, you know, if we have n things and we want to figure out the number of ways to permute them into k spaces, it's going to be, it's going to be n factorial over n minus, over n minus k factorial. Now we've also said that if we had, if we had n things that we want to permute into n places, well this really should just be n factorial. The first place has, you know, let's just do this. So this is the first place, this is the second place, this is the third place, all the way you get to the nth place. Well, there would be n possibilities for who's in the first position or which object is in the first position."}, {"video_title": "Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Now we've also said that if we had, if we had n things that we want to permute into n places, well this really should just be n factorial. The first place has, you know, let's just do this. So this is the first place, this is the second place, this is the third place, all the way you get to the nth place. Well, there would be n possibilities for who's in the first position or which object is in the first position. And then for each of those possibilities, there would be n minus one possibilities for which object you choose to put in the second position because you've already put one into that position. Now for each of these n times n minus one possibilities where you've placed two things, there would be n minus two possibilities of what goes in the third position and then you would just go all the way down to one. And this thing right over here is exactly what we wrote over here."}, {"video_title": "Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "Well, there would be n possibilities for who's in the first position or which object is in the first position. And then for each of those possibilities, there would be n minus one possibilities for which object you choose to put in the second position because you've already put one into that position. Now for each of these n times n minus one possibilities where you've placed two things, there would be n minus two possibilities of what goes in the third position and then you would just go all the way down to one. And this thing right over here is exactly what we wrote over here. This is equal to, this is equal to n factorial. But if we directly applied this formula, if we applied this formula, this would need to be n factorial over n minus n factorial. And then you might see why this is interesting because this is going to be n factorial over zero factorial."}, {"video_title": "Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And this thing right over here is exactly what we wrote over here. This is equal to, this is equal to n factorial. But if we directly applied this formula, if we applied this formula, this would need to be n factorial over n minus n factorial. And then you might see why this is interesting because this is going to be n factorial over zero factorial. So in order for this formula to apply for even in the case where k is equal to n, even in the case where k is equal to n, which is this one right over here, and for that to be consistent with just plain old logic, zero factorial needs to be equal to one. And so the mathematics community has decided, hey, this thing we've constructed called factorial, you know, we've said, hey, you put an exclamation mark behind something. In all of our heads, we say you kind of count down that number all the way to one and you keep multiplying them."}, {"video_title": "Zero factorial or 0! Probability and combinatorics Probability and Statistics Khan Academy.mp3", "Sentence": "And then you might see why this is interesting because this is going to be n factorial over zero factorial. So in order for this formula to apply for even in the case where k is equal to n, even in the case where k is equal to n, which is this one right over here, and for that to be consistent with just plain old logic, zero factorial needs to be equal to one. And so the mathematics community has decided, hey, this thing we've constructed called factorial, you know, we've said, hey, you put an exclamation mark behind something. In all of our heads, we say you kind of count down that number all the way to one and you keep multiplying them. For zero, we're just gonna define this. We're just gonna define, make a mathematical definition. We're just gonna say zero factorial is equal to one."}, {"video_title": "Probability distributions from empirical data Probability & combinatorics.mp3", "Sentence": "She decides to offer a discount on appetizers to attract more customers, and she's curious about the probability that a customer orders a large number of appetizers. Jada tracked how many appetizers were in each of the past 500 orders. All right, so the number of appetizers, so 40 out of the 500 ordered zero appetizers, and for example, 120 out of the 500 ordered three appetizers and so on and so forth. Let X represent the number of appetizers in a random order. Based on these results, construct an approximate probability distribution of X. Pause this video and see if you can have a go at this before we do this together. All right, so they're telling us an approximate probability distribution because we don't know the actual probability."}, {"video_title": "Probability distributions from empirical data Probability & combinatorics.mp3", "Sentence": "Let X represent the number of appetizers in a random order. Based on these results, construct an approximate probability distribution of X. Pause this video and see if you can have a go at this before we do this together. All right, so they're telling us an approximate probability distribution because we don't know the actual probability. We can't get into people's minds and figure out the probability that their neurons fire in exactly the right way to order appetizers, but what we can do is look at past results, empirical data right over here to approximate the distribution. So what we can do is look at the last 500, and for each of the outcomes, think about what fraction of the last 500 had that outcome, and that will be our approximation. And so the outcomes here, we could have zero appetizers, one, two, three, four, five, or six."}, {"video_title": "Probability distributions from empirical data Probability & combinatorics.mp3", "Sentence": "All right, so they're telling us an approximate probability distribution because we don't know the actual probability. We can't get into people's minds and figure out the probability that their neurons fire in exactly the right way to order appetizers, but what we can do is look at past results, empirical data right over here to approximate the distribution. So what we can do is look at the last 500, and for each of the outcomes, think about what fraction of the last 500 had that outcome, and that will be our approximation. And so the outcomes here, we could have zero appetizers, one, two, three, four, five, or six. Now, the approximate probability of zero appetizers is going to be 40 over 500, which is the same thing as four over 50, which is the same thing as two over 25. So I'll write 2 25ths right over there. The probability of one appetizer, well, that's going to be 90 over 500, which is the same thing as nine over 50."}, {"video_title": "Probability distributions from empirical data Probability & combinatorics.mp3", "Sentence": "And so the outcomes here, we could have zero appetizers, one, two, three, four, five, or six. Now, the approximate probability of zero appetizers is going to be 40 over 500, which is the same thing as four over 50, which is the same thing as two over 25. So I'll write 2 25ths right over there. The probability of one appetizer, well, that's going to be 90 over 500, which is the same thing as nine over 50. I think that's already in lowest terms. Then 160 over 500 is the same thing as 16 over 50, which is the same thing as eight over 25. And we just keep going."}, {"video_title": "Probability distributions from empirical data Probability & combinatorics.mp3", "Sentence": "The probability of one appetizer, well, that's going to be 90 over 500, which is the same thing as nine over 50. I think that's already in lowest terms. Then 160 over 500 is the same thing as 16 over 50, which is the same thing as eight over 25. And we just keep going. 120 out of 500 is the same thing as 12 out of 50 or six out of 25, six out of 25. And then 50 out of 500, well, that's one out of every 10, so I'll just write it like that. 30 out of 500 is the same thing as three out of 50."}, {"video_title": "Probability distributions from empirical data Probability & combinatorics.mp3", "Sentence": "And we just keep going. 120 out of 500 is the same thing as 12 out of 50 or six out of 25, six out of 25. And then 50 out of 500, well, that's one out of every 10, so I'll just write it like that. 30 out of 500 is the same thing as three out of 50. So I'll just write it like that. And then last but not least, 10 out of 500 is the same thing as one in 50. And we're done."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "So you have all of the seniors, I'm assuming there's more than 100 of them, and then they sampled 100 of them. So this is the sample. So the population is all of the seniors at the school. That's the population, all of the seniors. And they sampled 100 of them. So the 100 seniors that they talked to, that is the sample. That is the sample."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "That's the population, all of the seniors. And they sampled 100 of them. So the 100 seniors that they talked to, that is the sample. That is the sample. So they tell us, identify the population and the sample in the setting. So let's just see which of these choices actually match up to what I just said. And like always, I encourage you to pause the video and see if you can work through it on your own."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "That is the sample. So they tell us, identify the population and the sample in the setting. So let's just see which of these choices actually match up to what I just said. And like always, I encourage you to pause the video and see if you can work through it on your own. So the population is all high school seniors in the world. The sample is all of the seniors at Riverview High. No, this is not right."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "And like always, I encourage you to pause the video and see if you can work through it on your own. So the population is all high school seniors in the world. The sample is all of the seniors at Riverview High. No, this is not right. We're not trying to figure out, we're not trying to get an indication of how all of the high school seniors in the world feel about the food at Riverview High School. We're trying to get an indication of how the seniors at Riverview High School feel about the lunch at the school's cafeteria. So they did a sample of 100 of them."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "No, this is not right. We're not trying to figure out, we're not trying to get an indication of how all of the high school seniors in the world feel about the food at Riverview High School. We're trying to get an indication of how the seniors at Riverview High School feel about the lunch at the school's cafeteria. So they did a sample of 100 of them. So this is definitely not going to be, let me cross this one out. The population is all students at Riverview High. The sample is all of the seniors at Riverview High."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "So they did a sample of 100 of them. So this is definitely not going to be, let me cross this one out. The population is all students at Riverview High. The sample is all of the seniors at Riverview High. Well, they clearly didn't sample all of the seniors. They sample 100 of the seniors. So this isn't gonna be right either."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "The sample is all of the seniors at Riverview High. Well, they clearly didn't sample all of the seniors. They sample 100 of the seniors. So this isn't gonna be right either. Let's hope that the third choice is working out. The population is all seniors at Riverview High. The sample is the 100 seniors surveyed."}, {"video_title": "Identifying a sample and population Study design AP Statistics Khan Academy.mp3", "Sentence": "So this isn't gonna be right either. Let's hope that the third choice is working out. The population is all seniors at Riverview High. The sample is the 100 seniors surveyed. Yep, that's exactly what we talked here. We're trying to get an indication about how all of the seniors at Riverview High feel about the food, the lunch offerings. We probably think it's impractical, or the administrators feel it's impractical to talk to everyone so they get exactly what the population thinks."}, {"video_title": "Can causality be established from this study Study design AP Statistics Khan Academy.mp3", "Sentence": "A gym that specializes in weight loss offers its members an optional dietary program for an extra fee. To study the effectiveness of the dietary program, a manager at the gym takes a random sample of 50 members who participate in the dietary program and 50 members who don't. The manager finds that those who participated in the dietary program, on average, lost significantly more weight than those who didn't participate in the program in the past three months. The manager concludes that the dietary program caused the increased weight loss for the gym's members during that time period. So pause this video and think about whether you think the manager is making a valid conclusion based on this study. All right, so let's first just make sure we understand what the manager is concluding. The manager is saying that there's a causal relationship between the dietary program, that being on the dietary program causes increased weight, weight loss."}, {"video_title": "Can causality be established from this study Study design AP Statistics Khan Academy.mp3", "Sentence": "The manager concludes that the dietary program caused the increased weight loss for the gym's members during that time period. So pause this video and think about whether you think the manager is making a valid conclusion based on this study. All right, so let's first just make sure we understand what the manager is concluding. The manager is saying that there's a causal relationship between the dietary program, that being on the dietary program causes increased weight, weight loss. And they would be able to make this conclusion if it's a well-designed experimental study. And you might say, well, what does an experimental study look like? Well, in an experimental study, you have a control group that wouldn't have the dietary program, and then you would have a treatment or experimental group that does have the dietary program."}, {"video_title": "Can causality be established from this study Study design AP Statistics Khan Academy.mp3", "Sentence": "The manager is saying that there's a causal relationship between the dietary program, that being on the dietary program causes increased weight, weight loss. And they would be able to make this conclusion if it's a well-designed experimental study. And you might say, well, what does an experimental study look like? Well, in an experimental study, you have a control group that wouldn't have the dietary program, and then you would have a treatment or experimental group that does have the dietary program. And then you would see, hey, if these folks are actually seeing more weight loss than if it's statistically significant, then maybe you can conclude the causal relationship that the dietary program is causing the weight loss. And this kind of looks like that until you think about how folks were assigned to either of these groups. It would be a well-designed experimental study if you took random samples from the broader population."}, {"video_title": "Can causality be established from this study Study design AP Statistics Khan Academy.mp3", "Sentence": "Well, in an experimental study, you have a control group that wouldn't have the dietary program, and then you would have a treatment or experimental group that does have the dietary program. And then you would see, hey, if these folks are actually seeing more weight loss than if it's statistically significant, then maybe you can conclude the causal relationship that the dietary program is causing the weight loss. And this kind of looks like that until you think about how folks were assigned to either of these groups. It would be a well-designed experimental study if you took random samples from the broader population. So you took a random sample. So that's a random sample from the broader population. And then from this group, you randomly assigned folks to either of the control or the treatment."}, {"video_title": "Can causality be established from this study Study design AP Statistics Khan Academy.mp3", "Sentence": "It would be a well-designed experimental study if you took random samples from the broader population. So you took a random sample. So that's a random sample from the broader population. And then from this group, you randomly assigned folks to either of the control or the treatment. So you're randomly assigning. And then the people who happened to be in the treatment group, not the people who chose to be in it, well, those people, you'd say you need to be on this diet, and you make sure that they are on that diet. And then if you see a statistically significant increased weight loss, then you might be able to make this conclusion."}, {"video_title": "Can causality be established from this study Study design AP Statistics Khan Academy.mp3", "Sentence": "And then from this group, you randomly assigned folks to either of the control or the treatment. So you're randomly assigning. And then the people who happened to be in the treatment group, not the people who chose to be in it, well, those people, you'd say you need to be on this diet, and you make sure that they are on that diet. And then if you see a statistically significant increased weight loss, then you might be able to make this conclusion. But that's not what happened over here. In this situation, in this situation, we didn't randomly select from the broader population and then randomly assign folks to these groups. What happened is people self-selected into either being in the dietary program or not being in the dietary program."}, {"video_title": "Can causality be established from this study Study design AP Statistics Khan Academy.mp3", "Sentence": "And then if you see a statistically significant increased weight loss, then you might be able to make this conclusion. But that's not what happened over here. In this situation, in this situation, we didn't randomly select from the broader population and then randomly assign folks to these groups. What happened is people self-selected into either being in the dietary program or not being in the dietary program. I'll say not the dietary program. And then what they did is is that they randomly sampled from the dietary program to put them in, I guess you could say the treatment group. And then they randomly sampled from this control group."}, {"video_title": "Can causality be established from this study Study design AP Statistics Khan Academy.mp3", "Sentence": "What happened is people self-selected into either being in the dietary program or not being in the dietary program. I'll say not the dietary program. And then what they did is is that they randomly sampled from the dietary program to put them in, I guess you could say the treatment group. And then they randomly sampled from this control group. And so in this situation, you have a significant confounding variable. People essentially self-selected themselves into the dietary program group. And that you could view as a confounding variable."}, {"video_title": "Can causality be established from this study Study design AP Statistics Khan Academy.mp3", "Sentence": "And then they randomly sampled from this control group. And so in this situation, you have a significant confounding variable. People essentially self-selected themselves into the dietary program group. And that you could view as a confounding variable. And so it doesn't allow you to make this causal connection. Because people who selected themselves into the dietary program, maybe they're more motivated to lose weight. Maybe they could afford the money, and that wealth association is associated with being able to eat better and maybe being able to lose more weight."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So let's keep thinking about different ways to sit multiple people in a certain number of chairs. So let's say we have six people. We have person A, we have person B, we have person C, person D, person E, and we have person F. So we have six people, and for the sake of this video, we're going to say, oh, we want to figure out all the scenarios, all the possibilities, all the permutations, all the ways that we could put them into three chairs. So that's chair number one, chair number two, and chair number three. This is all a review, this is covered in the permutations video, but it'll be very instructive as we move into a new concept. So what are all of the permutations of putting six different people into three chairs? Well, like we've seen before, we could start with the first chair, and we could say, look, if we haven't seated anyone yet, how many different people could we put in chair number one?"}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So that's chair number one, chair number two, and chair number three. This is all a review, this is covered in the permutations video, but it'll be very instructive as we move into a new concept. So what are all of the permutations of putting six different people into three chairs? Well, like we've seen before, we could start with the first chair, and we could say, look, if we haven't seated anyone yet, how many different people could we put in chair number one? Well, there's six different people who could be in chair number one. Let me do that in a different color. There are six people who could be in chair number one."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "Well, like we've seen before, we could start with the first chair, and we could say, look, if we haven't seated anyone yet, how many different people could we put in chair number one? Well, there's six different people who could be in chair number one. Let me do that in a different color. There are six people who could be in chair number one. Six different scenarios for who sits in chair number one. Now, for each of those six scenarios, how many people, how many different people could sit in chair number two? Well, each of those six scenarios, we've taken one of the six people to sit in chair number one, so that means you have five out of the six people left to sit in chair number two."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "There are six people who could be in chair number one. Six different scenarios for who sits in chair number one. Now, for each of those six scenarios, how many people, how many different people could sit in chair number two? Well, each of those six scenarios, we've taken one of the six people to sit in chair number one, so that means you have five out of the six people left to sit in chair number two. Or another way to think about it is there's six scenarios of someone in chair number one, and for each of those six, you have five scenarios for who's in chair number two. So you have a total of 30 scenarios where you have seated six people in the first two chairs. And now, if you want to say, well, what about for the three chairs?"}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "Well, each of those six scenarios, we've taken one of the six people to sit in chair number one, so that means you have five out of the six people left to sit in chair number two. Or another way to think about it is there's six scenarios of someone in chair number one, and for each of those six, you have five scenarios for who's in chair number two. So you have a total of 30 scenarios where you have seated six people in the first two chairs. And now, if you want to say, well, what about for the three chairs? Well, for each of these 30 scenarios, how many different people could you put in chair number three? Well, you're still going to have four people standing up, not in chairs. So for each of these 30 scenarios, you have four people who you could put in chair number three."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "And now, if you want to say, well, what about for the three chairs? Well, for each of these 30 scenarios, how many different people could you put in chair number three? Well, you're still going to have four people standing up, not in chairs. So for each of these 30 scenarios, you have four people who you could put in chair number three. So your total number of scenarios, or your total number of permutations, where we care who's sitting in which chair, is six times five times four, which is equal to 120 permutations. Permutations. Now, permutations."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So for each of these 30 scenarios, you have four people who you could put in chair number three. So your total number of scenarios, or your total number of permutations, where we care who's sitting in which chair, is six times five times four, which is equal to 120 permutations. Permutations. Now, permutations. Now, it's worth thinking about what permutations are counting. Now, remember, we care, when we're talking about permutations, we care about who's sitting in which chair. So, for example, for example, this is one permutation, and this would be counted as another permutation."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "Now, permutations. Now, it's worth thinking about what permutations are counting. Now, remember, we care, when we're talking about permutations, we care about who's sitting in which chair. So, for example, for example, this is one permutation, and this would be counted as another permutation. And this would be counted as another permutation. This would be counted as another permutation. So notice, these are all the same three people, but we're putting them in different chairs."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So, for example, for example, this is one permutation, and this would be counted as another permutation. And this would be counted as another permutation. This would be counted as another permutation. So notice, these are all the same three people, but we're putting them in different chairs. And this counted that. That's counted in this 120. I could keep going."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So notice, these are all the same three people, but we're putting them in different chairs. And this counted that. That's counted in this 120. I could keep going. We could have that, or we could have that. So when we're thinking in the permutation world, we would count all of these, or we would count this as six different permutations. These are going towards this 120."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "I could keep going. We could have that, or we could have that. So when we're thinking in the permutation world, we would count all of these, or we would count this as six different permutations. These are going towards this 120. And of course, we have other permutations where we involve other people, where we have, it could be FBCFCBFACFF. Actually, let me do it this way. I'll be a little bit more systematic."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "These are going towards this 120. And of course, we have other permutations where we involve other people, where we have, it could be FBCFCBFACFF. Actually, let me do it this way. I'll be a little bit more systematic. F, let me do it, BBFCBCF, and obviously, I could keep going. I could do 120 of these. I'll do two more."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "I'll be a little bit more systematic. F, let me do it, BBFCBCF, and obviously, I could keep going. I could do 120 of these. I'll do two more. You could have CFB, and then you could have CBF. So in the permutation world, these are literally 12 of the 120 permutations. But what if all we cared about is the three people we're choosing to sit down, but we don't care in what order that they're sitting or in which chair they're sitting?"}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "I'll do two more. You could have CFB, and then you could have CBF. So in the permutation world, these are literally 12 of the 120 permutations. But what if all we cared about is the three people we're choosing to sit down, but we don't care in what order that they're sitting or in which chair they're sitting? So in that world, these would all be one. This is all the same set of three people if we don't care which chair they're sitting in. This would also be the same set of three people."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "But what if all we cared about is the three people we're choosing to sit down, but we don't care in what order that they're sitting or in which chair they're sitting? So in that world, these would all be one. This is all the same set of three people if we don't care which chair they're sitting in. This would also be the same set of three people. And so this question, if I have six people sitting in three chairs, how many ways can I choose three people out of the six where I don't care which chair they sit on? And I encourage you to pause the video and try to think of what that number would actually be. Well, a big clue was, when we essentially wrote all of the permutations where we've picked a group of three people, we see that there's six ways of arranging the three people."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "This would also be the same set of three people. And so this question, if I have six people sitting in three chairs, how many ways can I choose three people out of the six where I don't care which chair they sit on? And I encourage you to pause the video and try to think of what that number would actually be. Well, a big clue was, when we essentially wrote all of the permutations where we've picked a group of three people, we see that there's six ways of arranging the three people. When you pick a certain group of three people, that turned into six permutations. And so if all you wanna do is care about, well, how many different ways are there to choose three from the six, you would take your whole permutations, you would take your number of permutations, you would take your number of permutations, and then you would divide it by the number of ways to arrange three people. Number of ways to arrange three people."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "Well, a big clue was, when we essentially wrote all of the permutations where we've picked a group of three people, we see that there's six ways of arranging the three people. When you pick a certain group of three people, that turned into six permutations. And so if all you wanna do is care about, well, how many different ways are there to choose three from the six, you would take your whole permutations, you would take your number of permutations, you would take your number of permutations, and then you would divide it by the number of ways to arrange three people. Number of ways to arrange three people. And we see that you can arrange three people, or even three letters, you can arrange it in six different ways. So this would be equal to 120 divided by six, or this would be equal to 20. So there were 120 permutations here."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "Number of ways to arrange three people. And we see that you can arrange three people, or even three letters, you can arrange it in six different ways. So this would be equal to 120 divided by six, or this would be equal to 20. So there were 120 permutations here. If you said, how many different arrangements are there of taking six people and putting them into three chairs? That's 120. But now we're asking another thing."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So there were 120 permutations here. If you said, how many different arrangements are there of taking six people and putting them into three chairs? That's 120. But now we're asking another thing. We're saying, if we start with 120 people, and we wanna choose, and we wanna, sorry, if we're starting with six people, and we wanna figure out how many ways, how many combinations, how many ways are there for us to choose three of them, then we end up with 20 combinations. Combinations of people. This right over here, once again, this right over here is just one combination."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "But now we're asking another thing. We're saying, if we start with 120 people, and we wanna choose, and we wanna, sorry, if we're starting with six people, and we wanna figure out how many ways, how many combinations, how many ways are there for us to choose three of them, then we end up with 20 combinations. Combinations of people. This right over here, once again, this right over here is just one combination. This is the combination A, B, C. I don't care what order they sit in. I've chosen them. I've chosen these three of the six."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "This right over here, once again, this right over here is just one combination. This is the combination A, B, C. I don't care what order they sit in. I've chosen them. I've chosen these three of the six. This is a combination of people. I don't care about the order. This right over here is another combination."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "I've chosen these three of the six. This is a combination of people. I don't care about the order. This right over here is another combination. It is F, C, and B. Once again, I don't care about the order. I just care that I've chosen these three people."}, {"video_title": "Introduction to combinations Probability and Statistics Khan Academy.mp3", "Sentence": "This right over here is another combination. It is F, C, and B. Once again, I don't care about the order. I just care that I've chosen these three people. So how many ways are there to choose three people out of six? It is 20. It's the total number of permutations, it's 120, divided by the number of ways you can arrange three people."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So there's going to be two bins of balls. So you're gonna have two bins of balls. One of them's gonna have 56 balls in it. So 56 in one bin. And then another bin is going to have 46 balls in it. So there are 46 balls in this bin right over here. And so what they're gonna do is they're gonna pick five balls from this bin right over here."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So 56 in one bin. And then another bin is going to have 46 balls in it. So there are 46 balls in this bin right over here. And so what they're gonna do is they're gonna pick five balls from this bin right over here. And you have to get the exact numbers of those five balls. It can be in any order. So let me just draw them."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And so what they're gonna do is they're gonna pick five balls from this bin right over here. And you have to get the exact numbers of those five balls. It can be in any order. So let me just draw them. So it's one ball. I'll shade it so it looks like a ball. Two balls."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So let me just draw them. So it's one ball. I'll shade it so it looks like a ball. Two balls. Three balls. Four balls. And five balls that they're going to pick."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Two balls. Three balls. Four balls. And five balls that they're going to pick. And you just have to get the numbers in any order. So this is from a bin of 56. From a bin of 56."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And five balls that they're going to pick. And you just have to get the numbers in any order. So this is from a bin of 56. From a bin of 56. And then you have to get the mega ball right. And then they're gonna just pick one ball from there, which they call the mega ball. They're gonna pick one ball from there."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "From a bin of 56. And then you have to get the mega ball right. And then they're gonna just pick one ball from there, which they call the mega ball. They're gonna pick one ball from there. And obviously this is just going to be picked, this is gonna be one of 46. So from a bin of 46. And so to figure out the probability of winning, it's essentially going to be one of all of the possible, all of the possibilities of numbers that you might be able to pick."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "They're gonna pick one ball from there. And obviously this is just going to be picked, this is gonna be one of 46. So from a bin of 46. And so to figure out the probability of winning, it's essentially going to be one of all of the possible, all of the possibilities of numbers that you might be able to pick. So essentially all of the combinations of the white balls times the 46 possibilities that you might get for the mega ball. So to think about the combinations for the white balls, there's a couple of ways you could do it. If you are used to thinking in combinatorics terms, it would essentially saying, well, out of a set of 56 things, I am going to choose five of them."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And so to figure out the probability of winning, it's essentially going to be one of all of the possible, all of the possibilities of numbers that you might be able to pick. So essentially all of the combinations of the white balls times the 46 possibilities that you might get for the mega ball. So to think about the combinations for the white balls, there's a couple of ways you could do it. If you are used to thinking in combinatorics terms, it would essentially saying, well, out of a set of 56 things, I am going to choose five of them. So this is literally, you could view this as 56 choose five. Or if you wanna think of it in more conceptual terms, the first ball I pick, there's 56 possibilities. Since we're not replacing the ball, the next ball I pick, there's going to be 55 possibilities."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "If you are used to thinking in combinatorics terms, it would essentially saying, well, out of a set of 56 things, I am going to choose five of them. So this is literally, you could view this as 56 choose five. Or if you wanna think of it in more conceptual terms, the first ball I pick, there's 56 possibilities. Since we're not replacing the ball, the next ball I pick, there's going to be 55 possibilities. The ball after that, there's going to be 54 possibilities. Ball after that, there's going to be 53 possibilities. And then the ball after that, there's going to be 52 possibilities."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Since we're not replacing the ball, the next ball I pick, there's going to be 55 possibilities. The ball after that, there's going to be 54 possibilities. Ball after that, there's going to be 53 possibilities. And then the ball after that, there's going to be 52 possibilities. 52, because I've already picked four balls out of that. Now this number right over here, when you multiply it out, this is the number of permutations if I cared about order. So if I got that exact combination."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And then the ball after that, there's going to be 52 possibilities. 52, because I've already picked four balls out of that. Now this number right over here, when you multiply it out, this is the number of permutations if I cared about order. So if I got that exact combination. But to win this, you don't have to write them down in the same order. You just have to get those numbers in any order. And so what you wanna do is you wanna divide this by the number of ways that five things can actually be ordered."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So if I got that exact combination. But to win this, you don't have to write them down in the same order. You just have to get those numbers in any order. And so what you wanna do is you wanna divide this by the number of ways that five things can actually be ordered. So what you wanna do is divide this by the way that five things can be ordered. And if you're ordering five things, the first of the five things can take five different positions. Then the next one will have four positions left."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And so what you wanna do is you wanna divide this by the number of ways that five things can actually be ordered. So what you wanna do is divide this by the way that five things can be ordered. And if you're ordering five things, the first of the five things can take five different positions. Then the next one will have four positions left. Then the one after that will have three positions left. One after that will have two positions. And then the fifth one will be completely determined because you've already placed the other four."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Then the next one will have four positions left. Then the one after that will have three positions left. One after that will have two positions. And then the fifth one will be completely determined because you've already placed the other four. So it's going to have only one position. So when we calculate this part right over here, this will tell us all of the combinations of just the white balls. And so let's calculate that."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And then the fifth one will be completely determined because you've already placed the other four. So it's going to have only one position. So when we calculate this part right over here, this will tell us all of the combinations of just the white balls. And so let's calculate that. So just the white balls, we have 55, sorry, 56 times 55 times 54 times 53 times 52. And we're gonna divide that by five times four times three times two. We don't have to multiply by one, but I'll just do that just to show what we're doing."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And so let's calculate that. So just the white balls, we have 55, sorry, 56 times 55 times 54 times 53 times 52. And we're gonna divide that by five times four times three times two. We don't have to multiply by one, but I'll just do that just to show what we're doing. And then that gives us about 3.8 million. So let me actually take that, let me put that off screen. So let me write that number down."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "We don't have to multiply by one, but I'll just do that just to show what we're doing. And then that gives us about 3.8 million. So let me actually take that, let me put that off screen. So let me write that number down. So this comes out to 3,819,816. So that's the number of possibilities here. So just your odds of picking just the white balls right are going to be one out of this, assuming you only have one entry."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So let me write that number down. So this comes out to 3,819,816. So that's the number of possibilities here. So just your odds of picking just the white balls right are going to be one out of this, assuming you only have one entry. And then there's 46 possibilities for the orange ball. So you're gonna multiply that times 46. And so that's going to get you, so when you multiply it times 46, bring the calculator back."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So just your odds of picking just the white balls right are going to be one out of this, assuming you only have one entry. And then there's 46 possibilities for the orange ball. So you're gonna multiply that times 46. And so that's going to get you, so when you multiply it times 46, bring the calculator back. So we're gonna multiply our previous answer times 46. And it just means my previous answer times 46. I get a little under 176 million."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And so that's going to get you, so when you multiply it times 46, bring the calculator back. So we're gonna multiply our previous answer times 46. And it just means my previous answer times 46. I get a little under 176 million. A little under 176 million. So that is, let me write that number down. So that gives us 175,711,536."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "I get a little under 176 million. A little under 176 million. So that is, let me write that number down. So that gives us 175,711,536. So your odds of winning it with one entry, because this is the number of possibilities and you are essentially for a dollar getting one of those possibilities, your odds of winning is going to be one over this. And to put this in a little bit of context, I looked it up on the internet, what your odds are of actually getting struck by lightning in your lifetime. And so your odds of getting struck by lightning in your lifetime is roughly one in 10,000."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So that gives us 175,711,536. So your odds of winning it with one entry, because this is the number of possibilities and you are essentially for a dollar getting one of those possibilities, your odds of winning is going to be one over this. And to put this in a little bit of context, I looked it up on the internet, what your odds are of actually getting struck by lightning in your lifetime. And so your odds of getting struck by lightning in your lifetime is roughly one in 10,000. One in 10,000 chance of getting struck by lightning in your lifetime. And we can roughly say your odds of getting struck by lightning twice in your lifetime, or another way of saying it is the odds of you and your best friend both independently being struck by lightning when you're not around each other is going to be one in 10,000 times one in 10,000. And so that will get you one in, and we're going to have now eight zeros."}, {"video_title": "Mega millions jackpot probability Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And so your odds of getting struck by lightning in your lifetime is roughly one in 10,000. One in 10,000 chance of getting struck by lightning in your lifetime. And we can roughly say your odds of getting struck by lightning twice in your lifetime, or another way of saying it is the odds of you and your best friend both independently being struck by lightning when you're not around each other is going to be one in 10,000 times one in 10,000. And so that will get you one in, and we're going to have now eight zeros. One, two, three, four, five, six, seven, eight. So that gives you one in 100 million. So you're actually twice almost, this is very rough, you're roughly twice as likely to get struck by lightning twice in your life than to win the mega jackpot."}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "For example, you could get a zero, a zero, a zero, and a zero, a zero, a zero, a zero, and a one, all the way up to 9,999. Four nines. Players can choose to play a straight bet where the player wins if they match all four digits in the correct order. The lottery pays $4,500 on a successful $1 straight bet. Let X represent a player's net gain on a $1 straight bet. Calculate the expected net gain. And they say, hint, the expected net gain can be negative."}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "The lottery pays $4,500 on a successful $1 straight bet. Let X represent a player's net gain on a $1 straight bet. Calculate the expected net gain. And they say, hint, the expected net gain can be negative. So why don't you pause this video and see if you can calculate the expected net gain. All right, so there's a couple of ways that we can approach this. One way is to just think about the two different outcomes."}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "And they say, hint, the expected net gain can be negative. So why don't you pause this video and see if you can calculate the expected net gain. All right, so there's a couple of ways that we can approach this. One way is to just think about the two different outcomes. There's a scenario where you win with your straight bet. There's a scenario where you lose with your straight bet. Now let's think about the net gain in either one of those scenarios."}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "One way is to just think about the two different outcomes. There's a scenario where you win with your straight bet. There's a scenario where you lose with your straight bet. Now let's think about the net gain in either one of those scenarios. The scenario where you win, you pay $1, we know it's a $1 straight bet, and you get $4,500. So what's the net gain? So it's going to be $4,500 minus one."}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "Now let's think about the net gain in either one of those scenarios. The scenario where you win, you pay $1, we know it's a $1 straight bet, and you get $4,500. So what's the net gain? So it's going to be $4,500 minus one. So your net gain is going to be $4,499. Now what about the net gain in the situation that you lose? Well, in the situation that you lose, you just lose a dollar."}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "So it's going to be $4,500 minus one. So your net gain is going to be $4,499. Now what about the net gain in the situation that you lose? Well, in the situation that you lose, you just lose a dollar. So this is just going to be negative $1 right over here. Now let's think about the probabilities of each of these situations. So the probability of a win we know is one in 10,000."}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "Well, in the situation that you lose, you just lose a dollar. So this is just going to be negative $1 right over here. Now let's think about the probabilities of each of these situations. So the probability of a win we know is one in 10,000. One in 10,000. And what's the probability of a loss? Well, that's going to be 9,999 out of 10,000."}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "So the probability of a win we know is one in 10,000. One in 10,000. And what's the probability of a loss? Well, that's going to be 9,999 out of 10,000. And so then our expected net gain is just going to be the weighted average of these two. So I could write our expected net gain is going to be 4,499 times the probability of that, one in 10,000, plus negative one times this. So I could just write that as minus 9,999 over 10,000."}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "Well, that's going to be 9,999 out of 10,000. And so then our expected net gain is just going to be the weighted average of these two. So I could write our expected net gain is going to be 4,499 times the probability of that, one in 10,000, plus negative one times this. So I could just write that as minus 9,999 over 10,000. And so this is going to be equal to, let's see, it's going to be 4,499 minus 9,999, all of that over 10,000. And let's see, this is going to be equal to negative 5,500 over 10,000. Negative 5,500 over 10,000, which is the same thing as negative 55 over 100."}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "So I could just write that as minus 9,999 over 10,000. And so this is going to be equal to, let's see, it's going to be 4,499 minus 9,999, all of that over 10,000. And let's see, this is going to be equal to negative 5,500 over 10,000. Negative 5,500 over 10,000, which is the same thing as negative 55 over 100. Or I could write it this way. This is equal to negative 55 hundredths. I could write it this way, 0.55."}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "Negative 5,500 over 10,000, which is the same thing as negative 55 over 100. Or I could write it this way. This is equal to negative 55 hundredths. I could write it this way, 0.55. So that's one way to calculate the expected net gain. Another way to approach it is to say, all right, what if we were to get 10,000 tickets? What is our expected net gain on the 10,000 tickets?"}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "I could write it this way, 0.55. So that's one way to calculate the expected net gain. Another way to approach it is to say, all right, what if we were to get 10,000 tickets? What is our expected net gain on the 10,000 tickets? Well, we would pay $10,000 and we would expect to win once. It's not a guarantee, but we would expect to win once. So expect 4,500 in payout."}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "What is our expected net gain on the 10,000 tickets? Well, we would pay $10,000 and we would expect to win once. It's not a guarantee, but we would expect to win once. So expect 4,500 in payout. And so you would then, let's see, you would have a net gain of, it would be negative $5,500. Negative $5,500. Now this is the net gain when you do 10,000 tickets."}, {"video_title": "Expected payoff example lottery ticket Probability & combinatorics Khan Academy.mp3", "Sentence": "So expect 4,500 in payout. And so you would then, let's see, you would have a net gain of, it would be negative $5,500. Negative $5,500. Now this is the net gain when you do 10,000 tickets. Now, if you wanted to find the expected net gain per ticket, you would then just divide by 10,000. And if you did that, you would get exactly what we just calculated the other way. So any way you try to approach this, this is not a great bet."}, {"video_title": "Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "The code maker gives hints about whether the colors are correct and in the right position. The possible colors are blue, yellow, white, red, orange, and green. How many four-color codes can be made if the colors cannot be repeated? To some degree, this whole paragraph in the beginning doesn't even matter. If we're just choosing from how many colors are there? There's one, two, three, four, five, six colors. And we're going to pick four of them."}, {"video_title": "Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "To some degree, this whole paragraph in the beginning doesn't even matter. If we're just choosing from how many colors are there? There's one, two, three, four, five, six colors. And we're going to pick four of them. How many four-color codes can be made if the colors cannot be repeated? And since these are codes, we're going to assume that blue, red, yellow, and green is different than green, red, yellow, and blue. We're going to assume that these are not the same code."}, {"video_title": "Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And we're going to pick four of them. How many four-color codes can be made if the colors cannot be repeated? And since these are codes, we're going to assume that blue, red, yellow, and green is different than green, red, yellow, and blue. We're going to assume that these are not the same code. Even though we've picked the same four colors, we're going to assume that these are two different codes. And that makes sense because we're dealing with codes. These are different codes."}, {"video_title": "Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "We're going to assume that these are not the same code. Even though we've picked the same four colors, we're going to assume that these are two different codes. And that makes sense because we're dealing with codes. These are different codes. This would count as two different codes right here, even though we've picked the same actual colors, the same four colors. We've picked them in different orders. With that out of the way, let's think about how many different ways we can pick four colors."}, {"video_title": "Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "These are different codes. This would count as two different codes right here, even though we've picked the same actual colors, the same four colors. We've picked them in different orders. With that out of the way, let's think about how many different ways we can pick four colors. Let's say we have four slots here. One slot, two slot, three slot, and four slots. At first, we care only about how many ways can we pick a color for that slot right there, that first slot."}, {"video_title": "Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "With that out of the way, let's think about how many different ways we can pick four colors. Let's say we have four slots here. One slot, two slot, three slot, and four slots. At first, we care only about how many ways can we pick a color for that slot right there, that first slot. We haven't picked any colors yet. We have six possible colors. One, two, three, four, five, six."}, {"video_title": "Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "At first, we care only about how many ways can we pick a color for that slot right there, that first slot. We haven't picked any colors yet. We have six possible colors. One, two, three, four, five, six. There's going to be six different possibilities for this slot right there. Let's put a six right there. Now, they told us that the colors cannot be repeated."}, {"video_title": "Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "One, two, three, four, five, six. There's going to be six different possibilities for this slot right there. Let's put a six right there. Now, they told us that the colors cannot be repeated. Whatever color is in this slot, we're going to take it out of the possible colors. Now that we've taken that color out, how many possibilities are when we go to this slot, when we go to the next slot? How many possibilities when we go to the next slot right here?"}, {"video_title": "Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Now, they told us that the colors cannot be repeated. Whatever color is in this slot, we're going to take it out of the possible colors. Now that we've taken that color out, how many possibilities are when we go to this slot, when we go to the next slot? How many possibilities when we go to the next slot right here? We took one of the six out for the first slot, so there's only five possibilities here. By the same logic, when we go to the third slot, we've used up two of the colors already. There will only be four possible colors left."}, {"video_title": "Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "How many possibilities when we go to the next slot right here? We took one of the six out for the first slot, so there's only five possibilities here. By the same logic, when we go to the third slot, we've used up two of the colors already. There will only be four possible colors left. Then for the last slot, we would have used up three of the colors, so there's only three possibilities left. If we think about all of the possibilities, all of the permutations, and permutations are when you think about all of the possibilities, and you do care about order, where you say that this is different than this. This is a different permutation than this."}, {"video_title": "Example Ways to arrange colors Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "There will only be four possible colors left. Then for the last slot, we would have used up three of the colors, so there's only three possibilities left. If we think about all of the possibilities, all of the permutations, and permutations are when you think about all of the possibilities, and you do care about order, where you say that this is different than this. This is a different permutation than this. All of the different permutations here, when you pick four colors out of a possible of six colors, it's going to be six possibilities for the first one, times five for the second bucket, times four for the third bucket of third position, times three. Six times five is 30, times four is times three, so 30 times 12. This is 30 times 12, which is equal to 360 possible four-color codes."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "And so it has various outcomes of those two free throws and then the corresponding probability, missing both free throws, 0.2, making exactly one free throw, 0.5, and making both free throws, 0.1. Is this a valid probability model? Pause this video and see if you can make a conclusion there. So let's think about what makes a valid probability model. One, the sum of the probabilities of all the scenarios need to add up to 100%, so we would definitely want to check that. And also, they would all have to be positive values, or I guess I should say, none of them can be negative values. You could have a scenario that has a 0% probability."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "So let's think about what makes a valid probability model. One, the sum of the probabilities of all the scenarios need to add up to 100%, so we would definitely want to check that. And also, they would all have to be positive values, or I guess I should say, none of them can be negative values. You could have a scenario that has a 0% probability. And so all of these look like positive probabilities, so we meet that second test, that all the probabilities are non-negative, but do they add up to 100%? So if I had 0.2 to 0.5, that is 0.7 plus 0.1, they add up to 0.8, or they add up to 80%. So this is not a valid probability model."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "You could have a scenario that has a 0% probability. And so all of these look like positive probabilities, so we meet that second test, that all the probabilities are non-negative, but do they add up to 100%? So if I had 0.2 to 0.5, that is 0.7 plus 0.1, they add up to 0.8, or they add up to 80%. So this is not a valid probability model. In order for it to be valid, they would all, all the various scenarios need to add up exactly to 100%. In this case, we only add up to 80%. If we added up to 1.1 or 110%, then we would also have a problem, but we can just write no."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "So this is not a valid probability model. In order for it to be valid, they would all, all the various scenarios need to add up exactly to 100%. In this case, we only add up to 80%. If we added up to 1.1 or 110%, then we would also have a problem, but we can just write no. Let's do another example. So here, we are told, you are a space alien. You visit planet Earth and abduct 97 chickens, 47 cows, and 77 humans."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "If we added up to 1.1 or 110%, then we would also have a problem, but we can just write no. Let's do another example. So here, we are told, you are a space alien. You visit planet Earth and abduct 97 chickens, 47 cows, and 77 humans. Then, you randomly select one Earth creature from your sample to experiment on. Each creature has an equal probability of getting selected. Create a probability model to show how likely you are to select each type of Earth creature."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "You visit planet Earth and abduct 97 chickens, 47 cows, and 77 humans. Then, you randomly select one Earth creature from your sample to experiment on. Each creature has an equal probability of getting selected. Create a probability model to show how likely you are to select each type of Earth creature. Input your answers as fractions or as decimals rounded to the nearest hundredth. So in the last example, we wanted to see whether a probability model was valid, whether it was legitimate. Here, we wanna construct a legitimate probability model."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "Create a probability model to show how likely you are to select each type of Earth creature. Input your answers as fractions or as decimals rounded to the nearest hundredth. So in the last example, we wanted to see whether a probability model was valid, whether it was legitimate. Here, we wanna construct a legitimate probability model. Well, how would we do that? Well, the estimated probability of getting a chicken is going to be the fraction that you're sampling from that is, are chickens, because any one of the animals are equally likely to be selected. 97 of the 97 plus 47 plus 77 animals are chickens."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "Here, we wanna construct a legitimate probability model. Well, how would we do that? Well, the estimated probability of getting a chicken is going to be the fraction that you're sampling from that is, are chickens, because any one of the animals are equally likely to be selected. 97 of the 97 plus 47 plus 77 animals are chickens. And so, what is this going to be? It's going to be 97 over 97, 47, and 77. You add them up."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "97 of the 97 plus 47 plus 77 animals are chickens. And so, what is this going to be? It's going to be 97 over 97, 47, and 77. You add them up. Three sevens is 21. And then, let's see, two plus nine is 11, plus four is 15, plus seven is 22, so 221. So 97 of the 221 animals are chickens."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "You add them up. Three sevens is 21. And then, let's see, two plus nine is 11, plus four is 15, plus seven is 22, so 221. So 97 of the 221 animals are chickens. And so, I'll just write 97, two 21s. They say that we can answer as fractions, so the problem's gonna go that way. What about cows?"}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "So 97 of the 221 animals are chickens. And so, I'll just write 97, two 21s. They say that we can answer as fractions, so the problem's gonna go that way. What about cows? Well, 47 of the 221 are cows. So there's a 47, two 21st probability of getting a cow. And then, last but not least, you have 77 of the 221s are human."}, {"video_title": "Valid discrete probability distribution examples Random variables AP Statistics Khan Academy.mp3", "Sentence": "What about cows? Well, 47 of the 221 are cows. So there's a 47, two 21st probability of getting a cow. And then, last but not least, you have 77 of the 221s are human. Is this a legitimate probability distribution? Well, add these up. If you add these three fractions up, the denominator's gonna be 221, and we already know that 97 plus 47 plus 77 is 221, so it definitely adds up to one, and none of these are negative."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "He uses a box and whisker plot to map his data, shown below. What is the range of tree ages that he surveyed? What is the median age of a tree in the forest? So first of all, let's make sure we understand what this box and whiskers plot is even about. This is really a way of seeing the spread of all of the different data points, which are the age of the trees, and to also give other information, like what is the median and where do most of the ages of the trees sit. So this whisker part, so you can see this black part is a whisker, this is the box, and then this is another whisker right over here. The whiskers tell us essentially the spread of all of the data."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So first of all, let's make sure we understand what this box and whiskers plot is even about. This is really a way of seeing the spread of all of the different data points, which are the age of the trees, and to also give other information, like what is the median and where do most of the ages of the trees sit. So this whisker part, so you can see this black part is a whisker, this is the box, and then this is another whisker right over here. The whiskers tell us essentially the spread of all of the data. So it says the lowest data point in this sample is an eight-year-old tree. I'm assuming that this axis down here is in years. And it says that the highest, the oldest tree right over here is 50 years."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "The whiskers tell us essentially the spread of all of the data. So it says the lowest data point in this sample is an eight-year-old tree. I'm assuming that this axis down here is in years. And it says that the highest, the oldest tree right over here is 50 years. So if we want the range, and when we think of range in a statistics point of view, we're thinking of the highest data point minus the lowest data point. So it's going to be 50 minus 8. So we have a range of 42."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And it says that the highest, the oldest tree right over here is 50 years. So if we want the range, and when we think of range in a statistics point of view, we're thinking of the highest data point minus the lowest data point. So it's going to be 50 minus 8. So we have a range of 42. So that's what the whiskers tell us. It tells us that everything falls between 8 and 50 years, including 8 years and 50 years. Now what the box does, the box starts at, well, let me explain it to you this way."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So we have a range of 42. So that's what the whiskers tell us. It tells us that everything falls between 8 and 50 years, including 8 years and 50 years. Now what the box does, the box starts at, well, let me explain it to you this way. This line right over here, this is the median. This right over here is the median. And so half of the ages are going to be less than this median."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Now what the box does, the box starts at, well, let me explain it to you this way. This line right over here, this is the median. This right over here is the median. And so half of the ages are going to be less than this median. We see right over here the median is 21. So this box and whiskers plot tells us that half of the ages of the trees are less than 21, and half are older than 21. And then these endpoints right over here, these are the medians for each of those sections."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And so half of the ages are going to be less than this median. We see right over here the median is 21. So this box and whiskers plot tells us that half of the ages of the trees are less than 21, and half are older than 21. And then these endpoints right over here, these are the medians for each of those sections. So this is the median for all of the trees that are less than the real median, or less than the main median. So this is the middle of all of the ages of trees that are less than 21. This is the middle age for all of the trees that are greater than 21, or older than 21."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And then these endpoints right over here, these are the medians for each of those sections. So this is the median for all of the trees that are less than the real median, or less than the main median. So this is the middle of all of the ages of trees that are less than 21. This is the middle age for all of the trees that are greater than 21, or older than 21. And so these essentially are splitting, we're actually splitting all of the data into four groups. This we would call the first quartile. So I'll call it Q1 for first quartile."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "This is the middle age for all of the trees that are greater than 21, or older than 21. And so these essentially are splitting, we're actually splitting all of the data into four groups. This we would call the first quartile. So I'll call it Q1 for first quartile. Maybe I'll do 1Q. This is the first quartile. Roughly, a fourth of the trees, because the way you calculate it, sometimes a tree ends up in one point or another."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So I'll call it Q1 for first quartile. Maybe I'll do 1Q. This is the first quartile. Roughly, a fourth of the trees, because the way you calculate it, sometimes a tree ends up in one point or another. About a fourth of the trees end up here. A fourth of the trees are between 14 and 21. A fourth are between 21 and it looks like 33."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Roughly, a fourth of the trees, because the way you calculate it, sometimes a tree ends up in one point or another. About a fourth of the trees end up here. A fourth of the trees are between 14 and 21. A fourth are between 21 and it looks like 33. And then a fourth are in this quartile. So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile. So to answer the question, we already did the range."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "A fourth are between 21 and it looks like 33. And then a fourth are in this quartile. So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile. So to answer the question, we already did the range. There's a 42-year spread between the oldest and the youngest tree. And then the median age of a tree in the forest is at 21. So even though you might have trees that are as old as 50, the median of the forest is actually closer to the lower end of our entire spectrum of all of the ages."}, {"video_title": "Box and whisker plot Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So to answer the question, we already did the range. There's a 42-year spread between the oldest and the youngest tree. And then the median age of a tree in the forest is at 21. So even though you might have trees that are as old as 50, the median of the forest is actually closer to the lower end of our entire spectrum of all of the ages. So if you view median as your central tendency measurement, it's only at 21 years. And you can even see it. It's closer to the left of the box and closer to the end of the left whisker than the end of the right whisker."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "The following data points represent the number of animal crackers in each kid's lunchbox. Sort the data from least to greatest, and then find the interquartile range of the data set. And I encourage you to do this before I take a shot at it. All right, so let's first sort it. And if we were actually doing this on the Khan Academy exercise, you could just drag these, you could just click and drag these numbers around to sort them, but I'll just do it by hand. So let's see, the lowest number here looks like it's a four. So I have that four, and then I have another four."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "All right, so let's first sort it. And if we were actually doing this on the Khan Academy exercise, you could just drag these, you could just click and drag these numbers around to sort them, but I'll just do it by hand. So let's see, the lowest number here looks like it's a four. So I have that four, and then I have another four. And then I have another four. And let's see, are there any fives? No fives, but there is a six."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So I have that four, and then I have another four. And then I have another four. And let's see, are there any fives? No fives, but there is a six. So then there's a six, and then there's a seven. There doesn't seem to be an eight or a nine, but then we get to a 10. And then we get to 11, 12, no 13, but then we get 14."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "No fives, but there is a six. So then there's a six, and then there's a seven. There doesn't seem to be an eight or a nine, but then we get to a 10. And then we get to 11, 12, no 13, but then we get 14. And then finally we have a 15. So the first thing we want to do is figure out the median here. So the median's the middle number."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then we get to 11, 12, no 13, but then we get 14. And then finally we have a 15. So the first thing we want to do is figure out the median here. So the median's the middle number. I have one, two, three, four, five, six, seven, eight, nine numbers, so there's going to be just one middle number. I have an odd number of numbers here. And it's going to be the number that has four to the left and four to the right, and that middle number, the median, is going to be 10."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So the median's the middle number. I have one, two, three, four, five, six, seven, eight, nine numbers, so there's going to be just one middle number. I have an odd number of numbers here. And it's going to be the number that has four to the left and four to the right, and that middle number, the median, is going to be 10. Notice I have four to the left and four to the right. And the interquartile range is all about figuring out the difference between the middle of the first half and the middle of the second half. It's a measure of spread, how far apart all of these data points are."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And it's going to be the number that has four to the left and four to the right, and that middle number, the median, is going to be 10. Notice I have four to the left and four to the right. And the interquartile range is all about figuring out the difference between the middle of the first half and the middle of the second half. It's a measure of spread, how far apart all of these data points are. And so let's figure out the middle of the first half. So we're going to ignore the median here and just look at these first four numbers. And so out of these first four numbers, I have, since I have an even number of numbers, I'm going to calculate the median using the middle two numbers."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It's a measure of spread, how far apart all of these data points are. And so let's figure out the middle of the first half. So we're going to ignore the median here and just look at these first four numbers. And so out of these first four numbers, I have, since I have an even number of numbers, I'm going to calculate the median using the middle two numbers. So I'm going to look at the middle two numbers here, and I'm going to take their average. So the average of four and six, halfway between four and six is five. Or you could say four plus six is, four plus six is equal to 10."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And so out of these first four numbers, I have, since I have an even number of numbers, I'm going to calculate the median using the middle two numbers. So I'm going to look at the middle two numbers here, and I'm going to take their average. So the average of four and six, halfway between four and six is five. Or you could say four plus six is, four plus six is equal to 10. But then I want to divide that by two. So this is going to be equal to five. So the middle of the first half is five."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Or you could say four plus six is, four plus six is equal to 10. But then I want to divide that by two. So this is going to be equal to five. So the middle of the first half is five. You can imagine it right over there. And in the middle of the second half, I'm going to have to do the same thing. I have four numbers, so I'm going to look at the middle two numbers."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So the middle of the first half is five. You can imagine it right over there. And in the middle of the second half, I'm going to have to do the same thing. I have four numbers, so I'm going to look at the middle two numbers. The middle two numbers are 12 and 14. The average of 12 and 14 is going to be 13. Is going to be 13."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "I have four numbers, so I'm going to look at the middle two numbers. The middle two numbers are 12 and 14. The average of 12 and 14 is going to be 13. Is going to be 13. If you took 12 plus 14 over two, that's going to be 26 over two, which is equal to 13. But an easier way for numbers like this, you say, hey, 13 is right exactly halfway between 12 and 14. So there you have it."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Is going to be 13. If you took 12 plus 14 over two, that's going to be 26 over two, which is equal to 13. But an easier way for numbers like this, you say, hey, 13 is right exactly halfway between 12 and 14. So there you have it. I have the middle of the first half, this five. I have the middle of the second half, 13. To calculate the interquartile range, I just have to find the difference between these two things."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So there you have it. I have the middle of the first half, this five. I have the middle of the second half, 13. To calculate the interquartile range, I just have to find the difference between these two things. So the interquartile range for this first example is going to be 13 minus five. The middle of the second half minus the middle of the first half, which is going to be equal to eight. Let's do some more of these."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "To calculate the interquartile range, I just have to find the difference between these two things. So the interquartile range for this first example is going to be 13 minus five. The middle of the second half minus the middle of the first half, which is going to be equal to eight. Let's do some more of these. This is strangely fun. Find the interquartile range of the data in the dot plot below. Songs on each album in Shane's collection."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's do some more of these. This is strangely fun. Find the interquartile range of the data in the dot plot below. Songs on each album in Shane's collection. And so let's see what's going on here. And like always, I encourage you to take a shot at it. So this is just representing the data in a different way, but we could write this again as an ordered list."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Songs on each album in Shane's collection. And so let's see what's going on here. And like always, I encourage you to take a shot at it. So this is just representing the data in a different way, but we could write this again as an ordered list. So let's do that. We have one song, or we have one album with seven songs, I guess you could say. So we have a seven."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So this is just representing the data in a different way, but we could write this again as an ordered list. So let's do that. We have one song, or we have one album with seven songs, I guess you could say. So we have a seven. We have two albums with nine songs. So we have two nines. Let me write those."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So we have a seven. We have two albums with nine songs. So we have two nines. Let me write those. We have two nines. Then we have three tens. Let's cross those out."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let me write those. We have two nines. Then we have three tens. Let's cross those out. So 10, 10, 10. Then we have an 11. We have an 11."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's cross those out. So 10, 10, 10. Then we have an 11. We have an 11. We have two 12s. Two 12s. And then finally we have, so I used those already, and then we have an album with 14 songs."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We have an 11. We have two 12s. Two 12s. And then finally we have, so I used those already, and then we have an album with 14 songs. 14. So all I did here is I wrote this data like this. So we could see, okay, this album has seven songs, this album has nine, this album has nine."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then finally we have, so I used those already, and then we have an album with 14 songs. 14. So all I did here is I wrote this data like this. So we could see, okay, this album has seven songs, this album has nine, this album has nine. And the way I wrote it, it's already in order. So I can immediately start calculating the median. Let's see, I have one, two, three, four, five, six, seven, eight, nine, 10 numbers."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So we could see, okay, this album has seven songs, this album has nine, this album has nine. And the way I wrote it, it's already in order. So I can immediately start calculating the median. Let's see, I have one, two, three, four, five, six, seven, eight, nine, 10 numbers. I have an even number of numbers. So to calculate the median, I'm gonna have to look at the middle two numbers. So the middle two numbers look like it's these two tens here because I have four to the left of them and then four to the right of them."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's see, I have one, two, three, four, five, six, seven, eight, nine, 10 numbers. I have an even number of numbers. So to calculate the median, I'm gonna have to look at the middle two numbers. So the middle two numbers look like it's these two tens here because I have four to the left of them and then four to the right of them. And so since I'm calculating the median using two numbers, it's going to be halfway between them. It's going to be the average of these two numbers. Well, the average of 10 and 10 is just going to be 10."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So the middle two numbers look like it's these two tens here because I have four to the left of them and then four to the right of them. And so since I'm calculating the median using two numbers, it's going to be halfway between them. It's going to be the average of these two numbers. Well, the average of 10 and 10 is just going to be 10. So the median is going to be 10. Median is going to be 10. And in a case like this where I calculated the median using the middle two numbers, I can now include this left 10 in the first half and I can include this right 10 in the second half."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, the average of 10 and 10 is just going to be 10. So the median is going to be 10. Median is going to be 10. And in a case like this where I calculated the median using the middle two numbers, I can now include this left 10 in the first half and I can include this right 10 in the second half. So let's do that. So the first half is going to be those five numbers and then the second half is going to be these five numbers. And it makes sense because we're literally just looking at first half, it's going to be five numbers, second half is going to be five numbers."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And in a case like this where I calculated the median using the middle two numbers, I can now include this left 10 in the first half and I can include this right 10 in the second half. So let's do that. So the first half is going to be those five numbers and then the second half is going to be these five numbers. And it makes sense because we're literally just looking at first half, it's going to be five numbers, second half is going to be five numbers. If I had a true middle number like the previous example, then we ignore that when we look at the first and second half or at least that's the way that we're doing it in these examples. But what's the median of this first half if we look at these five numbers? Well, if you have five numbers, if you have an odd number of numbers, you're going to have one middle number and it's going to be the one that has two on either sides."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And it makes sense because we're literally just looking at first half, it's going to be five numbers, second half is going to be five numbers. If I had a true middle number like the previous example, then we ignore that when we look at the first and second half or at least that's the way that we're doing it in these examples. But what's the median of this first half if we look at these five numbers? Well, if you have five numbers, if you have an odd number of numbers, you're going to have one middle number and it's going to be the one that has two on either sides. This has two to the left and it has two to the right. So the median of the first half, the middle of the first half is nine right over here. And the middle of the second half, I have one, two, three, four, five numbers and this 12 is right in the middle."}, {"video_title": "How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, if you have five numbers, if you have an odd number of numbers, you're going to have one middle number and it's going to be the one that has two on either sides. This has two to the left and it has two to the right. So the median of the first half, the middle of the first half is nine right over here. And the middle of the second half, I have one, two, three, four, five numbers and this 12 is right in the middle. You have two to the left and two to the right. So the median of the second half is 12. Interquartile range is just going to be the median of the second half, 12, minus the median of the first half, nine, which is going to be equal to three."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So what you're concerned with, if we imagine the entire country that we've already talked about, especially if we're talking about a country like the United States, but pretty much any country, is a very large population. In the United States, we're talking about on the order of 300 million people. So ideally, if you could somehow magically do it, you would survey or somehow observe all 300 million people and take the mean of how many hours of TV they watch on a given day. And then that will give you the parameter, the population mean. But we've already talked about, in a case like this, that's very impractical. Even if you tried to do it, by the time you did it, your data might be stale because some people might have passed away, other people might have been born. Who knows what might have happened?"}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And then that will give you the parameter, the population mean. But we've already talked about, in a case like this, that's very impractical. Even if you tried to do it, by the time you did it, your data might be stale because some people might have passed away, other people might have been born. Who knows what might have happened? And so this is a truth that is out there. There is a theoretical population mean for the amount of the average or the mean hours of TV watched per day by Americans. There is a truth here at any given point in time."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Who knows what might have happened? And so this is a truth that is out there. There is a theoretical population mean for the amount of the average or the mean hours of TV watched per day by Americans. There is a truth here at any given point in time. It's just pretty much impossible to come up with the exact answer, to come up with this exact truth. But you don't give up. You say, well, maybe I don't have to survey all 300 million or observe all 300 million."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "There is a truth here at any given point in time. It's just pretty much impossible to come up with the exact answer, to come up with this exact truth. But you don't give up. You say, well, maybe I don't have to survey all 300 million or observe all 300 million. Instead, I'm just going to observe a sample. I'm just going to observe a sample right over here. And let's say, for the sake, make the computation simple."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "You say, well, maybe I don't have to survey all 300 million or observe all 300 million. Instead, I'm just going to observe a sample. I'm just going to observe a sample right over here. And let's say, for the sake, make the computation simple. You do a sample of six. And we'll talk about later why six might not be as large of a sample as you would like. But you survey how much TV these folks watch."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And let's say, for the sake, make the computation simple. You do a sample of six. And we'll talk about later why six might not be as large of a sample as you would like. But you survey how much TV these folks watch. And you get that you find one person who watched 1 and 1 half hours, another person watched 2 and 1 half hours, another person watched 4 hours. And then you get one person who watched 2 hours. And then you get two people who watched 1 hour each."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But you survey how much TV these folks watch. And you get that you find one person who watched 1 and 1 half hours, another person watched 2 and 1 half hours, another person watched 4 hours. And then you get one person who watched 2 hours. And then you get two people who watched 1 hour each. So given this data from your sample, what do you get as your sample mean? Well, the sample mean, which we would denote by lowercase x with a bar over it, it's just the sum of all of these divided by the number of data points we have. So let's see, we have 1.5 plus 2.5 plus 4 plus 2 plus 1 plus 1."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And then you get two people who watched 1 hour each. So given this data from your sample, what do you get as your sample mean? Well, the sample mean, which we would denote by lowercase x with a bar over it, it's just the sum of all of these divided by the number of data points we have. So let's see, we have 1.5 plus 2.5 plus 4 plus 2 plus 1 plus 1. And all of that divided by 6, which gives us, see, the numerator 1.5 plus 2.5 is 4. Plus 4 is 8. Plus 2 is 10."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So let's see, we have 1.5 plus 2.5 plus 4 plus 2 plus 1 plus 1. And all of that divided by 6, which gives us, see, the numerator 1.5 plus 2.5 is 4. Plus 4 is 8. Plus 2 is 10. Plus 2 more is 12. So this is going to be 12 over 6, which is equal to 2 hours of television. So at least for your sample, you say my sample mean is 2 hours of television."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Plus 2 is 10. Plus 2 more is 12. So this is going to be 12 over 6, which is equal to 2 hours of television. So at least for your sample, you say my sample mean is 2 hours of television. It's an estimate. It's a statistic that is trying to estimate this parameter, this thing that's very hard to know. But it's our best shot."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So at least for your sample, you say my sample mean is 2 hours of television. It's an estimate. It's a statistic that is trying to estimate this parameter, this thing that's very hard to know. But it's our best shot. Maybe we'll get a better answer if we get more data points. But this is what we have so far. Now, the next question you ask yourself is, well, I don't want to just estimate my population mean."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But it's our best shot. Maybe we'll get a better answer if we get more data points. But this is what we have so far. Now, the next question you ask yourself is, well, I don't want to just estimate my population mean. I also want to estimate another parameter. I also am interested in estimating my population variance. So once again, since we can't survey everyone in the population, this is pretty much impossible to know."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Now, the next question you ask yourself is, well, I don't want to just estimate my population mean. I also want to estimate another parameter. I also am interested in estimating my population variance. So once again, since we can't survey everyone in the population, this is pretty much impossible to know. But we're going to attempt to estimate this parameter. We attempted to estimate the mean. Now we will also attempt to estimate this parameter, this variance parameter."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So once again, since we can't survey everyone in the population, this is pretty much impossible to know. But we're going to attempt to estimate this parameter. We attempted to estimate the mean. Now we will also attempt to estimate this parameter, this variance parameter. So how would you do it? Well, reasonable logic would say, well, maybe we'll do the same thing with the sample as we would have done with the population. When you're doing the population variance, you would take each data point in the population, find the distance between that and the population mean, take the square of that difference, and then add up all those squares of those difference, and then divide by the number of data points you have."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Now we will also attempt to estimate this parameter, this variance parameter. So how would you do it? Well, reasonable logic would say, well, maybe we'll do the same thing with the sample as we would have done with the population. When you're doing the population variance, you would take each data point in the population, find the distance between that and the population mean, take the square of that difference, and then add up all those squares of those difference, and then divide by the number of data points you have. So let's try that over here. So let's try to take each of these data points and find the difference. Let me do that in a different color."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "When you're doing the population variance, you would take each data point in the population, find the distance between that and the population mean, take the square of that difference, and then add up all those squares of those difference, and then divide by the number of data points you have. So let's try that over here. So let's try to take each of these data points and find the difference. Let me do that in a different color. Each of these data points and find the difference between that data point and our sample mean, not the population mean. We don't know what the population mean. The sample mean, so that's that first data point plus the second data point plus the second data point."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Let me do that in a different color. Each of these data points and find the difference between that data point and our sample mean, not the population mean. We don't know what the population mean. The sample mean, so that's that first data point plus the second data point plus the second data point. So it's 4 minus 2 squared plus 1 minus 2 squared. And this is what you would have done if you were taking a population variance. If this was your entire population, this is how you would find a population mean here if this was your entire population."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "The sample mean, so that's that first data point plus the second data point plus the second data point. So it's 4 minus 2 squared plus 1 minus 2 squared. And this is what you would have done if you were taking a population variance. If this was your entire population, this is how you would find a population mean here if this was your entire population. And you would find the squared distances from each of those data points and then divide by the number of data points. So let's just think about this a little bit. 1 minus 2 squared, then you have 2.5 minus 2 squared."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "If this was your entire population, this is how you would find a population mean here if this was your entire population. And you would find the squared distances from each of those data points and then divide by the number of data points. So let's just think about this a little bit. 1 minus 2 squared, then you have 2.5 minus 2 squared. 2.5 minus 2, 2 being the sample mean, squared plus, let me see this green color, plus 2 minus 2 squared, plus 1 minus 2 squared. And then maybe you would divide by the number of data points that you have, where you have the number of data points. So in this case, we're dividing by 6."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "1 minus 2 squared, then you have 2.5 minus 2 squared. 2.5 minus 2, 2 being the sample mean, squared plus, let me see this green color, plus 2 minus 2 squared, plus 1 minus 2 squared. And then maybe you would divide by the number of data points that you have, where you have the number of data points. So in this case, we're dividing by 6. And what would we get in this circumstance? Well, if we just do the computation, 1.5 minus 2 is negative 0.5. We square that."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So in this case, we're dividing by 6. And what would we get in this circumstance? Well, if we just do the computation, 1.5 minus 2 is negative 0.5. We square that. This becomes a positive 0.25. 4 minus 2 squared is going to be 2 squared, which is 4. 1 minus 2 squared, well, that's negative 1 squared, which is just 1."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "We square that. This becomes a positive 0.25. 4 minus 2 squared is going to be 2 squared, which is 4. 1 minus 2 squared, well, that's negative 1 squared, which is just 1. 2.5 minus 2 is 0.5 squared is 0.25. 2 minus 2 squared, well, this is 0. And then 1 minus 2 squared is 1."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "1 minus 2 squared, well, that's negative 1 squared, which is just 1. 2.5 minus 2 is 0.5 squared is 0.25. 2 minus 2 squared, well, this is 0. And then 1 minus 2 squared is 1. It's negative 1 squared, so we just get 1. And if we add all of this up, let's see. Let me add the whole numbers first."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And then 1 minus 2 squared is 1. It's negative 1 squared, so we just get 1. And if we add all of this up, let's see. Let me add the whole numbers first. 4 plus 1 is 5, plus 1 is 6. And then we have 0.25. So this is going to be equal to 6.5."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Let me add the whole numbers first. 4 plus 1 is 5, plus 1 is 6. And then we have 0.25. So this is going to be equal to 6.5. Let me write this in a neutral color. So this is going to be 6.5 over this 6 right over here. And we could write this as, well, there's a couple of ways that we could write this, but I'll just get the calculator out and we can just calculate it."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So this is going to be equal to 6.5. Let me write this in a neutral color. So this is going to be 6.5 over this 6 right over here. And we could write this as, well, there's a couple of ways that we could write this, but I'll just get the calculator out and we can just calculate it. So 6.5 divided by 6 gets us, if we round, it's approximately 1.08. So it's approximately 1.08 is this calculation. Now, what we have to think about is whether this is the best calculation, whether this is the best estimate for the population variance, given the data that we have."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And we could write this as, well, there's a couple of ways that we could write this, but I'll just get the calculator out and we can just calculate it. So 6.5 divided by 6 gets us, if we round, it's approximately 1.08. So it's approximately 1.08 is this calculation. Now, what we have to think about is whether this is the best calculation, whether this is the best estimate for the population variance, given the data that we have. You can always argue that we could have more data. But given the data we have, is this the best calculation that we can make to estimate the population variance? And I'll have you think about that for a second."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Now, what we have to think about is whether this is the best calculation, whether this is the best estimate for the population variance, given the data that we have. You can always argue that we could have more data. But given the data we have, is this the best calculation that we can make to estimate the population variance? And I'll have you think about that for a second. Well, it turns out that this is close. This is close to the best calculation, the best estimate that we can make, given the data we have. And sometimes this will be called the sample variance."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And I'll have you think about that for a second. Well, it turns out that this is close. This is close to the best calculation, the best estimate that we can make, given the data we have. And sometimes this will be called the sample variance. But it's a particular type of sample variance where we just divide by the number of data points we have. And so people will write just an n over here. So this is one way to define a sample variance."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And sometimes this will be called the sample variance. But it's a particular type of sample variance where we just divide by the number of data points we have. And so people will write just an n over here. So this is one way to define a sample variance. In an attempt to estimate our population variance. But it turns out, and in the next video I'll give you an intuitive explanation of why it turns out this way. And then I would also like to write a computer simulation that at least experimentally makes you feel a little bit better."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So this is one way to define a sample variance. In an attempt to estimate our population variance. But it turns out, and in the next video I'll give you an intuitive explanation of why it turns out this way. And then I would also like to write a computer simulation that at least experimentally makes you feel a little bit better. But it turns out you're going to get a better estimate. And it's a little bit weird and voodooish at first when you first think about it. You're going to get a better estimate for your population variance."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And then I would also like to write a computer simulation that at least experimentally makes you feel a little bit better. But it turns out you're going to get a better estimate. And it's a little bit weird and voodooish at first when you first think about it. You're going to get a better estimate for your population variance. If you don't divide by 6, if you don't divide by the number of data points you have, but you divide by 1 less than the number of data points you have. So how would we do that? And we can denote that as sample variance."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "You're going to get a better estimate for your population variance. If you don't divide by 6, if you don't divide by the number of data points you have, but you divide by 1 less than the number of data points you have. So how would we do that? And we can denote that as sample variance. So when most people talk about the sample variance, they're talking about a sample variance where you do this calculation. But instead of dividing by 6, you were to divide by 5. So they'd say you divide by n minus 1."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And we can denote that as sample variance. So when most people talk about the sample variance, they're talking about a sample variance where you do this calculation. But instead of dividing by 6, you were to divide by 5. So they'd say you divide by n minus 1. So what would we get in those circumstances? Well, the top part's going to be the exact same thing. We're going to get 6.5."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So they'd say you divide by n minus 1. So what would we get in those circumstances? Well, the top part's going to be the exact same thing. We're going to get 6.5. But then our denominator, our n is 6. We have 6 data points. But we're going to divide by 1 less than 6."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "We're going to get 6.5. But then our denominator, our n is 6. We have 6 data points. But we're going to divide by 1 less than 6. We're going to divide by 5. And 6.5 divided by 5 is equal to 1.3. So when we calculate our sample variance, this technique, which is the more mainstream technique."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "But we're going to divide by 1 less than 6. We're going to divide by 5. And 6.5 divided by 5 is equal to 1.3. So when we calculate our sample variance, this technique, which is the more mainstream technique. And I know it seems voodoo. Why are we dividing by n minus 1? Well, for population variance, we divide by n. But remember, we're trying to estimate the population variance."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "So when we calculate our sample variance, this technique, which is the more mainstream technique. And I know it seems voodoo. Why are we dividing by n minus 1? Well, for population variance, we divide by n. But remember, we're trying to estimate the population variance. And it turns out that this is a better estimate. Because this calculation is underestimating what the population variance is. This is a better estimate."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "Well, for population variance, we divide by n. But remember, we're trying to estimate the population variance. And it turns out that this is a better estimate. Because this calculation is underestimating what the population variance is. This is a better estimate. We don't know for sure what it is. These both could be way off. It could be just by chance what we happen to sample."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "This is a better estimate. We don't know for sure what it is. These both could be way off. It could be just by chance what we happen to sample. But over many samples, and there's many ways to think about it, this is going to be a better calculation. It's going to give you a better estimate. And so how would we write this down?"}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "It could be just by chance what we happen to sample. But over many samples, and there's many ways to think about it, this is going to be a better calculation. It's going to give you a better estimate. And so how would we write this down? How would we write this down with mathematical notation? Well, we could, remember, we're taking the sum. And we're taking each of the data points."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And so how would we write this down? How would we write this down with mathematical notation? Well, we could, remember, we're taking the sum. And we're taking each of the data points. So we'll start with the first data point, all the way to the nth data point. This lowercase n says that, hey, we're looking at the sample. If I said uppercase N, that usually denotes that we're trying to sum up everything in the population."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And we're taking each of the data points. So we'll start with the first data point, all the way to the nth data point. This lowercase n says that, hey, we're looking at the sample. If I said uppercase N, that usually denotes that we're trying to sum up everything in the population. Here we're looking at a sample of size lowercase n. And we're taking each data point. So each x sub i. And from it, we're subtracting the sample mean."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "If I said uppercase N, that usually denotes that we're trying to sum up everything in the population. Here we're looking at a sample of size lowercase n. And we're taking each data point. So each x sub i. And from it, we're subtracting the sample mean. And then we're squaring it. We're taking the sum of the squared distances. And then we're dividing not by the number of data points we have, but by 1 less than the number of data points we have."}, {"video_title": "Sample variance Descriptive statistics Probability and Statistics Khan Academy.mp3", "Sentence": "And from it, we're subtracting the sample mean. And then we're squaring it. We're taking the sum of the squared distances. And then we're dividing not by the number of data points we have, but by 1 less than the number of data points we have. So this calculation, where we summed up all of this, and then we divided by 5, not by 6, this is the standard definition of sample variance. So I'll leave you there. In the next video, I will attempt to give you an intuition of why we're dividing by n minus 1 instead of dividing by n."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "Is it an observational study? Is it an experiment? And then also think about what type of conclusions can you make based on the information in this study. All right, now let's work on this together. British researchers were interested in the relationship between farmers' approach to their cows and cows' milk yield. They prepared a survey questionnaire regarding the farmers' perception of the cows' mental capacity, the treatment they give to the cows, and the cows' yield. The survey was filled by all the farms in Great Britain."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "All right, now let's work on this together. British researchers were interested in the relationship between farmers' approach to their cows and cows' milk yield. They prepared a survey questionnaire regarding the farmers' perception of the cows' mental capacity, the treatment they give to the cows, and the cows' yield. The survey was filled by all the farms in Great Britain. After analyzing their results, they found that on farms where cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. All right, so they're making a connection between two variables. One was whether cows called by name, whether cows named, all right, whether cows named, and this would be a categorical variable because for any given farmer, it's gonna be a yes or a no that the cows are named."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "The survey was filled by all the farms in Great Britain. After analyzing their results, they found that on farms where cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. All right, so they're making a connection between two variables. One was whether cows called by name, whether cows named, all right, whether cows named, and this would be a categorical variable because for any given farmer, it's gonna be a yes or a no that the cows are named. And so they're trying to form a connection between whether the cows are named and milk yield. And this would be a quantitative variable because you're measuring it in terms of number of liters. Milk yield, whether we are drawing a connection."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "One was whether cows called by name, whether cows named, all right, whether cows named, and this would be a categorical variable because for any given farmer, it's gonna be a yes or a no that the cows are named. And so they're trying to form a connection between whether the cows are named and milk yield. And this would be a quantitative variable because you're measuring it in terms of number of liters. Milk yield, whether we are drawing a connection. And they're able to draw some form of a connection. They're saying, hey, when the cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. So first, let's just think about what type of statistical study this is."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "Milk yield, whether we are drawing a connection. And they're able to draw some form of a connection. They're saying, hey, when the cows were called by name, milk yield was 258 liters higher on average than on farms when this was not the case. So first, let's just think about what type of statistical study this is. And we could think, okay, is this a sample study? Is this a sample study? Is this an observational study?"}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "So first, let's just think about what type of statistical study this is. And we could think, okay, is this a sample study? Is this a sample study? Is this an observational study? Observational, or is this an experiment? Now, a sample study, an experiment, a sample study, you would be trying to estimate a parameter for a broader population. Here, it's not so much that they're estimating the parameter they're trying to see the connection between two variables."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "Is this an observational study? Observational, or is this an experiment? Now, a sample study, an experiment, a sample study, you would be trying to estimate a parameter for a broader population. Here, it's not so much that they're estimating the parameter they're trying to see the connection between two variables. And that brings us to observational study because that's what an observational study is all about. Can we draw a connection? Can we draw a positive or a negative correlation between variables based on observations?"}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "Here, it's not so much that they're estimating the parameter they're trying to see the connection between two variables. And that brings us to observational study because that's what an observational study is all about. Can we draw a connection? Can we draw a positive or a negative correlation between variables based on observations? So we've surveyed a population here, the farmers in Great Britain, and we are able to draw some type of connection between these variables. And so this is clearly an observational study. Now, this is not an experiment."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "Can we draw a positive or a negative correlation between variables based on observations? So we've surveyed a population here, the farmers in Great Britain, and we are able to draw some type of connection between these variables. And so this is clearly an observational study. Now, this is not an experiment. If there was an experiment, we would take the farmers and we would randomly assign them into one or two groups. And in one group, we would say, don't name, no name, no naming. And in the other group, we would say, name your cows."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "Now, this is not an experiment. If there was an experiment, we would take the farmers and we would randomly assign them into one or two groups. And in one group, we would say, don't name, no name, no naming. And in the other group, we would say, name your cows. And then we would wait some period of time and we would see the average milk production going into the experiment in the no naming group and the naming group. And then we will see, we wait some period of time, six months, a year, and then we will see the average milk production after either not naming or naming the cows for six months. So that's not what occurred here."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "And in the other group, we would say, name your cows. And then we would wait some period of time and we would see the average milk production going into the experiment in the no naming group and the naming group. And then we will see, we wait some period of time, six months, a year, and then we will see the average milk production after either not naming or naming the cows for six months. So that's not what occurred here. Here, we just did the survey to everybody and we just asked them this question and we were able to find this connection between whether the cows were named and the actual milk yield. So clearly, not an experiment. This was an observational study."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "So that's not what occurred here. Here, we just did the survey to everybody and we just asked them this question and we were able to find this connection between whether the cows were named and the actual milk yield. So clearly, not an experiment. This was an observational study. Now, the next thing is what can we conclude here? We know when, they're telling us that when the cows were named, it looks like there was a 258 liter higher yield on average. So the conclusion that we can strictly make here is like, well, for farmers in Great Britain, there is a correlation, a positive correlation between whether cows are named and the milk yield."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "This was an observational study. Now, the next thing is what can we conclude here? We know when, they're telling us that when the cows were named, it looks like there was a 258 liter higher yield on average. So the conclusion that we can strictly make here is like, well, for farmers in Great Britain, there is a correlation, a positive correlation between whether cows are named and the milk yield. So that we can say for sure. So let me write that down. So for Great Britain, for Great Britain farmers, Great Britain farmers, we have a positive correlation."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "So the conclusion that we can strictly make here is like, well, for farmers in Great Britain, there is a correlation, a positive correlation between whether cows are named and the milk yield. So that we can say for sure. So let me write that down. So for Great Britain, for Great Britain farmers, Great Britain farmers, we have a positive correlation. Positive correlation between naming cows between naming cows and milk yield. And milk yield. That's pretty much what we can say here."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "So for Great Britain, for Great Britain farmers, Great Britain farmers, we have a positive correlation. Positive correlation between naming cows between naming cows and milk yield. And milk yield. That's pretty much what we can say here. Now, some people might be tempted to try to draw causality. You'll see this all the time where you see these observational studies and people try to hint that maybe there's a causal relationship here. Maybe the naming is actually what makes the milk yield go up or maybe it's the other way."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "That's pretty much what we can say here. Now, some people might be tempted to try to draw causality. You'll see this all the time where you see these observational studies and people try to hint that maybe there's a causal relationship here. Maybe the naming is actually what makes the milk yield go up or maybe it's the other way. The cows produce a lot of milk, the farmers like them more and they wanna name them. It's like, hey, that's my high milk producing cow. So there's a lot of temptation to say naming, that maybe there's a causality that naming causes more milk, more milk, or that maybe more milk causes naming."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "Maybe the naming is actually what makes the milk yield go up or maybe it's the other way. The cows produce a lot of milk, the farmers like them more and they wanna name them. It's like, hey, that's my high milk producing cow. So there's a lot of temptation to say naming, that maybe there's a causality that naming causes more milk, more milk, or that maybe more milk causes naming. You or the farmers really like that cow so they start naming them or whatever it might be. But you can't make this causal relationship based on this observational study. You might have been able to do it with a well-constructed experiment but not with an observational study."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "So there's a lot of temptation to say naming, that maybe there's a causality that naming causes more milk, more milk, or that maybe more milk causes naming. You or the farmers really like that cow so they start naming them or whatever it might be. But you can't make this causal relationship based on this observational study. You might have been able to do it with a well-constructed experiment but not with an observational study. And that's because there could be some confounding variable that is driving both of them. So for example, that confounding variable might just be a nice farmer. A nice farmer."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "You might have been able to do it with a well-constructed experiment but not with an observational study. And that's because there could be some confounding variable that is driving both of them. So for example, that confounding variable might just be a nice farmer. A nice farmer. And we can define nice in a lot of ways. They're gentle. And a nice farmer is more likely to name and a nice farmer is more likely to get, it gets a higher yield."}, {"video_title": "Worked example identifying observational study Study design AP Statistics Khan Academy.mp3", "Sentence": "A nice farmer. And we can define nice in a lot of ways. They're gentle. And a nice farmer is more likely to name and a nice farmer is more likely to get, it gets a higher yield. And the reason why this is a confounding variable, if you were to control for that, if you just take, well, let's just control for nice farmers and then see if naming makes a difference, it might not make a difference. If the farmer is petting the cows and treating them humanely and doing other things, it might not matter whether the farmer names them or not. Likewise, if you take some less nice farmers who hit their cows and they have really inhumane conditions, it might not make a difference whether they name the cows or not."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "So let's ask ourselves some interesting questions about alphabets in the English language. And in case you don't remember, or are in the mood to count, there are 26 alphabets. So if you go A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, and Z, you get 26. 26 alphabets. Now let's ask some interesting questions. So given that there are 26 alphabets in the English language, how many possible three-letter words are there? And we're not gonna be thinking about phonetics or how hard it is to pronounce it."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "26 alphabets. Now let's ask some interesting questions. So given that there are 26 alphabets in the English language, how many possible three-letter words are there? And we're not gonna be thinking about phonetics or how hard it is to pronounce it. So for example, the word, the word zigget would be a legitimate word in this example. Or the word, the word, the word skudge would be a legitimate word in this example. So how many possible three-letter words are there in the English language?"}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "And we're not gonna be thinking about phonetics or how hard it is to pronounce it. So for example, the word, the word zigget would be a legitimate word in this example. Or the word, the word, the word skudge would be a legitimate word in this example. So how many possible three-letter words are there in the English language? I encourage you to pause the video and try to think about it. All right, I assume you've had a go at it. So let's just think about it."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "So how many possible three-letter words are there in the English language? I encourage you to pause the video and try to think about it. All right, I assume you've had a go at it. So let's just think about it. For three-letter words, there's three spaces. So how many possibilities are there for the first one? Well, there's 26 possible letters for the first one."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "So let's just think about it. For three-letter words, there's three spaces. So how many possibilities are there for the first one? Well, there's 26 possible letters for the first one. Anything from A to Z would be completely fine. Now how many possibilities for the second one? And I intentionally ask this to you to be a distractor, because you might be, you know, we've seen a lot of examples where we're saying, oh, there's 26 possibilities for the first one, and then maybe there's 25 for the second one, and then 24 for the third."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "Well, there's 26 possible letters for the first one. Anything from A to Z would be completely fine. Now how many possibilities for the second one? And I intentionally ask this to you to be a distractor, because you might be, you know, we've seen a lot of examples where we're saying, oh, there's 26 possibilities for the first one, and then maybe there's 25 for the second one, and then 24 for the third. But that's not the case right over here, because we can repeat letters. I didn't say that all the letters had to be different. So for example, the word, the word, ha, would also be a legitimate word in our example right over here."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "And I intentionally ask this to you to be a distractor, because you might be, you know, we've seen a lot of examples where we're saying, oh, there's 26 possibilities for the first one, and then maybe there's 25 for the second one, and then 24 for the third. But that's not the case right over here, because we can repeat letters. I didn't say that all the letters had to be different. So for example, the word, the word, ha, would also be a legitimate word in our example right over here. So we have 26 possibilities for the second letter, and we have 26 possibilities for the third letter. So we're going to have, and I don't know what this is, 26 to the third power possibilities, or 26 times 26 times 26, and you can figure out what that is. That is how many possible three-letter words we can have for the English language if we didn't care about how pronounceable they are, if they meant anything, and if we repeated letters."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "So for example, the word, the word, ha, would also be a legitimate word in our example right over here. So we have 26 possibilities for the second letter, and we have 26 possibilities for the third letter. So we're going to have, and I don't know what this is, 26 to the third power possibilities, or 26 times 26 times 26, and you can figure out what that is. That is how many possible three-letter words we can have for the English language if we didn't care about how pronounceable they are, if they meant anything, and if we repeated letters. Now let's ask a different question. What if we said, how many possible three-letter words are there if we want all different letters? So we want all different letters."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "That is how many possible three-letter words we can have for the English language if we didn't care about how pronounceable they are, if they meant anything, and if we repeated letters. Now let's ask a different question. What if we said, how many possible three-letter words are there if we want all different letters? So we want all different letters. So these all have to be different letters. Different, different letters. And once again, pause the video and see if you can think it through."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "So we want all different letters. So these all have to be different letters. Different, different letters. And once again, pause the video and see if you can think it through. All right. So this is where permutations start to be useful. Although I, you know, I think a lot of things like this, it's always best to reason through than to try to figure out whether some formula applies to it."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "And once again, pause the video and see if you can think it through. All right. So this is where permutations start to be useful. Although I, you know, I think a lot of things like this, it's always best to reason through than to try to figure out whether some formula applies to it. So in this situation, well, if we went in order, we could have 26 different letters for the first one, 26 different possibilities for the first one. You know, I'm always starting with that one, but there's nothing special about the one on the left. We could say that the one on the right, there's 26 possibilities."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "Although I, you know, I think a lot of things like this, it's always best to reason through than to try to figure out whether some formula applies to it. So in this situation, well, if we went in order, we could have 26 different letters for the first one, 26 different possibilities for the first one. You know, I'm always starting with that one, but there's nothing special about the one on the left. We could say that the one on the right, there's 26 possibilities. Well, for each of those possibilities, for each of those 26 possibilities, there might be 25 possibilities for what we put in the middle one if we say we're gonna figure out the middle one next. And then for each of these 25 times 26 possibilities for where we figured out two of the letters, there's 24 possibilities because we've already used two letters for the last bucket that we haven't filled. And the only reason why I went 26, 25, 24 is to show you there's nothing special about always filling in the leftmost letter or the leftmost chair first."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "We could say that the one on the right, there's 26 possibilities. Well, for each of those possibilities, for each of those 26 possibilities, there might be 25 possibilities for what we put in the middle one if we say we're gonna figure out the middle one next. And then for each of these 25 times 26 possibilities for where we figured out two of the letters, there's 24 possibilities because we've already used two letters for the last bucket that we haven't filled. And the only reason why I went 26, 25, 24 is to show you there's nothing special about always filling in the leftmost letter or the leftmost chair first. It's just about, well, let's just think in terms of, let's fill out one of the buckets first. Hey, we have the most possibilities for that. Once we use something up, then for each of those possibilities, we'll have one less for the next bucket."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "And the only reason why I went 26, 25, 24 is to show you there's nothing special about always filling in the leftmost letter or the leftmost chair first. It's just about, well, let's just think in terms of, let's fill out one of the buckets first. Hey, we have the most possibilities for that. Once we use something up, then for each of those possibilities, we'll have one less for the next bucket. And so I could do 24 times 25 times 26, but just so I don't fully confuse you, I'll go back to what I have been doing. 26 possibilities for the leftmost one. For each of those, you would have 25 possibilities for the next one that you're going to try to figure out because you've already used one letter and they have to be different."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "Once we use something up, then for each of those possibilities, we'll have one less for the next bucket. And so I could do 24 times 25 times 26, but just so I don't fully confuse you, I'll go back to what I have been doing. 26 possibilities for the leftmost one. For each of those, you would have 25 possibilities for the next one that you're going to try to figure out because you've already used one letter and they have to be different. And then for the last bucket, you're going to have 24 possibilities. So this is going to be 26 times 25 times 24, whatever that happens to be. And if we wanted to write it in the notation of permutations, we would say that this is equal to, we're taking 26 things, sorry, not 2p, 20, my brain is malfunctioning, 26, we're figuring out how many permutations are there for putting 26 different things into three different spaces."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "For each of those, you would have 25 possibilities for the next one that you're going to try to figure out because you've already used one letter and they have to be different. And then for the last bucket, you're going to have 24 possibilities. So this is going to be 26 times 25 times 24, whatever that happens to be. And if we wanted to write it in the notation of permutations, we would say that this is equal to, we're taking 26 things, sorry, not 2p, 20, my brain is malfunctioning, 26, we're figuring out how many permutations are there for putting 26 different things into three different spaces. And this is 26 if we just blindly apply the formula, which I never suggest doing. It'd be 26 factorial over 26 minus three factorial, which would be 26 factorial over 23 factorial, which is going to be exactly this right over here because the 23 times 22 times 21 all the way down to one is going to cancel with the 23 factorial. And so the whole point of this video, there's two points, is one, as soon as someone says, oh, how many different letters could you form or something like that, you just don't blindly do permutations or combinations."}, {"video_title": "Possible three letter words Probability and Statistics Khan Academy.mp3", "Sentence": "And if we wanted to write it in the notation of permutations, we would say that this is equal to, we're taking 26 things, sorry, not 2p, 20, my brain is malfunctioning, 26, we're figuring out how many permutations are there for putting 26 different things into three different spaces. And this is 26 if we just blindly apply the formula, which I never suggest doing. It'd be 26 factorial over 26 minus three factorial, which would be 26 factorial over 23 factorial, which is going to be exactly this right over here because the 23 times 22 times 21 all the way down to one is going to cancel with the 23 factorial. And so the whole point of this video, there's two points, is one, as soon as someone says, oh, how many different letters could you form or something like that, you just don't blindly do permutations or combinations. You think about, well, what is being asked in the question here? I really just have to take 26 times 26 times 26. The other thing I want to point out, and I know I keep pointing it out and it's probably getting tiring to you, is even when permutations are applicable, in my brain at least, it's always more valuable to just try to reason through the problem as opposed to just saying, oh, there's this formula that I remember from weeks or years ago in my life that had n factorial and k factorial and I have to memorize it, I have to look it up, always much more useful to just reason it through."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "If we only have a few experiments, it's very possible that our experimental probability could be different than our theoretical probability or even very different. But as we have many, many more experiments, thousands, millions, billions of experiments, the probability that the experimental and the theoretical probabilities are very different goes down dramatically. But let's get an intuitive sense for it. This right over here is a simulation created by Macmillan USA. I'll provide the link as an annotation. And what it does is it allows us to simulate many coin flips and figure out the proportion that are heads. So right over here, we can decide if we want our coin to be fair or not."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "This right over here is a simulation created by Macmillan USA. I'll provide the link as an annotation. And what it does is it allows us to simulate many coin flips and figure out the proportion that are heads. So right over here, we can decide if we want our coin to be fair or not. Right now it says that we have a 50% probability of getting heads. We can make it unfair by changing this, but I'll stick with the 50% probability. We want to show that on this graph here."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So right over here, we can decide if we want our coin to be fair or not. Right now it says that we have a 50% probability of getting heads. We can make it unfair by changing this, but I'll stick with the 50% probability. We want to show that on this graph here. We can plot it. And what this says is at a time, how many tosses do we want to take? So let's say, let's just start with 10 tosses."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "We want to show that on this graph here. We can plot it. And what this says is at a time, how many tosses do we want to take? So let's say, let's just start with 10 tosses. So what this is going to do is take 10 simulated flips of coins with each one having a 50% chance of being heads. And then as we flip, we're gonna see our total proportion that are heads. So let's just talk through this together."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So let's say, let's just start with 10 tosses. So what this is going to do is take 10 simulated flips of coins with each one having a 50% chance of being heads. And then as we flip, we're gonna see our total proportion that are heads. So let's just talk through this together. So starting to toss. And so what's going on here after 10 flips? So as you see, the first flip actually came out heads."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So let's just talk through this together. So starting to toss. And so what's going on here after 10 flips? So as you see, the first flip actually came out heads. And if you wanted to say, what's your experimental probability after that one flip? You'd say, well, with only one experiment, I got one head. So it looks like 100% were heads."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So as you see, the first flip actually came out heads. And if you wanted to say, what's your experimental probability after that one flip? You'd say, well, with only one experiment, I got one head. So it looks like 100% were heads. But in the second flip, it looks like it was a tails because now the proportion that was heads after two flips was 50%. But in the third flip, it looks like it was tails again because now only one out of three, or 33% of the flips have resulted in heads. Now by the fourth flip, we got a heads again, getting us back to 50th percentile."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So it looks like 100% were heads. But in the second flip, it looks like it was a tails because now the proportion that was heads after two flips was 50%. But in the third flip, it looks like it was tails again because now only one out of three, or 33% of the flips have resulted in heads. Now by the fourth flip, we got a heads again, getting us back to 50th percentile. Now at the fifth flip, it looks like we got another heads. And so now we have three out of five, or 60% being heads. And so the general takeaway here is when you have one, two, three, four, five, or six experiments, it's completely plausible that your experimental proportion, your experimental probability, diverges from the real probability."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "Now by the fourth flip, we got a heads again, getting us back to 50th percentile. Now at the fifth flip, it looks like we got another heads. And so now we have three out of five, or 60% being heads. And so the general takeaway here is when you have one, two, three, four, five, or six experiments, it's completely plausible that your experimental proportion, your experimental probability, diverges from the real probability. And this even continues all the way until we get to our ninth or 10th tosses. But what happens if we do way more tosses? So now I'm gonna do another, well, let's just do another 200 tosses and see what happens."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "And so the general takeaway here is when you have one, two, three, four, five, or six experiments, it's completely plausible that your experimental proportion, your experimental probability, diverges from the real probability. And this even continues all the way until we get to our ninth or 10th tosses. But what happens if we do way more tosses? So now I'm gonna do another, well, let's just do another 200 tosses and see what happens. So I'm just gonna keep tossing here. And you can see, wow, look at this. There was a big run of getting a lot of heads right over here."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So now I'm gonna do another, well, let's just do another 200 tosses and see what happens. So I'm just gonna keep tossing here. And you can see, wow, look at this. There was a big run of getting a lot of heads right over here. And then it looks like there's actually a run of getting a bunch of tails right over here, and then a little run of heads, tails, and then another run of heads. And notice, even after 215 tosses, our experimental probability is still reasonably different than our theoretical probability. So let's do another 200 and see if we can converge these over time."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "There was a big run of getting a lot of heads right over here. And then it looks like there's actually a run of getting a bunch of tails right over here, and then a little run of heads, tails, and then another run of heads. And notice, even after 215 tosses, our experimental probability is still reasonably different than our theoretical probability. So let's do another 200 and see if we can converge these over time. And what we're seeing in real time here should be the law of large numbers. As our number of tosses get larger and larger and larger, the probability that these two are very different goes down and down and down. Yes, you will get moments where you could even get 10 heads in a row or even 20 heads in a row, but over time, those will be balanced by the times where you're getting disproportionate number of tails."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "So let's do another 200 and see if we can converge these over time. And what we're seeing in real time here should be the law of large numbers. As our number of tosses get larger and larger and larger, the probability that these two are very different goes down and down and down. Yes, you will get moments where you could even get 10 heads in a row or even 20 heads in a row, but over time, those will be balanced by the times where you're getting disproportionate number of tails. So I'm just gonna keep going. We're now at almost 800 tosses, and you see now we are converging. Now this is, we're gonna cross 1,000 tosses soon."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "Yes, you will get moments where you could even get 10 heads in a row or even 20 heads in a row, but over time, those will be balanced by the times where you're getting disproportionate number of tails. So I'm just gonna keep going. We're now at almost 800 tosses, and you see now we are converging. Now this is, we're gonna cross 1,000 tosses soon. And you can see that our proportion here is now 51%. It's getting close now. We're at 50.6%, and I could just keep tossing."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "Now this is, we're gonna cross 1,000 tosses soon. And you can see that our proportion here is now 51%. It's getting close now. We're at 50.6%, and I could just keep tossing. This is 1,100. We're gonna approach 1,200 or 1,300 flips right over here. But as you can see, as we get many, many, many more flips, it was actually valuable to see even after 200 flips that there was a difference in the proportion between what we got from the experiment and what you would theoretically expect."}, {"video_title": "Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3", "Sentence": "We're at 50.6%, and I could just keep tossing. This is 1,100. We're gonna approach 1,200 or 1,300 flips right over here. But as you can see, as we get many, many, many more flips, it was actually valuable to see even after 200 flips that there was a difference in the proportion between what we got from the experiment and what you would theoretically expect. But as we get to many, many more flips, now we're at 1,210, we're getting pretty close to 50% of them turning out heads, but we could keep tossing it more and more and more. And what we'll see is as we get larger and larger and larger, it is likely that we're gonna get closer and closer and closer to 50%. It's not to say that it's impossible that we diverge again, but the likelihood of diverging gets lower and lower and lower the more tosses, the more experiments you make."}, {"video_title": "Interpreting two-way tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "They give us this, as they say, the two-way table of column relative frequencies. So for example, this column right over here is men. The column total is one, or you could say 100%. And we could see that 0.42 of the men, or 42% of the men, voted for Obama. We can see 52% of the men, or 0.52 of the men, voted for Romney. And we can see that the other, that the neither Obama, or 6% went for neither Obama or, nor Romney. And for women, 52% went for Obama, 43% went for Romney, 5% went for other."}, {"video_title": "Interpreting two-way tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And we could see that 0.42 of the men, or 42% of the men, voted for Obama. We can see 52% of the men, or 0.52 of the men, voted for Romney. And we can see that the other, that the neither Obama, or 6% went for neither Obama or, nor Romney. And for women, 52% went for Obama, 43% went for Romney, 5% went for other. And then these, this 52 plus 43 plus five will add up to 100% of the women. During the 2012 United States presidential election, were male voters more likely to vote for Romney than female voters? So let's see."}, {"video_title": "Interpreting two-way tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And for women, 52% went for Obama, 43% went for Romney, 5% went for other. And then these, this 52 plus 43 plus five will add up to 100% of the women. During the 2012 United States presidential election, were male voters more likely to vote for Romney than female voters? So let's see. If we, there's a couple of ways you could think about it. Well, actually, let's do it this way. Male voters, if you were a man, 52% of them voted for Romney, while for the women, 43% of them voted for Romney."}, {"video_title": "Interpreting two-way tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So let's see. If we, there's a couple of ways you could think about it. Well, actually, let's do it this way. Male voters, if you were a man, 52% of them voted for Romney, while for the women, 43% of them voted for Romney. So a man was more likely. There's a, if you randomly picked a man who voted, there was a 52% chance they voted for Romney, while if you randomly picked a woman, there was a 43%, a woman who voted, there was a 43% chance that she voted for Romney. So yes, male voters were more likely to vote for Romney than female voters."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So you might be wondering why I went off into permutations and combinations in the probability playlist. And I think you'll learn in this video. So let's say I want to figure out the probability. I'm going to flip a coin eight times, and it's a fair coin. And I want to figure out the probability of getting exactly three out of eight heads. So I say three eight heads. But three of my flips are going to be heads, and the rest are going to be tails."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "I'm going to flip a coin eight times, and it's a fair coin. And I want to figure out the probability of getting exactly three out of eight heads. So I say three eight heads. But three of my flips are going to be heads, and the rest are going to be tails. So how do I think about that? Well, let's go back to one of the early definitions we use for probability. And that says, the probability of anything happening is the probability of the number of equally probable events in which what we're stating is true."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "But three of my flips are going to be heads, and the rest are going to be tails. So how do I think about that? Well, let's go back to one of the early definitions we use for probability. And that says, the probability of anything happening is the probability of the number of equally probable events in which what we're stating is true. So the number of events, I guess trials or situations, in which we get three heads. And exactly three heads. We're not saying greater than three heads."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "And that says, the probability of anything happening is the probability of the number of equally probable events in which what we're stating is true. So the number of events, I guess trials or situations, in which we get three heads. And exactly three heads. We're not saying greater than three heads. So four heads won't count, and two heads won't count, five heads won't. Only three heads. And then over the total number of equally probable trials."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "We're not saying greater than three heads. So four heads won't count, and two heads won't count, five heads won't. Only three heads. And then over the total number of equally probable trials. So total number of equally possible outcomes. I should be using the word outcomes. So just with the word outcomes, it should be the total number of outcomes in which what we're saying happens, so we get three heads, over the total of possible outcomes."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "And then over the total number of equally probable trials. So total number of equally possible outcomes. I should be using the word outcomes. So just with the word outcomes, it should be the total number of outcomes in which what we're saying happens, so we get three heads, over the total of possible outcomes. So let's do the bottom part first. What are the total possible outcomes if I'm flipping a fair coin eight times? Well, the first time I flip it, I either get heads or tails, so I get two outcomes."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So just with the word outcomes, it should be the total number of outcomes in which what we're saying happens, so we get three heads, over the total of possible outcomes. So let's do the bottom part first. What are the total possible outcomes if I'm flipping a fair coin eight times? Well, the first time I flip it, I either get heads or tails, so I get two outcomes. And then when I flip it again, I get two more outcomes for the second one. And then how many total outcomes? Well, that's 2 times 2, because I've got 2 in the first, 2 in the second flip."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "Well, the first time I flip it, I either get heads or tails, so I get two outcomes. And then when I flip it again, I get two more outcomes for the second one. And then how many total outcomes? Well, that's 2 times 2, because I've got 2 in the first, 2 in the second flip. And then essentially, we multiply 2 times the number of flips. So that's 5, 6, 7, 8. And that equals 2 to the 8th."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "Well, that's 2 times 2, because I've got 2 in the first, 2 in the second flip. And then essentially, we multiply 2 times the number of flips. So that's 5, 6, 7, 8. And that equals 2 to the 8th. So the number of outcomes is just going to be 2 to the total number of flips. And hopefully that makes sense to you. If not, you might want to re-watch some of the earlier videos."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "And that equals 2 to the 8th. So the number of outcomes is just going to be 2 to the total number of flips. And hopefully that makes sense to you. If not, you might want to re-watch some of the earlier videos. But that's the easy part. So there's 2 to the 8th possible outcomes when you flip a fair coin eight times. So how many of those outcomes are going to result in exactly three heads?"}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "If not, you might want to re-watch some of the earlier videos. But that's the easy part. So there's 2 to the 8th possible outcomes when you flip a fair coin eight times. So how many of those outcomes are going to result in exactly three heads? Let's think of it this way. Let's give a name to each of our flips. Let's give a name to them."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So how many of those outcomes are going to result in exactly three heads? Let's think of it this way. Let's give a name to each of our flips. Let's give a name to them. So let me make a little column. Call these the flips. This is my flips column."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "Let's give a name to them. So let me make a little column. Call these the flips. This is my flips column. And I don't know, I could name them anything. I could name them Larry, Curly Moe. I could name them the, well, I would need five more names for them."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "This is my flips column. And I don't know, I could name them anything. I could name them Larry, Curly Moe. I could name them the, well, I would need five more names for them. But I could name them the seven dwarves, or the eight dwarves, really, because I have eight flips. But I could, you know, I'll just name, I'll number the flips. One, two, three, four, five, six, seven, eight."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "I could name them the, well, I would need five more names for them. But I could name them the seven dwarves, or the eight dwarves, really, because I have eight flips. But I could, you know, I'll just name, I'll number the flips. One, two, three, four, five, six, seven, eight. And I'm the god of probability. And essentially, I need to just pick three of these flips that are going to result in heads. So another way to think about it is, these could be eight people."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "One, two, three, four, five, six, seven, eight. And I'm the god of probability. And essentially, I need to just pick three of these flips that are going to result in heads. So another way to think about it is, these could be eight people. And I could pick which of these, you know, how many ways can I pick three of these people to put into the car? Or how many ways can I pick three of these people to sit in chairs? And it doesn't matter the order that I pick them in, right?"}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So another way to think about it is, these could be eight people. And I could pick which of these, you know, how many ways can I pick three of these people to put into the car? Or how many ways can I pick three of these people to sit in chairs? And it doesn't matter the order that I pick them in, right? It doesn't matter if I say the people that are going to get in the car are going to be people one, two, and three. Or if I say three, two, and one. Or if I say two, three, and one."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "And it doesn't matter the order that I pick them in, right? It doesn't matter if I say the people that are going to get in the car are going to be people one, two, and three. Or if I say three, two, and one. Or if I say two, three, and one. Those are all the same combination, right? So similarly, if I'm just picking flips, and I have to say, OK, three of these flips are going to get into the heads car, or are going to sit on, you know, heads is like they're people sitting down. I don't want to confuse you too much."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "Or if I say two, three, and one. Those are all the same combination, right? So similarly, if I'm just picking flips, and I have to say, OK, three of these flips are going to get into the heads car, or are going to sit on, you know, heads is like they're people sitting down. I don't want to confuse you too much. But essentially, I'm just going to choose three things out of the eight. So I'm essentially just saying, how many combinations can I get where I pick three out of these eight? And so that should immediately ring a bell that we're essentially saying, out of eight things, eight, we're going to choose three."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "I don't want to confuse you too much. But essentially, I'm just going to choose three things out of the eight. So I'm essentially just saying, how many combinations can I get where I pick three out of these eight? And so that should immediately ring a bell that we're essentially saying, out of eight things, eight, we're going to choose three. And that's, you know, how many combinations of three can we pick of eight? And that's, we went over in the last video. And let's do it with the formula first."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "And so that should immediately ring a bell that we're essentially saying, out of eight things, eight, we're going to choose three. And that's, you know, how many combinations of three can we pick of eight? And that's, we went over in the last video. And let's do it with the formula first. So let me write the formula up here, just so you remember it. But I also want to give you the intuition again for the formula. So in general, we said n choose k, that is equal to n factorial over k factorial times n minus k factorial."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "And let's do it with the formula first. So let me write the formula up here, just so you remember it. But I also want to give you the intuition again for the formula. So in general, we said n choose k, that is equal to n factorial over k factorial times n minus k factorial. So in this situation, it would be, that would equal 8 factorial over 3 factorial times what? 8 minus k times 5 factorial. Or another way of writing this, this would be 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 over, I'll just write 3 factorial for a second, then times 5 times 4 times 3 times 2 times 1."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So in general, we said n choose k, that is equal to n factorial over k factorial times n minus k factorial. So in this situation, it would be, that would equal 8 factorial over 3 factorial times what? 8 minus k times 5 factorial. Or another way of writing this, this would be 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 over, I'll just write 3 factorial for a second, then times 5 times 4 times 3 times 2 times 1. And of course, that and that cancel out. And all you're left with is 8 times 7 times 6 over 3 factorial. And I did this for a reason."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "Or another way of writing this, this would be 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 over, I'll just write 3 factorial for a second, then times 5 times 4 times 3 times 2 times 1. And of course, that and that cancel out. And all you're left with is 8 times 7 times 6 over 3 factorial. And I did this for a reason. Because I want you to re-get the intuition for, at least for this part of the formula. That's essentially just saying, how many permutations, can I, you know, how many ways can I pick three things out of eight? And that's essentially saying, well, before I pick anything, I can pick one of eight."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "And I did this for a reason. Because I want you to re-get the intuition for, at least for this part of the formula. That's essentially just saying, how many permutations, can I, you know, how many ways can I pick three things out of eight? And that's essentially saying, well, before I pick anything, I can pick one of eight. Then I have seven left to pick from for the second spot. And then I have six left to pick for the third spot. And so that's essentially the number of permutations."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "And that's essentially saying, well, before I pick anything, I can pick one of eight. Then I have seven left to pick from for the second spot. And then I have six left to pick for the third spot. And so that's essentially the number of permutations. But since we don't care, you know, if we picked it, what order we pick them in, we need to divide by the number of ways we can rearrange three things. And that's where the 3 factorial comes from. And so we're just, hopefully I didn't confuse you, but you know, if I did, you can go back to this formula for the binomial coefficient."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "And so that's essentially the number of permutations. But since we don't care, you know, if we picked it, what order we pick them in, we need to divide by the number of ways we can rearrange three things. And that's where the 3 factorial comes from. And so we're just, hopefully I didn't confuse you, but you know, if I did, you can go back to this formula for the binomial coefficient. But it's good to have the intuition. And then, well, once we're at this point, we can just calculate this. Well, what's this?"}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "And so we're just, hopefully I didn't confuse you, but you know, if I did, you can go back to this formula for the binomial coefficient. But it's good to have the intuition. And then, well, once we're at this point, we can just calculate this. Well, what's this? This is 8 times 7 times 6 over 3 factorial is 3 times 2 times 1, so that's 6. So 6 cancels out, so it's 8 times 7. So there's 8 times 7, or what is that, 56."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "Well, what's this? This is 8 times 7 times 6 over 3 factorial is 3 times 2 times 1, so that's 6. So 6 cancels out, so it's 8 times 7. So there's 8 times 7, or what is that, 56. Right, 56. Yeah, that's equal to 56. So there's 56 different ways to pick three things out of eight."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So there's 8 times 7, or what is that, 56. Right, 56. Yeah, that's equal to 56. So there's 56 different ways to pick three things out of eight. Or if I have eight people, there's 56 ways of picking three people to sit in the car, or however you want to view it. But if I have eight flips, there's 56 ways of picking three of those flips to be heads. So let's go to our original probability problem."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So there's 56 different ways to pick three things out of eight. Or if I have eight people, there's 56 ways of picking three people to sit in the car, or however you want to view it. But if I have eight flips, there's 56 ways of picking three of those flips to be heads. So let's go to our original probability problem. What is the probability that I get three out of eight heads? Well, it's the probability, it's the number of ways I can pick three out of those eight, so it equals 56, over the total number of outcomes, right? The total number of outcomes is 2 to the 8th."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So let's go to our original probability problem. What is the probability that I get three out of eight heads? Well, it's the probability, it's the number of ways I can pick three out of those eight, so it equals 56, over the total number of outcomes, right? The total number of outcomes is 2 to the 8th. Another way I could write that, 56, let me unseparate, that's 8 times 7 over 2 to the 8th. 8 is 2 to the 3rd, right? Let me erase some of this."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "The total number of outcomes is 2 to the 8th. Another way I could write that, 56, let me unseparate, that's 8 times 7 over 2 to the 8th. 8 is 2 to the 3rd, right? Let me erase some of this. Not with that color. Let me erase all of this so I have space. I will switch colors for variety."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "Let me erase some of this. Not with that color. Let me erase all of this so I have space. I will switch colors for variety. OK, so I'm back. All right, so 8 is the same thing as 2 to the 3rd times 7. This is all just mathematical simplification, but it's useful, over 2 to the 8th."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "I will switch colors for variety. OK, so I'm back. All right, so 8 is the same thing as 2 to the 3rd times 7. This is all just mathematical simplification, but it's useful, over 2 to the 8th. And so if we just divide both sides, the numerator and the denominator by 2 to the 3rd, this becomes 1, this becomes 2 to the 5th, and so it becomes 7 over 32. Is that right? So if I were to pick 3 out of 8, yep, I think that is right."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "This is all just mathematical simplification, but it's useful, over 2 to the 8th. And so if we just divide both sides, the numerator and the denominator by 2 to the 3rd, this becomes 1, this becomes 2 to the 5th, and so it becomes 7 over 32. Is that right? So if I were to pick 3 out of 8, yep, I think that is right. And so what does that turn out to be? Let me get my calculator. I often make careless mistakes."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "So if I were to pick 3 out of 8, yep, I think that is right. And so what does that turn out to be? Let me get my calculator. I often make careless mistakes. Let me see, my calculator seems to have disappeared. Let me get it back. There it is."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "I often make careless mistakes. Let me see, my calculator seems to have disappeared. Let me get it back. There it is. OK, 7 divided by 32 is equal to 0.21875, which is equal to 21, if I were to round, roughly, 21.9% chance. So there's a little bit better than 1 in 5 chance that I get exactly 3 out of the 8 flips as heads. Hopefully I didn't confuse you, and now you can apply that to pretty much anything."}, {"video_title": "Probability using combinations Probability and Statistics Khan Academy.mp3", "Sentence": "There it is. OK, 7 divided by 32 is equal to 0.21875, which is equal to 21, if I were to round, roughly, 21.9% chance. So there's a little bit better than 1 in 5 chance that I get exactly 3 out of the 8 flips as heads. Hopefully I didn't confuse you, and now you can apply that to pretty much anything. You could say, well, what is the probability of getting, if I flip a fair coin, of getting exactly 7 out of 8 heads? Or you could say, what's the probability of getting 2 out of 100 heads? And you could use it the exact same way we did this problem."}, {"video_title": "Probability with combinations example choosing cards Probability & combinatorics.mp3", "Sentence": "Suppose that Luis randomly draws four cards without replacement. What is the probability that Luis gets two aces and two kings in any order? So like always, pause this video and see if you can work through this. All right, now to figure out this probability, we can think about this as going to be the number of, let's call them draws, with exactly two aces and two kings. Two aces and two kings. And that's going to be over the total number of possible draws of four cards. So number of possible draws of four cards."}, {"video_title": "Probability with combinations example choosing cards Probability & combinatorics.mp3", "Sentence": "All right, now to figure out this probability, we can think about this as going to be the number of, let's call them draws, with exactly two aces and two kings. Two aces and two kings. And that's going to be over the total number of possible draws of four cards. So number of possible draws of four cards. Now for many of y'all, this bottom, the denominator here, might be a little bit easier to think about. We know that there's 52 total cards of which we are choosing four. So we could say 52 choose four, and that will tell us the total number of possible draws of four cards."}, {"video_title": "Probability with combinations example choosing cards Probability & combinatorics.mp3", "Sentence": "So number of possible draws of four cards. Now for many of y'all, this bottom, the denominator here, might be a little bit easier to think about. We know that there's 52 total cards of which we are choosing four. So we could say 52 choose four, and that will tell us the total number of possible draws of four cards. How many combinations of four cards can we get when we're picking from 52? But the top here might be a little bit more of a stumper. We can think we have exactly two spots for aces."}, {"video_title": "Probability with combinations example choosing cards Probability & combinatorics.mp3", "Sentence": "So we could say 52 choose four, and that will tell us the total number of possible draws of four cards. How many combinations of four cards can we get when we're picking from 52? But the top here might be a little bit more of a stumper. We can think we have exactly two spots for aces. So we're choosing two aces out of how many possible aces? Well, there's four total aces. So if we say four choose two, this is the total number of ways, when you don't care about order, that you can have two out of your four aces picked."}, {"video_title": "Probability with combinations example choosing cards Probability & combinatorics.mp3", "Sentence": "We can think we have exactly two spots for aces. So we're choosing two aces out of how many possible aces? Well, there's four total aces. So if we say four choose two, this is the total number of ways, when you don't care about order, that you can have two out of your four aces picked. And then separately, we can use similar logic to say, all right, there's also four choose two ways of picking two kings out of four possible kings. And now the total number of draws with two aces and two kings, this is going to be the product of these two. And if you're wondering why you can just multiply it, think about it."}, {"video_title": "Probability with combinations example choosing cards Probability & combinatorics.mp3", "Sentence": "So if we say four choose two, this is the total number of ways, when you don't care about order, that you can have two out of your four aces picked. And then separately, we can use similar logic to say, all right, there's also four choose two ways of picking two kings out of four possible kings. And now the total number of draws with two aces and two kings, this is going to be the product of these two. And if you're wondering why you can just multiply it, think about it. For every scenario that you have these two aces, you have four choose two scenarios of which kings you're dealing with. So you would take the product of them. And we've already done many examples of computing combinatorics like this."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "In an experiment aimed at studying the effect of advertising on eating behavior in children, a group of 500 children, 7 to 11 years old, were randomly assigned to two different groups. After randomization, each child was asked to watch a cartoon in a private room, containing a large bowl of Goldfish crackers. The cartoon included two commercial breaks. The first group watched food commercials, mostly snacks, while the second group watched non-food commercials, games and entertainment products. Once the child finished watching the cartoon, the conductors of the experiment weighed the cracker bowls to measure how many grams of crackers the child ate. They found that the mean amount of crackers eaten by the children who watched food commercials is 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. So let's just think about what happens up to this point."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "The first group watched food commercials, mostly snacks, while the second group watched non-food commercials, games and entertainment products. Once the child finished watching the cartoon, the conductors of the experiment weighed the cracker bowls to measure how many grams of crackers the child ate. They found that the mean amount of crackers eaten by the children who watched food commercials is 10 grams greater than the mean amount of crackers eaten by the children who watched non-food commercials. So let's just think about what happens up to this point. They took 500 children and then they randomly assigned them to two different groups. So you have group 1 over here and you have group 2. So let's say that this right over here is the first group."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So let's just think about what happens up to this point. They took 500 children and then they randomly assigned them to two different groups. So you have group 1 over here and you have group 2. So let's say that this right over here is the first group. The first group watched food commercials. So this is group number 1. So they watched food commercials."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So let's say that this right over here is the first group. The first group watched food commercials. So this is group number 1. So they watched food commercials. We could call this the treatment group. We're trying to see what's the effect of watching food commercials. And then they tell us the second group watched non-food commercials."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So they watched food commercials. We could call this the treatment group. We're trying to see what's the effect of watching food commercials. And then they tell us the second group watched non-food commercials. So this is the control group. So number 2, this is non-food commercials. So this is the control right over here."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "And then they tell us the second group watched non-food commercials. So this is the control group. So number 2, this is non-food commercials. So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So this is the control right over here. Once the child finished watching the cartoon, for each child they weighed how much of the crackers they ate and then they took the mean of it and they found that the mean here, that the kids ate 10 grams greater on average than this group right over here. Which, just looking at that data, makes you believe that something maybe happened over here. That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "That maybe the treatment from watching the food commercials made the students eat more of the goldfish crackers. But the question that you always have to ask yourself in a situation like this, well, isn't there some probability that this would have happened by chance? That even if you didn't make them watch the commercials, if these were just two random groups and you didn't make either group watch a commercial, you made them all watch the same commercials, there's some chance that the mean of one group could be dramatically different than the other one. It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "It just happened to be in this experiment that the mean here, that it looks like the kids ate 10 grams more. So how do you figure out what's the probability that this could have happened, that the 10 grams greater in mean amount eaten here, that that could have just happened by chance? Well, the way you do it is what they do right over here. Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "Using a simulator, they re-randomized the results into two new groups and measured the difference between the means of the new groups. They repeated the simulation 150 times and plotted the resulting differences as given below. So what they did is they said, okay, they have 500 kids, and each kid, they had 500 children, so number 1, 2, 3, all the way up to 500. And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "And for each child, they measured how much was the weight of the crackers that they ate. So maybe child 1 ate 2 grams, and child 2 ate 4 grams, and child 3 ate, I don't know, ate 12 grams, all the way to child number 500, ate, I don't know, maybe they didn't eat anything at all, ate 0 grams. And we already know, let's say, the first time around, 1 through 200, and 1 through, I guess we should, you know, the first half was in the treatment group, when we're just ranking them like this, and then the second, they were randomly assigned into these groups, and then the second half was in the control group. But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "But what they're doing now is they're taking these same results and they're re-randomizing it. So now they're saying, okay, let's maybe put this person in group number 2, put this person in group number 2, and this person stays in group number 2, and this person stays in group number 1, and this person stays in group number 1. So now they're completely mixing up all of the results that they had, so it's completely random of whether the student had watched the food commercial or the non-food commercial, and then they're testing what's the mean of the new number 1 group and the new number 2 group, and then they're saying, well, what is the distribution of the differences in means? So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So they see when they did it this way, when they're essentially just completely randomly taking these results and putting them into two new buckets, you have a bunch of cases where you get no difference in the mean. So out of the 150 times that they repeated the simulation doing this little exercise here, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, I'm having trouble counting this, let's see, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, I keep, it's so small, I'm aging, but it looks like there's about, I don't know, high teens, about 8 times when there was actually no noticeable difference in the means of the groups where you just randomly, when you just randomly allocate the results amongst the two groups. So when you look at this, if it was just, if you just randomly put people into two groups, the probability or the situations where you get a 10-gram difference are actually very unlikely. So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So this is, let's see, is this the difference, the difference between the means of the new groups. So it's not clear whether this is group 1 minus group 2 or group 2 minus group 1, but in either case, the situations where you have a 10-gram difference in mean, it's only 2 out of the 150 times. So when you do it randomly, when you just randomly put these results into two groups, the probability of the means being this different, it only happens 2 out of the 150 times. There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "There's 150 dots here. So that is on the order of 2%, or actually it's less than 2%. It's between 1 and 2%. And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times. It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "And if you know that this is group, let's say that the situation we're talking about, let's say that this is group 1 minus group 2 in terms of how much was eaten, and so you're looking at this situation right over here, then that's only 1 out of 150 times. It happened less frequently than 1 in 100 times. It happened only 1 in 150 times. So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times. So it seems like this was very, it's very unlikely that this was purely due to chance. If this was just a chance event, this would only happen roughly 1 in 150 times."}, {"video_title": "Statistical significance of experiment Probability and Statistics Khan Academy.mp3", "Sentence": "So if you look at that, you say, well, the probability, if this was just random, the probability of getting the results that you got is less than 1%. So to me, and then to most statisticians, that tells us that our experiment was significant, that the probability of getting the results that you got, so the children who watch food commercials being 10 grams greater than the mean amount of crackers eaten by the children who watch non-food commercials, if you just randomly put 500 kids into 2 different buckets based on the simulation results, it looks like there's only, if you run the simulation 150 times, that only happened 1 out of 150 times. So it seems like this was very, it's very unlikely that this was purely due to chance. If this was just a chance event, this would only happen roughly 1 in 150 times. But the fact that this happened in your experiment makes you feel pretty confident that your experiment is significant. In most studies, in most experiments, the threshold that they think about is the probability of something statistically significant, if the probability of that happening by chance is less than 5%, so this is less than 1%. So I would definitely say that the experiment is significant."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Complete the following two-way table of column relative frequencies, so that's what they're talking here, this is a two-way table of column relative frequencies, if necessary, round your answers to the nearest hundred. So let's see what they're saying. They're saying, let's see, of the accidents within the last year, 28 were the people were driving an SUV, a sport utility vehicle, and 35 were in a sports car. Of the no accidents in the last year, 97 were an SUV, and 104 were a sports car. Or another way you could think of it, of the sport utility vehicles that were driven, and the total, let's see, it's 28 plus 97, which is going to be 125, of that 125, 28 had an accident within the last year, and 97 did not have an accident in the last year. Similarly, you could say of the 139 sports cars, 35 had an accident in the last year, 104 did not have an accident in the last year. And so what they want us to do is put those relative frequencies in here."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Of the no accidents in the last year, 97 were an SUV, and 104 were a sports car. Or another way you could think of it, of the sport utility vehicles that were driven, and the total, let's see, it's 28 plus 97, which is going to be 125, of that 125, 28 had an accident within the last year, and 97 did not have an accident in the last year. Similarly, you could say of the 139 sports cars, 35 had an accident in the last year, 104 did not have an accident in the last year. And so what they want us to do is put those relative frequencies in here. So the way we could think about it, one right over here, this represents all of the sport utility vehicles. So one way you could think about it, that represents the whole universe of the sport utility vehicles, or at least the universe that this table shows. So that's really representative of the 28 plus 97."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And so what they want us to do is put those relative frequencies in here. So the way we could think about it, one right over here, this represents all of the sport utility vehicles. So one way you could think about it, that represents the whole universe of the sport utility vehicles, or at least the universe that this table shows. So that's really representative of the 28 plus 97. And so in each of these, we want to put the relative frequency. So this right over here is going to be 28, 28 divided by the total. Notice over here it was 28, but we want this number to be the fraction of the total."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So that's really representative of the 28 plus 97. And so in each of these, we want to put the relative frequency. So this right over here is going to be 28, 28 divided by the total. Notice over here it was 28, but we want this number to be the fraction of the total. Well, the fraction of the total is going to be 28 over 97 plus 28, which of course is going to be 125. Actually, let me just write them all like that first. This one right over here is going to be 97 over 125."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Notice over here it was 28, but we want this number to be the fraction of the total. Well, the fraction of the total is going to be 28 over 97 plus 28, which of course is going to be 125. Actually, let me just write them all like that first. This one right over here is going to be 97 over 125. And of course, when you add this one and this one, it should add up to one. Likewise, this one's going to be 35 over 139, 35 plus 104, so 139. And this is going to be 104 over 104 plus 35, which is 139."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "This one right over here is going to be 97 over 125. And of course, when you add this one and this one, it should add up to one. Likewise, this one's going to be 35 over 139, 35 plus 104, so 139. And this is going to be 104 over 104 plus 35, which is 139. And so let me just calculate each of them using this calculator. So let me scroll down a little bit. And so if I do 28 divided by 125, I get 0.224."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And this is going to be 104 over 104 plus 35, which is 139. And so let me just calculate each of them using this calculator. So let me scroll down a little bit. And so if I do 28 divided by 125, I get 0.224. They said round your answers to the nearest hundredth. So this is 0.22. No accident within the last year, 97 divided by 125."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And so if I do 28 divided by 125, I get 0.224. They said round your answers to the nearest hundredth. So this is 0.22. No accident within the last year, 97 divided by 125. So 97 divided by 125 is equal to, let's see, here if I round to the nearest hundredth, I'm going to round up 0.78. So this is 0.78. Then 35 divided by 139, 35 divided by 139 is equal to, round to the nearest hundredth, 0.25."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "No accident within the last year, 97 divided by 125. So 97 divided by 125 is equal to, let's see, here if I round to the nearest hundredth, I'm going to round up 0.78. So this is 0.78. Then 35 divided by 139, 35 divided by 139 is equal to, round to the nearest hundredth, 0.25. 0.25. And then 104 divided by 139. 104 divided by 139 gets me, if I round to the nearest hundredth, 0.75."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Then 35 divided by 139, 35 divided by 139 is equal to, round to the nearest hundredth, 0.25. 0.25. And then 104 divided by 139. 104 divided by 139 gets me, if I round to the nearest hundredth, 0.75. 0.75. And I can check my answer, and I got it right. But the key thing here is to make sure we understand what's going on here."}, {"video_title": "Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3", "Sentence": "104 divided by 139 gets me, if I round to the nearest hundredth, 0.75. 0.75. And I can check my answer, and I got it right. But the key thing here is to make sure we understand what's going on here. So one way to think about this is 22% of the sport utility vehicles had an accident within the last year, or you could say 0.22 of them. And you could say 78%, or 0.78, of the sport utility vehicles had no accidents. Likewise, you could say 25% of the sports cars had an accident within the last year, and 75% did not have an accident in the last year."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Which interval contains the median sleep amount? And so they're saying is it this interval on the histogram from six to 6.5 or this one or this one or any of these? Which of these intervals contain the median? Pause this video and see if you can figure that out. All right, now let's work through this together and let's just remind ourselves how we find the median. If I had the data points 11, nine, seven, three, and two, the way that we find the median is we can order it from least to greatest, or actually you could do it from greatest to least, but let's do least to greatest. So two, three, seven, nine, 11."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Pause this video and see if you can figure that out. All right, now let's work through this together and let's just remind ourselves how we find the median. If I had the data points 11, nine, seven, three, and two, the way that we find the median is we can order it from least to greatest, or actually you could do it from greatest to least, but let's do least to greatest. So two, three, seven, nine, 11. And the median would be the middle number. And I have a clear middle number because I have five data points. If I have an even number of data points, I still would wanna order them from least to greatest."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So two, three, seven, nine, 11. And the median would be the middle number. And I have a clear middle number because I have five data points. If I have an even number of data points, I still would wanna order them from least to greatest. So let's say that I have a one, one, three, and a seven. But here you don't have a clear middle, so the median would be the mean of the middle two numbers. So in this situation, Miguel has an even number of data points."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "If I have an even number of data points, I still would wanna order them from least to greatest. So let's say that I have a one, one, three, and a seven. But here you don't have a clear middle, so the median would be the mean of the middle two numbers. So in this situation, Miguel has an even number of data points. So the median would be the mean of the 25th and 26th data point. These would be the middle two data points. So which interval here contains the 25th and the 26th data point?"}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So in this situation, Miguel has an even number of data points. So the median would be the mean of the 25th and 26th data point. These would be the middle two data points. So which interval here contains the 25th and the 26th data point? Well, we can start at the bottom. So we have, actually, let's just look at each interval and think about how many data points they have in it. This one has two."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So which interval here contains the 25th and the 26th data point? Well, we can start at the bottom. So we have, actually, let's just look at each interval and think about how many data points they have in it. This one has two. This one has nine. This one has 12. And I'm just reading out how many data points there are in each of these intervals."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "This one has two. This one has nine. This one has 12. And I'm just reading out how many data points there are in each of these intervals. This one has 12. This one has 11. I see that there."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "And I'm just reading out how many data points there are in each of these intervals. This one has 12. This one has 11. I see that there. This one has two, and this one has two. So if we look at just this, we have the two lowest. If we look at the two bottom intervals combined, we have the 11 lowest."}, {"video_title": "Median in a histogram Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "I see that there. This one has two, and this one has two. So if we look at just this, we have the two lowest. If we look at the two bottom intervals combined, we have the 11 lowest. If we look at the three bottom intervals, we have the 11 plus 12, you have the 23 lowest. So this is the 23 lowest data points. And so the 24th, 25th, 26th, the next 12 data points starting from the bottom, starting from the lowest, are going to be in this next interval here."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Assuming the trend shown in the data has been consistent since 1945, use the graph to estimate the percentage of American adults who smoked in 1945. So let's see what's going on here. The horizontal axis here, they say years since 1965. So this point right over here, this is zero years since 1965, so this really represents 1965. And we see it looks like around, let's see if I were to eyeball it, it looks like it's around 42% of Americans, just looking at this graph, I know that's not an exact number, roughly 41 or 42% of Americans smoked in 1965 based on this graph. And then five years later, this would be 1970, 10 years later, that would be 1975. And they don't sample the data, or we don't have data from every given year."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So this point right over here, this is zero years since 1965, so this really represents 1965. And we see it looks like around, let's see if I were to eyeball it, it looks like it's around 42% of Americans, just looking at this graph, I know that's not an exact number, roughly 41 or 42% of Americans smoked in 1965 based on this graph. And then five years later, this would be 1970, 10 years later, that would be 1975. And they don't sample the data, or we don't have data from every given year. This is just from some of the years that we happen to have. But what is clear, it looks like we have a negative linear relationship right over here, that it would not be difficult to fit a line, so let me try to do that. So I'm just gonna eyeball it and try to fit a line to this data."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And they don't sample the data, or we don't have data from every given year. This is just from some of the years that we happen to have. But what is clear, it looks like we have a negative linear relationship right over here, that it would not be difficult to fit a line, so let me try to do that. So I'm just gonna eyeball it and try to fit a line to this data. So our line might look something like that. So it looks like a pretty strong negative linear relationship. When I say it's a negative linear relationship, we see that as time increases, the percentage of smokers in the US is decreasing."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So I'm just gonna eyeball it and try to fit a line to this data. So our line might look something like that. So it looks like a pretty strong negative linear relationship. When I say it's a negative linear relationship, we see that as time increases, the percentage of smokers in the US is decreasing. So that's what makes it a negative relationship. Now what are they asking? They want us to estimate the percentage of American adults who smoked in 1945."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "When I say it's a negative linear relationship, we see that as time increases, the percentage of smokers in the US is decreasing. So that's what makes it a negative relationship. Now what are they asking? They want us to estimate the percentage of American adults who smoked in 1945. Well 1945 would be to the left of zero. So we could even think of it as if 1945 is 20 years before 1965. So let me see if I can draw that."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "They want us to estimate the percentage of American adults who smoked in 1945. Well 1945 would be to the left of zero. So we could even think of it as if 1945 is 20 years before 1965. So let me see if I can draw that. So 20 years before 1965. Let's see, this would be five years before 1965, 10 years, 15 years, 20 years before 1965. So I could even put that as negative 20 right over here."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So let me see if I can draw that. So 20 years before 1965. Let's see, this would be five years before 1965, 10 years, 15 years, 20 years before 1965. So I could even put that as negative 20 right over here. Negative 20 years since 1965, you could view as 20 years before 1965, so that would represent 1945 right over there. And one thing that we could do is very roughly just try to extend this negative linear relationship backwards, and they allow us to do that by saying assuming the trend shown in the data has been consistent. So the trend has been consistent."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So I could even put that as negative 20 right over here. Negative 20 years since 1965, you could view as 20 years before 1965, so that would represent 1945 right over there. And one thing that we could do is very roughly just try to extend this negative linear relationship backwards, and they allow us to do that by saying assuming the trend shown in the data has been consistent. So the trend has been consistent. This line represents the trend. So let's just keep going backwards. Keep going backwards at the same rate."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So the trend has been consistent. This line represents the trend. So let's just keep going backwards. Keep going backwards at the same rate. So something like that. I wanna make sure that it looks at the same, looks like it's the same rate right over here. And you could just try to eyeball it."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Keep going backwards at the same rate. So something like that. I wanna make sure that it looks at the same, looks like it's the same rate right over here. And you could just try to eyeball it. You could say well see, 20 years ago, 1945, if I were to extend that line backwards, it looks like there were about 52% of the population was smoking. This seems like we're about 52% right over here. Another way to think about it would be to actually try to calculate the rate of decline."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And you could just try to eyeball it. You could say well see, 20 years ago, 1945, if I were to extend that line backwards, it looks like there were about 52% of the population was smoking. This seems like we're about 52% right over here. Another way to think about it would be to actually try to calculate the rate of decline. And let's say we do it over every 20 years because that'll be useful because we're going 20 years back. So if we go 20 years from this point, so this is 1965, you go 20 years in the future. So that is 10 years and then that is 20 years."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Another way to think about it would be to actually try to calculate the rate of decline. And let's say we do it over every 20 years because that'll be useful because we're going 20 years back. So if we go 20 years from this point, so this is 1965, you go 20 years in the future. So that is 10 years and then that is 20 years. So my change in the horizontal is 20 years. What's the change in the vertical? Well it looks like we have a decrease of a little bit more than 10%."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So that is 10 years and then that is 20 years. So my change in the horizontal is 20 years. What's the change in the vertical? Well it looks like we have a decrease of a little bit more than 10%. Looks like it's 11 or 12% decrease. So I'll just say minus 11% right there. And let's see if that's consistent."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Well it looks like we have a decrease of a little bit more than 10%. Looks like it's 11 or 12% decrease. So I'll just say minus 11% right there. And let's see if that's consistent. If we were to go another 20 years. So if we go another 20 years, it looks like once again, we've gone down by about 10%. So that looks like roughly 10%."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And let's see if that's consistent. If we were to go another 20 years. So if we go another 20 years, it looks like once again, we've gone down by about 10%. So that looks like roughly 10%. If we're following the line, it should actually be the same number. So let me write it this way. It's approximately down 10%."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So that looks like roughly 10%. If we're following the line, it should actually be the same number. So let me write it this way. It's approximately down 10%. So that little squiggly line, I'm just saying approximately negative 10% every 20 years. Negative 10% every 20 years. So if you go back 20 years, you should increase your percentage by 20%."}, {"video_title": "Smoking in 1945 Data and modeling 8th grade Khan Academy.mp3", "Sentence": "It's approximately down 10%. So that little squiggly line, I'm just saying approximately negative 10% every 20 years. Negative 10% every 20 years. So if you go back 20 years, you should increase your percentage by 20%. So this should go up by, or you should increase your percentage by 10% I should say. So if we started at 41 or 42, once again, this is what we saw when we just eyeballed it. You should get to 51 or 52%."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "And you walk up to a table, and on that table there is an empty bag. And the guy who runs the table says, look, I've got some marbles here, three green marbles, two orange marbles, and I'm going to stick them in the bag. And he literally sticks them into the empty bag to show you that it's truly three green marbles and two orange marbles. And he says, the game that I want you to play, or that if you choose to play, is you're going to look away, stick your hand in this bag. The bag is not transparent. Feel around the marbles. All the marbles feel exactly the same."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "And he says, the game that I want you to play, or that if you choose to play, is you're going to look away, stick your hand in this bag. The bag is not transparent. Feel around the marbles. All the marbles feel exactly the same. And if you're able to pick two green marbles, if you're able to take one marble out of the bag, it's green, you put it down on the table, then put your hand back in the bag and take another marble. And if that one is also green, then you're going to win the prize. You're going to win the prize of $1 if you get two greens."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "All the marbles feel exactly the same. And if you're able to pick two green marbles, if you're able to take one marble out of the bag, it's green, you put it down on the table, then put your hand back in the bag and take another marble. And if that one is also green, then you're going to win the prize. You're going to win the prize of $1 if you get two greens. We say, well, this sounds like an interesting game. How much does it cost to play? And the guy tells you it is $0.35 to play."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "You're going to win the prize of $1 if you get two greens. We say, well, this sounds like an interesting game. How much does it cost to play? And the guy tells you it is $0.35 to play. So obviously, fairly low stakes casino. So my question to you is, would you want to play this game? And don't put the fun factor into it."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "And the guy tells you it is $0.35 to play. So obviously, fairly low stakes casino. So my question to you is, would you want to play this game? And don't put the fun factor into it. Just economically, does it make sense for you to actually play this game? Well, let's think through the probabilities a little bit. So first of all, what's the probability that the first marble you pick is green?"}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "And don't put the fun factor into it. Just economically, does it make sense for you to actually play this game? Well, let's think through the probabilities a little bit. So first of all, what's the probability that the first marble you pick is green? What's the probability that first marble is green? Actually, let me just write first green. Probability first green."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "So first of all, what's the probability that the first marble you pick is green? What's the probability that first marble is green? Actually, let me just write first green. Probability first green. Well, the total possible outcomes, there's five marbles here, all equally likely. So there's five possible outcomes. Three of them satisfy your event that the first is green."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Probability first green. Well, the total possible outcomes, there's five marbles here, all equally likely. So there's five possible outcomes. Three of them satisfy your event that the first is green. So there's a 3 5th probability that the first is green. So you have a 3 5th chance, 3 5th probability, I should say, that after that first pick, you're kind of still in the game. Now, what we really care about is your probability of winning the game."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Three of them satisfy your event that the first is green. So there's a 3 5th probability that the first is green. So you have a 3 5th chance, 3 5th probability, I should say, that after that first pick, you're kind of still in the game. Now, what we really care about is your probability of winning the game. You want the first to be green and the second green. Well, let's think about this a little bit. What is the probability that the first is green?"}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Now, what we really care about is your probability of winning the game. You want the first to be green and the second green. Well, let's think about this a little bit. What is the probability that the first is green? First, I'll just write g for green. And the second is green. Now, you might be tempted to say, oh, well, maybe the second being green is the same probability."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "What is the probability that the first is green? First, I'll just write g for green. And the second is green. Now, you might be tempted to say, oh, well, maybe the second being green is the same probability. It's 3 5ths. I can just multiply 3 5ths times 3 5ths, and I'll get 9 over 25. Seems like a pretty straightforward thing."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Now, you might be tempted to say, oh, well, maybe the second being green is the same probability. It's 3 5ths. I can just multiply 3 5ths times 3 5ths, and I'll get 9 over 25. Seems like a pretty straightforward thing. But the realization here is what you do with that first green marble. You don't take that first green marble out, look at it, and put it back in the bag. So when you take that second pick, the number of green marbles that are in the bag depends on what you got on the first pick."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Seems like a pretty straightforward thing. But the realization here is what you do with that first green marble. You don't take that first green marble out, look at it, and put it back in the bag. So when you take that second pick, the number of green marbles that are in the bag depends on what you got on the first pick. Remember, we take the marble out. If it's a green marble, whatever marble it is, at whatever after the first pick, we leave it on the table. We are not replacing it."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "So when you take that second pick, the number of green marbles that are in the bag depends on what you got on the first pick. Remember, we take the marble out. If it's a green marble, whatever marble it is, at whatever after the first pick, we leave it on the table. We are not replacing it. So there's not any replacement here. So these are not independent events. Let me make this clear."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "We are not replacing it. So there's not any replacement here. So these are not independent events. Let me make this clear. Not independent. Or in particular, the second pick is dependent on the first. If the first pick is green, then you don't have three green marbles in a bag of five."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Let me make this clear. Not independent. Or in particular, the second pick is dependent on the first. If the first pick is green, then you don't have three green marbles in a bag of five. If the first pick is green, you now have two green marbles in a bag of four. So the way that we would refer to this, the probability of both of these happening, yes, it's definitely equal to the probability of the first green times, now this is kind of the new idea, the probability of the second green given, this little line right over here, just this straight up vertical line, just means given that the first was green. Now what is the probability that the second marble is green given that the first marble was green?"}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "If the first pick is green, then you don't have three green marbles in a bag of five. If the first pick is green, you now have two green marbles in a bag of four. So the way that we would refer to this, the probability of both of these happening, yes, it's definitely equal to the probability of the first green times, now this is kind of the new idea, the probability of the second green given, this little line right over here, just this straight up vertical line, just means given that the first was green. Now what is the probability that the second marble is green given that the first marble was green? Well, we draw through the scenario right over here. If the first marble is green, there are four possible outcomes, not five anymore. And two of them satisfy your criteria."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "Now what is the probability that the second marble is green given that the first marble was green? Well, we draw through the scenario right over here. If the first marble is green, there are four possible outcomes, not five anymore. And two of them satisfy your criteria. So the probability of the first marble being green and the second marble being green is going to be the probability that your first is green, so it's going to be 3 5ths, times the probability that the second is green given that the first was green. Now you have one less marble in the bag, and we're assuming that the first pick was green, so you only have two green marbles left. And so what does this give us for our total probability?"}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "And two of them satisfy your criteria. So the probability of the first marble being green and the second marble being green is going to be the probability that your first is green, so it's going to be 3 5ths, times the probability that the second is green given that the first was green. Now you have one less marble in the bag, and we're assuming that the first pick was green, so you only have two green marbles left. And so what does this give us for our total probability? Well, let's see, 3 5ths times 2 4ths. Well, 2 4ths is the same thing as 1 half. This is going to be equal to 3 5ths times 1 half, which is equal to 3 tenths."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "And so what does this give us for our total probability? Well, let's see, 3 5ths times 2 4ths. Well, 2 4ths is the same thing as 1 half. This is going to be equal to 3 5ths times 1 half, which is equal to 3 tenths. Or we could write that as 0.30. Or we could say there's a 30% chance of picking two green marbles when we are not replacing. So given that, let me ask you the question again."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "This is going to be equal to 3 5ths times 1 half, which is equal to 3 tenths. Or we could write that as 0.30. Or we could say there's a 30% chance of picking two green marbles when we are not replacing. So given that, let me ask you the question again. Would you want to play this game? Well, if you played this game many, many, many, many, many times, on average, you have a 30% chance of winning $1. And we haven't covered this yet, but so your expected value is really going to be 30% times $1."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "So given that, let me ask you the question again. Would you want to play this game? Well, if you played this game many, many, many, many, many times, on average, you have a 30% chance of winning $1. And we haven't covered this yet, but so your expected value is really going to be 30% times $1. This gives you a little bit of a preview, which is going to be $0.30. 30% chance of winning $1. You would expect, on average, if you play this many, many, many times, that playing the game is going to give you $0.30."}, {"video_title": "Dependent probability introduction Probability and Statistics Khan Academy.mp3", "Sentence": "And we haven't covered this yet, but so your expected value is really going to be 30% times $1. This gives you a little bit of a preview, which is going to be $0.30. 30% chance of winning $1. You would expect, on average, if you play this many, many, many times, that playing the game is going to give you $0.30. Now, would you want to give someone $0.35 to get, on average, $0.30? No, you would not want to play this game. Now, one thing I will let you think about is, would you want to play this game if you could replace the green marble, the first pick after the first pick?"}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Represent the following data using a box and whiskers plot. Exclude the median when computing the quartiles. All right, let's see if we can do this. So we have a bunch of data here, and they say if it helps, you might drag the numbers to put them in a different order so we can drag these numbers around, which is useful because we will want to order them. The order isn't checked with your answer. I'm doing this off of the Khan Academy exercises, so I don't have my drawing tablet here. I just have my mouse and I'm interacting with the exercise, which I encourage you to do too because the best way to learn any of this stuff is to actually practice it, and at Khan Academy we have 150,000 exercises for you to practice with."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So we have a bunch of data here, and they say if it helps, you might drag the numbers to put them in a different order so we can drag these numbers around, which is useful because we will want to order them. The order isn't checked with your answer. I'm doing this off of the Khan Academy exercises, so I don't have my drawing tablet here. I just have my mouse and I'm interacting with the exercise, which I encourage you to do too because the best way to learn any of this stuff is to actually practice it, and at Khan Academy we have 150,000 exercises for you to practice with. Anyway, so let's do this. Let's order this thing so we can figure out the range of numbers. What's the lowest and what's the highest?"}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "I just have my mouse and I'm interacting with the exercise, which I encourage you to do too because the best way to learn any of this stuff is to actually practice it, and at Khan Academy we have 150,000 exercises for you to practice with. Anyway, so let's do this. Let's order this thing so we can figure out the range of numbers. What's the lowest and what's the highest? So let's see, there's a seven here. Then let's see, we have some eights. We've got some eights going on."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "What's the lowest and what's the highest? So let's see, there's a seven here. Then let's see, we have some eights. We've got some eights going on. And then we have some nines. Actually, we have a bunch of nines. We have four nines here."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We've got some eights going on. And then we have some nines. Actually, we have a bunch of nines. We have four nines here. We have some nines. And then let's see, 13 is the largest number. There we go, we've ordered the numbers."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We have four nines here. We have some nines. And then let's see, 13 is the largest number. There we go, we've ordered the numbers. So our smallest number is seven. And this is what the whiskers are useful for, for helping us figure out the entire range of numbers. Our smallest number is seven."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There we go, we've ordered the numbers. So our smallest number is seven. And this is what the whiskers are useful for, for helping us figure out the entire range of numbers. Our smallest number is seven. Our largest number is 13. So we know the range. Now let's plot the median."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Our smallest number is seven. Our largest number is 13. So we know the range. Now let's plot the median. And this will help us. One's getting this center line of our box, but then also we need to do that to figure out what these other lines are that kind of define the box, to define the middle two fourths of our number, of our data, or the middle two quartiles, roughly the middle two quartiles. It depends how some of the numbers work out."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Now let's plot the median. And this will help us. One's getting this center line of our box, but then also we need to do that to figure out what these other lines are that kind of define the box, to define the middle two fourths of our number, of our data, or the middle two quartiles, roughly the middle two quartiles. It depends how some of the numbers work out. But this middle number, this middle line is going to be the median of our entire data set. Now the median is just the middle number. If we sort them in order, median is just the middle number."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It depends how some of the numbers work out. But this middle number, this middle line is going to be the median of our entire data set. Now the median is just the middle number. If we sort them in order, median is just the middle number. We have 11 numbers here. So the middle one is gonna have five on either side. So it's gonna be this nine."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "If we sort them in order, median is just the middle number. We have 11 numbers here. So the middle one is gonna have five on either side. So it's gonna be this nine. If we had 10 numbers here, if we had an even number of numbers, you actually would have had two middle numbers. And then to find the median, you would have found the mean of those two. If that last sentence was confusing, watch the videos on Khan Academy on median."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So it's gonna be this nine. If we had 10 numbers here, if we had an even number of numbers, you actually would have had two middle numbers. And then to find the median, you would have found the mean of those two. If that last sentence was confusing, watch the videos on Khan Academy on median. And I go into much more detail on that. But here I have 11 numbers. So my median is going to be the middle one."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "If that last sentence was confusing, watch the videos on Khan Academy on median. And I go into much more detail on that. But here I have 11 numbers. So my median is going to be the middle one. It has five larger, five less. It's this nine right over here. If I had my pen tablet, I would circle it."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So my median is going to be the middle one. It has five larger, five less. It's this nine right over here. If I had my pen tablet, I would circle it. So it's this nine. That is the median. And now we need to figure out, well, what number is halfway, or let me put it this way, what number is the median of the numbers in this bottom half?"}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "If I had my pen tablet, I would circle it. So it's this nine. That is the median. And now we need to figure out, well, what number is halfway, or let me put it this way, what number is the median of the numbers in this bottom half? And they told us to exclude the median when we compute the quartiles. So this was the median. Let's ignore that."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And now we need to figure out, well, what number is halfway, or let me put it this way, what number is the median of the numbers in this bottom half? And they told us to exclude the median when we compute the quartiles. So this was the median. Let's ignore that. So let's look at all the numbers below that. So this nine, eight, eight, eight, and seven. So we have five numbers."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's ignore that. So let's look at all the numbers below that. So this nine, eight, eight, eight, and seven. So we have five numbers. What's the median of these five numbers? Well, the median's the middle number. That is eight."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So we have five numbers. What's the median of these five numbers? Well, the median's the middle number. That is eight. So the beginning of our second quartile is gonna be at eight right over there. And we do the same thing for our third quartile. Remember, this was our median of our entire data set."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "That is eight. So the beginning of our second quartile is gonna be at eight right over there. And we do the same thing for our third quartile. Remember, this was our median of our entire data set. Let's exclude it. Let's look at the top half of the numbers, so to speak. And there's five numbers here in order."}, {"video_title": "Box and whisker plot exercise example Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Remember, this was our median of our entire data set. Let's exclude it. Let's look at the top half of the numbers, so to speak. And there's five numbers here in order. So the middle one, the median of this, is 10. So that's gonna be the top of our second quartile. And just like that, we're done."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "And what I want to do is think about the outliers. And to help us with that, let's actually visualize this, the distribution of actual numbers. So let us do that. So here on a number line, I have all the numbers from one to 19. And let's see. We have two ones. So I could say that's one one, and then two ones."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So here on a number line, I have all the numbers from one to 19. And let's see. We have two ones. So I could say that's one one, and then two ones. We have one six, so let's put that six there. We have got two 13s. So we're going to go up here."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So I could say that's one one, and then two ones. We have one six, so let's put that six there. We have got two 13s. So we're going to go up here. One 13 and two 13s. Let's see. We have three 14s."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So we're going to go up here. One 13 and two 13s. Let's see. We have three 14s. So 14, 14, and 14. We have a couple of 15s. 15, 15."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "We have three 14s. So 14, 14, and 14. We have a couple of 15s. 15, 15. So 15, 15. We have one 16. So that's our 16 there."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "15, 15. So 15, 15. We have one 16. So that's our 16 there. We have three 18s. One, two, three. So one, two, and then three."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So that's our 16 there. We have three 18s. One, two, three. So one, two, and then three. And then we have a 19. Then we have a 19. So when you look visually at the distribution of numbers, it looks like the meat of the distribution, so to speak, is in this area right over here."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So one, two, and then three. And then we have a 19. Then we have a 19. So when you look visually at the distribution of numbers, it looks like the meat of the distribution, so to speak, is in this area right over here. And so some people might say, OK, we have three outliers. These two ones and the six. Some people might say, well, the six is kind of close enough."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So when you look visually at the distribution of numbers, it looks like the meat of the distribution, so to speak, is in this area right over here. And so some people might say, OK, we have three outliers. These two ones and the six. Some people might say, well, the six is kind of close enough. Maybe only these two ones are outliers. And those would actually be both reasonable things to say. Now to get on the same page, statisticians will use a rule sometimes."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Some people might say, well, the six is kind of close enough. Maybe only these two ones are outliers. And those would actually be both reasonable things to say. Now to get on the same page, statisticians will use a rule sometimes. We say, well, anything that is more than 1 and 1 1 times the interquartile range from below Q1 or above Q3, well, those are going to be outliers. Well, what am I talking about? Well, actually, let's figure out the median, Q1 and Q3 here."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Now to get on the same page, statisticians will use a rule sometimes. We say, well, anything that is more than 1 and 1 1 times the interquartile range from below Q1 or above Q3, well, those are going to be outliers. Well, what am I talking about? Well, actually, let's figure out the median, Q1 and Q3 here. Then we can figure out the interquartile range. And then we can figure out by that definition what is going to be an outlier. And if that all made sense to you so far, I encourage you to pause this video and try to work through it on your own."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Well, actually, let's figure out the median, Q1 and Q3 here. Then we can figure out the interquartile range. And then we can figure out by that definition what is going to be an outlier. And if that all made sense to you so far, I encourage you to pause this video and try to work through it on your own. Or I'll do it for you right now. All right, so what's the median here? Well, the median is the middle number."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "And if that all made sense to you so far, I encourage you to pause this video and try to work through it on your own. Or I'll do it for you right now. All right, so what's the median here? Well, the median is the middle number. We have 15 numbers. So the middle number is going to be whatever number has seven on either side. So that's going to be the eighth number."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Well, the median is the middle number. We have 15 numbers. So the middle number is going to be whatever number has seven on either side. So that's going to be the eighth number. 1, 2, 3, 4, 5, 6, 7. Is that right? Yep, 6, 7."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So that's going to be the eighth number. 1, 2, 3, 4, 5, 6, 7. Is that right? Yep, 6, 7. So that's the median. And you have 1, 2, 3, 4, 5, 6, 7 numbers on the right side, too. So that is the median, sometimes called Q2."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Yep, 6, 7. So that's the median. And you have 1, 2, 3, 4, 5, 6, 7 numbers on the right side, too. So that is the median, sometimes called Q2. That is our median. Now, what is Q1? Well, Q1 is going to be the middle of this first group."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So that is the median, sometimes called Q2. That is our median. Now, what is Q1? Well, Q1 is going to be the middle of this first group. This first group has seven numbers in it. And so the middle is going to be the fourth number. It has 3 and 3."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Well, Q1 is going to be the middle of this first group. This first group has seven numbers in it. And so the middle is going to be the fourth number. It has 3 and 3. 3 to the left, 3 to the right. So that is Q1. And then Q3 is going to be the middle of this upper group."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "It has 3 and 3. 3 to the left, 3 to the right. So that is Q1. And then Q3 is going to be the middle of this upper group. Well, that also has seven numbers in it. So the middle is going to be right over there. It has 3 on either side."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "And then Q3 is going to be the middle of this upper group. Well, that also has seven numbers in it. So the middle is going to be right over there. It has 3 on either side. So that is Q3. Now, what is the interquartile range going to be? Interquartile range is going to be equal to Q3 minus Q1."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "It has 3 on either side. So that is Q3. Now, what is the interquartile range going to be? Interquartile range is going to be equal to Q3 minus Q1. The difference between 18 and 13. Between 18 and 13, well, that is going to be 18 minus 13, which is equal to 5. Now, to figure out outliers, well, outliers are going to be anything that is below."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Interquartile range is going to be equal to Q3 minus Q1. The difference between 18 and 13. Between 18 and 13, well, that is going to be 18 minus 13, which is equal to 5. Now, to figure out outliers, well, outliers are going to be anything that is below. So outliers are going to be less than our Q1 minus 1.5 times our interquartile range. And once again, this isn't some rule of the universe. This is something that statisticians have kind of said, well, if we want to have a better definition for outliers, let's just agree that it's something that's more than 1.5 times the interquartile range below Q1."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Now, to figure out outliers, well, outliers are going to be anything that is below. So outliers are going to be less than our Q1 minus 1.5 times our interquartile range. And once again, this isn't some rule of the universe. This is something that statisticians have kind of said, well, if we want to have a better definition for outliers, let's just agree that it's something that's more than 1.5 times the interquartile range below Q1. Or an outlier could be greater than Q3 plus 1.5 times the interquartile range. And once again, this is somewhat, people just decided it felt right. One could argue it should be 1.6."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "This is something that statisticians have kind of said, well, if we want to have a better definition for outliers, let's just agree that it's something that's more than 1.5 times the interquartile range below Q1. Or an outlier could be greater than Q3 plus 1.5 times the interquartile range. And once again, this is somewhat, people just decided it felt right. One could argue it should be 1.6. Or one could argue it should be 1 or 2 or whatever. But this is what people have tended to agree on. So let's think about what these numbers are."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "One could argue it should be 1.6. Or one could argue it should be 1 or 2 or whatever. But this is what people have tended to agree on. So let's think about what these numbers are. Q1, we already know. So this is going to be 13 minus 1.5 times our interquartile range. Our interquartile range here is 5."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So let's think about what these numbers are. Q1, we already know. So this is going to be 13 minus 1.5 times our interquartile range. Our interquartile range here is 5. So it's 1.5 times 5, which is 7.5. So this is 7.5. 13 minus 7.5 is what?"}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Our interquartile range here is 5. So it's 1.5 times 5, which is 7.5. So this is 7.5. 13 minus 7.5 is what? 13 minus 7 is 6. And then you subtract another 0.5 is 5.5. So we have outliers would be less than 5.5."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "13 minus 7.5 is what? 13 minus 7 is 6. And then you subtract another 0.5 is 5.5. So we have outliers would be less than 5.5. Or Q3 is 18. This is, once again, 7.5. 18 plus 7.5 is 25.5."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So we have outliers would be less than 5.5. Or Q3 is 18. This is, once again, 7.5. 18 plus 7.5 is 25.5. Or outliers greater than 25.5. So based on this, we have a numerical definition for what's an outlier. We're not just subjectively saying, oh, this feels right or that feels right."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "18 plus 7.5 is 25.5. Or outliers greater than 25.5. So based on this, we have a numerical definition for what's an outlier. We're not just subjectively saying, oh, this feels right or that feels right. And based on this, we only have two outliers, that only these two ones are less than 5.5. This is the cutoff right over here. So this dot just happened to make it."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "We're not just subjectively saying, oh, this feels right or that feels right. And based on this, we only have two outliers, that only these two ones are less than 5.5. This is the cutoff right over here. So this dot just happened to make it. And we don't have any outliers on the high side. Now, another thing to think about is drawing box and whiskers plots based on Q1, our median, our range, all the range of numbers. And you could do it either taking in consideration your outliers or not taking into consideration your outliers."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So this dot just happened to make it. And we don't have any outliers on the high side. Now, another thing to think about is drawing box and whiskers plots based on Q1, our median, our range, all the range of numbers. And you could do it either taking in consideration your outliers or not taking into consideration your outliers. So there's a couple of ways that we can do it. So let me actually clear all of this. We've figured out all of this stuff."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "And you could do it either taking in consideration your outliers or not taking into consideration your outliers. So there's a couple of ways that we can do it. So let me actually clear all of this. We've figured out all of this stuff. So let me clear all of that out. And let's actually draw a box and whiskers plot. So I'll put another, actually, let me do two here."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "We've figured out all of this stuff. So let me clear all of that out. And let's actually draw a box and whiskers plot. So I'll put another, actually, let me do two here. That's one. And then let me put another one down there. This is another."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So I'll put another, actually, let me do two here. That's one. And then let me put another one down there. This is another. Now, if we were to just draw a classic box and whiskers plot here, we would say, all right, our median's at 14. And actually, I'll do it both ways. Median's at 14."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "This is another. Now, if we were to just draw a classic box and whiskers plot here, we would say, all right, our median's at 14. And actually, I'll do it both ways. Median's at 14. Q1's at 13. Q3 is at 18. So that's the box part."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Median's at 14. Q1's at 13. Q3 is at 18. So that's the box part. And let me draw that as an actual, let me actually draw that as a box. So my best attempt. There you go."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So that's the box part. And let me draw that as an actual, let me actually draw that as a box. So my best attempt. There you go. That's the box. And this is also a box. So far, I'm doing the exact same thing."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "There you go. That's the box. And this is also a box. So far, I'm doing the exact same thing. Now, if we don't want to consider outliers, we would say, well, what's the entire range here? Well, we have things that go from 1 all the way to 19. So one way to do it is to say, hey, we start at 1."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So far, I'm doing the exact same thing. Now, if we don't want to consider outliers, we would say, well, what's the entire range here? Well, we have things that go from 1 all the way to 19. So one way to do it is to say, hey, we start at 1. And so our entire range, we go, actually, let me draw it a little bit better than that. We're going all the way from 1 to 19. Now, in this one, we're including everything."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "So one way to do it is to say, hey, we start at 1. And so our entire range, we go, actually, let me draw it a little bit better than that. We're going all the way from 1 to 19. Now, in this one, we're including everything. We're including even these two outliers. But if we don't want to include those outliers, we want to make it clear that they're outliers, well, let's not include them. And what we can do instead is say, all right, including our non-outliers, we would start at 6."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "Now, in this one, we're including everything. We're including even these two outliers. But if we don't want to include those outliers, we want to make it clear that they're outliers, well, let's not include them. And what we can do instead is say, all right, including our non-outliers, we would start at 6. Because 6, we're saying, is in our data set. But it is not an outlier. Let me make this look better."}, {"video_title": "Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3", "Sentence": "And what we can do instead is say, all right, including our non-outliers, we would start at 6. Because 6, we're saying, is in our data set. But it is not an outlier. Let me make this look better. So we are going to start at 6 and go all the way to 19. And then to say that we have these outliers, we would put this, we have outliers over there. So once again, this is a box and whiskers plot of the same data set without outliers."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "He decides to start with an online poll. He asks his listeners to visit his website and participate in the poll. The poll shows that 89% of about 200 respondents love his show. What is the most concerning source of bias in this scenario? And like always, pause this video and see if you can figure it out on your own and then we'll work through it together. Let's think about what's going on. He has this population of listeners right over here."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "What is the most concerning source of bias in this scenario? And like always, pause this video and see if you can figure it out on your own and then we'll work through it together. Let's think about what's going on. He has this population of listeners right over here. I'll assume that the number of listeners is more than 200. And he says, hey, I wanna find a sample and I can't ask all of my listeners. Who knows, maybe he has 10,000 listeners."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "He has this population of listeners right over here. I'll assume that the number of listeners is more than 200. And he says, hey, I wanna find a sample and I can't ask all of my listeners. Who knows, maybe he has 10,000 listeners. They don't tell us that. But let's say there's 10,000 listeners here and he says, well, I wanna get an indication of what percent like my show. So I need a sample."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "Who knows, maybe he has 10,000 listeners. They don't tell us that. But let's say there's 10,000 listeners here and he says, well, I wanna get an indication of what percent like my show. So I need a sample. But instead of taking a truly random sample, he asks them to volunteer. He asks his listeners to visit his website. So that's classic volunteer response sampling."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "So I need a sample. But instead of taking a truly random sample, he asks them to volunteer. He asks his listeners to visit his website. So that's classic volunteer response sampling. This is non-random because who decides to go to his website and listen to what he just said and maybe even has access to a computer? That's not random. In fact, the people more likely to do that, so these are the people out of the 10,000."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "So that's classic volunteer response sampling. This is non-random because who decides to go to his website and listen to what he just said and maybe even has access to a computer? That's not random. In fact, the people more likely to do that, so these are the people out of the 10,000. So these are the 200 responses here who decide to do it. These are more likely to be the people who already like David or like to listen to what he tells them to do. The people, the listeners who are not into David or don't wanna do what he tells them to do, well, they're unlikely to be in the say, oh, I'm not really into David and I don't like him telling me what to do, but hey, I'm gonna go to his website anyway."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "In fact, the people more likely to do that, so these are the people out of the 10,000. So these are the 200 responses here who decide to do it. These are more likely to be the people who already like David or like to listen to what he tells them to do. The people, the listeners who are not into David or don't wanna do what he tells them to do, well, they're unlikely to be in the say, oh, I'm not really into David and I don't like him telling me what to do, but hey, I'm gonna go to his website anyway. I'm gonna fill out that poll. That's less likely. Or you might get extreme."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "The people, the listeners who are not into David or don't wanna do what he tells them to do, well, they're unlikely to be in the say, oh, I'm not really into David and I don't like him telling me what to do, but hey, I'm gonna go to his website anyway. I'm gonna fill out that poll. That's less likely. Or you might get extreme. So people who really don't like him might say, I'm gonna definitely go there. But in this case, I would say that it's more likely your fans are gonna do what you ask them to do and go to your website and spend time on your website. And because of that, that 89% is probably an overestimate."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "Or you might get extreme. So people who really don't like him might say, I'm gonna definitely go there. But in this case, I would say that it's more likely your fans are gonna do what you ask them to do and go to your website and spend time on your website. And because of that, that 89% is probably an overestimate. 89% is probably an overestimate of the number of listeners who really love his show, because you're more likely to get the ones who love him to show up and fill out that actual survey. Now, these other forms of bias, response bias. This is when you're asking something that people don't necessarily wanna answer truthfully or the way that it's phrased, it might make someone respond, you say, in a biased way."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "And because of that, that 89% is probably an overestimate. 89% is probably an overestimate of the number of listeners who really love his show, because you're more likely to get the ones who love him to show up and fill out that actual survey. Now, these other forms of bias, response bias. This is when you're asking something that people don't necessarily wanna answer truthfully or the way that it's phrased, it might make someone respond, you say, in a biased way. Classic examples of this are like, have you lied to your parents in the past week? Or have you ever cheated on your spouse? Or something like that."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "This is when you're asking something that people don't necessarily wanna answer truthfully or the way that it's phrased, it might make someone respond, you say, in a biased way. Classic examples of this are like, have you lied to your parents in the past week? Or have you ever cheated on your spouse? Or something like that. Or have you smoked? Any of these things that people might not wanna answer completely truthfully or they might be hiding from the world and might not just wanna answer that truthfully on a survey. And so you're gonna have response bias."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "Or something like that. Or have you smoked? Any of these things that people might not wanna answer completely truthfully or they might be hiding from the world and might not just wanna answer that truthfully on a survey. And so you're gonna have response bias. But that's not the case right over here. And undercoverage is when the way that you're sampling, you're definitely missing out on an important constituency. You know, voluntary response, we're likely missing out on some important constituencies, on some people who might not be into going to your website."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "And so you're gonna have response bias. But that's not the case right over here. And undercoverage is when the way that you're sampling, you're definitely missing out on an important constituency. You know, voluntary response, we're likely missing out on some important constituencies, on some people who might not be into going to your website. But undercoverage is where it's a little bit more clear that that is happening. Now let's do another case. Let's do another case, maybe an alternate reality, where David's trying to figure this out again."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "You know, voluntary response, we're likely missing out on some important constituencies, on some people who might not be into going to your website. But undercoverage is where it's a little bit more clear that that is happening. Now let's do another case. Let's do another case, maybe an alternate reality, where David's trying to figure this out again. He's still hosting a podcast, and he's still curious how much his listeners like his show, but he tries to take a different sample. He decides, in this case, to poll the next 100 listeners who send him fan emails. They don't all respond, but 94 out of the 97 listeners polled say they loved his show."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "Let's do another case, maybe an alternate reality, where David's trying to figure this out again. He's still hosting a podcast, and he's still curious how much his listeners like his show, but he tries to take a different sample. He decides, in this case, to poll the next 100 listeners who send him fan emails. They don't all respond, but 94 out of the 97 listeners polled say they loved his show. What is the most concerning source of bias in this scenario? Well, this is a classic, hey, I have a group, I have a sample sitting in front of me, it's in my inbox on my email, let me just go to them. Isn't that convenient?"}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "They don't all respond, but 94 out of the 97 listeners polled say they loved his show. What is the most concerning source of bias in this scenario? Well, this is a classic, hey, I have a group, I have a sample sitting in front of me, it's in my inbox on my email, let me just go to them. Isn't that convenient? So this is classic convenience sample. And this isn't just like, hey, you know, these are the first 100 people to walk through the door, and there's, you know, a lot of times you could argue why that might be not so random, but these are the next 100 listeners who sent him fan emails. So this is convenience sampling, and the sample that you happen to use out of convenience is one that's going to be very skewed to liking you."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "Isn't that convenient? So this is classic convenience sample. And this isn't just like, hey, you know, these are the first 100 people to walk through the door, and there's, you know, a lot of times you could argue why that might be not so random, but these are the next 100 listeners who sent him fan emails. So this is convenience sampling, and the sample that you happen to use out of convenience is one that's going to be very skewed to liking you. So once again, this is overestimating, overestimating the percent, the percent that love his show. Now, nonresponse is when you ask a certain number of people to fill out a survey, or to answer a questionnaire, and for some reason, some percent do not fill it out, and you're like, well, who were those people? Maybe they would have said something important, and maybe their viewpoint is not properly represented in the overall number that actually did fill it out."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "So this is convenience sampling, and the sample that you happen to use out of convenience is one that's going to be very skewed to liking you. So once again, this is overestimating, overestimating the percent, the percent that love his show. Now, nonresponse is when you ask a certain number of people to fill out a survey, or to answer a questionnaire, and for some reason, some percent do not fill it out, and you're like, well, who were those people? Maybe they would have said something important, and maybe their viewpoint is not properly represented in the overall number that actually did fill it out. And there is some nonresponse going on here. He asks 100 people who sent fan emails to fill out the survey, to say whether they love it or not, 97 fill it out, so there were three people who did not fill out the survey. So there is some nonresponse going on that would be a source of bias, but it's not the most concerning."}, {"video_title": "Examples of bias in surveys Study design AP Statistics Khan Academy.mp3", "Sentence": "Maybe they would have said something important, and maybe their viewpoint is not properly represented in the overall number that actually did fill it out. And there is some nonresponse going on here. He asks 100 people who sent fan emails to fill out the survey, to say whether they love it or not, 97 fill it out, so there were three people who did not fill out the survey. So there is some nonresponse going on that would be a source of bias, but it's not the most concerning. You know, right over here, they're asking us, fill out the most concerning source of bias, and the convenience sampling is definitely the biggest deal here. There were three people who didn't respond, but that's not as big of a deal. Voluntary response sampling, well, he didn't ask people, like in the last example, like, hey, if you can go here and fill it out, I guess there is actually, actually, no, take that back."}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "The dot plot shows the number of hours of daily driving time for 14 school bus drivers. Each dot represents a driver. So for example, one driver drives one hour a day. Two drivers drive two hours a day. One driver drives three hours a day. It looks like there's five drivers that drive seven hours a day. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours?"}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Two drivers drive two hours a day. One driver drives three hours a day. It looks like there's five drivers that drive seven hours a day. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? And then they give us some choices. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? So pause the video and see if you can figure out which of these percentiles is the closest estimate to the percentile rank of a driver with a daily driving time of six hours, looking at this data right over here."}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? And then they give us some choices. Which of the following is the closest estimate to the percentile rank for the driver with a daily driving time of six hours? So pause the video and see if you can figure out which of these percentiles is the closest estimate to the percentile rank of a driver with a daily driving time of six hours, looking at this data right over here. All right, now let's work through this together. So when you think about percentile, you really wanna think about, so let me write this down. When we're talking about percentile, we're really saying the percentage of the data that, and there's actually two ways that you could compute it."}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So pause the video and see if you can figure out which of these percentiles is the closest estimate to the percentile rank of a driver with a daily driving time of six hours, looking at this data right over here. All right, now let's work through this together. So when you think about percentile, you really wanna think about, so let me write this down. When we're talking about percentile, we're really saying the percentage of the data that, and there's actually two ways that you could compute it. One is the percentage of the data that is below the amount in question, amount in question. The other possibility is the percent of the data that is at or below, that is at or below the amount, the amount in question. So if we look at this right over here, let's just figure out how many data points, what percentage of the data points are below six hours per day?"}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "When we're talking about percentile, we're really saying the percentage of the data that, and there's actually two ways that you could compute it. One is the percentage of the data that is below the amount in question, amount in question. The other possibility is the percent of the data that is at or below, that is at or below the amount, the amount in question. So if we look at this right over here, let's just figure out how many data points, what percentage of the data points are below six hours per day? So let's see, there are, I'm just gonna count them. One, two, three, four, five, six, seven. So seven of the 14 are below six hours."}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So if we look at this right over here, let's just figure out how many data points, what percentage of the data points are below six hours per day? So let's see, there are, I'm just gonna count them. One, two, three, four, five, six, seven. So seven of the 14 are below six hours. So we could just say seven, if we use this first technique, we would have seven of the 14 are below six hours per day, and so that would get us a number of 50%, that six hours is at the 50th percentile. If we wanna say what percentage is at that number or below, then we would also count this one. So we would say eight or eight out of 14, eight out of 14, which is the same thing as four out of seven and if we wanna write that as a decimal, let's see, seven goes into 4.000, we just need to estimate."}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So seven of the 14 are below six hours. So we could just say seven, if we use this first technique, we would have seven of the 14 are below six hours per day, and so that would get us a number of 50%, that six hours is at the 50th percentile. If we wanna say what percentage is at that number or below, then we would also count this one. So we would say eight or eight out of 14, eight out of 14, which is the same thing as four out of seven and if we wanna write that as a decimal, let's see, seven goes into 4.000, we just need to estimate. So seven goes into 45 times 35, we subtract, we get a five, bring down a zero, goes five times, I guess it's just gonna be.5 repeating. So 55.5555%. So either of these would actually be a legitimate response to the percentile rank for the driver with a daily driving time of six hours."}, {"video_title": "Calculating percentile Modeling data distributions AP Statistics Khan Academy.mp3", "Sentence": "So we would say eight or eight out of 14, eight out of 14, which is the same thing as four out of seven and if we wanna write that as a decimal, let's see, seven goes into 4.000, we just need to estimate. So seven goes into 45 times 35, we subtract, we get a five, bring down a zero, goes five times, I guess it's just gonna be.5 repeating. So 55.5555%. So either of these would actually be a legitimate response to the percentile rank for the driver with a daily driving time of six hours. It depends on whether you include the six hours or not. So you could say either the 50th percentile or the roughly the 55th, well actually the 56th percentile if you wanted to round to the nearest percentile. Now if you look at these choices here, lucky for us, there's only one choice that's reasonably close to either one of those and that's the 55th percentile."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "A set of philosophy exam scores are normally distributed with a mean of 40 points and a standard deviation of three points. Ludwig got a score of 47.5 points on the exam. What proportion of exam scores are higher than Ludwig's score? Give your answer correct to four decimal places. So let's just visualize what's going on here. So the scores are normally distributed. So it would look like this."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "Give your answer correct to four decimal places. So let's just visualize what's going on here. So the scores are normally distributed. So it would look like this. So the distribution would look something like that. Trying to make that pretty symmetric looking. The mean is 40 points."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So it would look like this. So the distribution would look something like that. Trying to make that pretty symmetric looking. The mean is 40 points. So that would be 40 points right over there. Standard deviation is three points. So this could be one standard deviation above the mean."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "The mean is 40 points. So that would be 40 points right over there. Standard deviation is three points. So this could be one standard deviation above the mean. That would be one standard deviation below the mean. And once again, this is just very rough. And so this would be 43."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So this could be one standard deviation above the mean. That would be one standard deviation below the mean. And once again, this is just very rough. And so this would be 43. This would be 37 right over here. And they say Ludwig got a score of 47.5 points on the exam. So Ludwig's score is going to be someplace around here."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "And so this would be 43. This would be 37 right over here. And they say Ludwig got a score of 47.5 points on the exam. So Ludwig's score is going to be someplace around here. So Ludwig got a 47.5 on the exam. And they're saying what proportion of exam scores are higher than Ludwig's score? So what we need to do is figure out what is the area under the normal distribution curve that is above 47.5?"}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So Ludwig's score is going to be someplace around here. So Ludwig got a 47.5 on the exam. And they're saying what proportion of exam scores are higher than Ludwig's score? So what we need to do is figure out what is the area under the normal distribution curve that is above 47.5? So the way we will tackle this is we will figure out the z-score for 47.5. How many standard deviations above the mean is that? Then we will look at a z-table to figure out what proportion is below that."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So what we need to do is figure out what is the area under the normal distribution curve that is above 47.5? So the way we will tackle this is we will figure out the z-score for 47.5. How many standard deviations above the mean is that? Then we will look at a z-table to figure out what proportion is below that. Because that's what z-tables give us. They give us the proportion that is below a certain z-score. And then we could take one minus that to figure out the proportion that is above."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "Then we will look at a z-table to figure out what proportion is below that. Because that's what z-tables give us. They give us the proportion that is below a certain z-score. And then we could take one minus that to figure out the proportion that is above. Remember, the entire area under the curve is one. So if we can figure out this orange area and take one minus that, we're gonna get the red area. So let's do that."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "And then we could take one minus that to figure out the proportion that is above. Remember, the entire area under the curve is one. So if we can figure out this orange area and take one minus that, we're gonna get the red area. So let's do that. So first of all, let's figure out the z-score for 47.5. So let's see. We would take 47.5 and we would subtract the mean."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So let's do that. So first of all, let's figure out the z-score for 47.5. So let's see. We would take 47.5 and we would subtract the mean. So this is his score. We'll subtract the mean minus 40. We know what that's gonna be."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "We would take 47.5 and we would subtract the mean. So this is his score. We'll subtract the mean minus 40. We know what that's gonna be. That's 7.5. So that's how much more above the mean. But how many standard deviations is that?"}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "We know what that's gonna be. That's 7.5. So that's how much more above the mean. But how many standard deviations is that? Well, each standard deviation is three. So what's 7.5 divided by three? This just means the previous answer divided by three."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "But how many standard deviations is that? Well, each standard deviation is three. So what's 7.5 divided by three? This just means the previous answer divided by three. So he has 2.5 standard deviations above the mean. So the z-score here, z-score here is a positive 2.5. If he was below the mean, it would be a negative."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "This just means the previous answer divided by three. So he has 2.5 standard deviations above the mean. So the z-score here, z-score here is a positive 2.5. If he was below the mean, it would be a negative. So now we can look at a z-table to figure out what proportion is less than 2.5 standard deviations above the mean. So that'll give us that orange and then we'll subtract that from one. So let's get our z-table."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "If he was below the mean, it would be a negative. So now we can look at a z-table to figure out what proportion is less than 2.5 standard deviations above the mean. So that'll give us that orange and then we'll subtract that from one. So let's get our z-table. So here we go. And we've already done this in previous videos. But what's going on here is this left column gives us our z-score up to the tenths place."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So let's get our z-table. So here we go. And we've already done this in previous videos. But what's going on here is this left column gives us our z-score up to the tenths place. And then these other columns give us the hundredths place. So what we wanna do is find 2.5 right over here on the left. And it's actually gonna be 2.50."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "But what's going on here is this left column gives us our z-score up to the tenths place. And then these other columns give us the hundredths place. So what we wanna do is find 2.5 right over here on the left. And it's actually gonna be 2.50. There's no, there's zero hundredths here. So we're gonna, we wanna look up 2.50. So let me scroll my z-table."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "And it's actually gonna be 2.50. There's no, there's zero hundredths here. So we're gonna, we wanna look up 2.50. So let me scroll my z-table. So I'm gonna go down to 2.5. Alright, I think I am there. So what I have here, so I have 2.5."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So let me scroll my z-table. So I'm gonna go down to 2.5. Alright, I think I am there. So what I have here, so I have 2.5. So I am going to be in this row. And it's now scrolled off, but this first column we saw, this is the hundredths place and this is zero hundredths. And so 2.50 puts us right over here."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So what I have here, so I have 2.5. So I am going to be in this row. And it's now scrolled off, but this first column we saw, this is the hundredths place and this is zero hundredths. And so 2.50 puts us right over here. Now you might be tempted to say, okay, that's the proportion that scores higher than Ludwig, but you'd be wrong. This is the proportion that scores lower than Ludwig. So what we wanna do is take one minus this value."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "And so 2.50 puts us right over here. Now you might be tempted to say, okay, that's the proportion that scores higher than Ludwig, but you'd be wrong. This is the proportion that scores lower than Ludwig. So what we wanna do is take one minus this value. So let me get my calculator out again. So what I'm going to do is I'm going to take one minus this. One minus 0.9938 is equal to, now this is, so this is the proportion that scores less than Ludwig."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So what we wanna do is take one minus this value. So let me get my calculator out again. So what I'm going to do is I'm going to take one minus this. One minus 0.9938 is equal to, now this is, so this is the proportion that scores less than Ludwig. One minus that is gonna be the proportion that scores more than him. The reason why we have to do this is because the z-table gives us the proportion less than a certain z-score. So this gives us right over here, 0.0062."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "One minus 0.9938 is equal to, now this is, so this is the proportion that scores less than Ludwig. One minus that is gonna be the proportion that scores more than him. The reason why we have to do this is because the z-table gives us the proportion less than a certain z-score. So this gives us right over here, 0.0062. So that's the proportion. If you thought of it in percent, it would be.62% scores higher than Ludwig. And that makes sense, because Ludwig scored over two standard deviations, two and a half standard deviations above the mean."}, {"video_title": "Standard normal table for proportion above AP Statistics Khan Academy.mp3", "Sentence": "So this gives us right over here, 0.0062. So that's the proportion. If you thought of it in percent, it would be.62% scores higher than Ludwig. And that makes sense, because Ludwig scored over two standard deviations, two and a half standard deviations above the mean. So our answer here is 0.0062. So this is going to be 0.0062. That's the proportion of exam scores higher than Ludwig's score."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "I think they have some great articles and they have some great information on their site. But what I want to do here is to think about what a lot of articles you might read or a lot of research you might read are implying and to think about whether they really imply what they claim to be implying. So this is an excerpt of an article. And the title of the article says, eating breakfast may beat teen obesity. So they're already trying to kind of create this cause and effect relationship. The title itself says, if you eat breakfast, then you're less likely or you won't be obese. You're not going to be obese."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "And the title of the article says, eating breakfast may beat teen obesity. So they're already trying to kind of create this cause and effect relationship. The title itself says, if you eat breakfast, then you're less likely or you won't be obese. You're not going to be obese. So the title right there already sets up this, that eating breakfast may beat teen obesity. And then they tell us about the study. In the study, published in Pediatrics, researchers analyzed the dietary and weight patterns of a group of 2,216 adolescents over a five-year period from public schools in Minneapolis, St. Paul, Minnesota."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "You're not going to be obese. So the title right there already sets up this, that eating breakfast may beat teen obesity. And then they tell us about the study. In the study, published in Pediatrics, researchers analyzed the dietary and weight patterns of a group of 2,216 adolescents over a five-year period from public schools in Minneapolis, St. Paul, Minnesota. And I won't talk too much about it. This looks like a good sample size. It was over a large period of time."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "In the study, published in Pediatrics, researchers analyzed the dietary and weight patterns of a group of 2,216 adolescents over a five-year period from public schools in Minneapolis, St. Paul, Minnesota. And I won't talk too much about it. This looks like a good sample size. It was over a large period of time. I'll just give the researchers the benefit of the doubt, assume that it was over a broad audience that they were able to control for a lot of variables. But then they go on to say, the researchers write that teens who ate breakfast regularly had a lower percentage of total calories from saturated fat and ate more fiber and carbohydrates. And to some degree, that first, then those who skipped breakfast, and to some degree this first sentence is obvious."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "It was over a large period of time. I'll just give the researchers the benefit of the doubt, assume that it was over a broad audience that they were able to control for a lot of variables. But then they go on to say, the researchers write that teens who ate breakfast regularly had a lower percentage of total calories from saturated fat and ate more fiber and carbohydrates. And to some degree, that first, then those who skipped breakfast, and to some degree this first sentence is obvious. Breakfast tends to be things like cereals, grains, you eat syrup, you eat waffles. That all tends to fall in the category of carbohydrates and sugars. And frankly, that's not even necessarily a good thing."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "And to some degree, that first, then those who skipped breakfast, and to some degree this first sentence is obvious. Breakfast tends to be things like cereals, grains, you eat syrup, you eat waffles. That all tends to fall in the category of carbohydrates and sugars. And frankly, that's not even necessarily a good thing. Not obvious to me whether bacon is more or less healthy than downing a bunch of syrup or Froot Loops or whatever else. But we'll let that be right here. In addition, regular breakfast eaters seemed more physically active than their breakfast skippers."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "And frankly, that's not even necessarily a good thing. Not obvious to me whether bacon is more or less healthy than downing a bunch of syrup or Froot Loops or whatever else. But we'll let that be right here. In addition, regular breakfast eaters seemed more physically active than their breakfast skippers. So over here, they're once again trying to create this other cause and effect relationship. Regular breakfast eaters seemed more physically active than their breakfast skippers. So the implication here is that breakfast makes you more active."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "In addition, regular breakfast eaters seemed more physically active than their breakfast skippers. So over here, they're once again trying to create this other cause and effect relationship. Regular breakfast eaters seemed more physically active than their breakfast skippers. So the implication here is that breakfast makes you more active. And then this last sentence right over here, they say, over time, researchers found teens who regularly ate breakfast tended to gain less weight and had a lower body mass index than breakfast skippers. So they're telling us that breakfast skipping, this is the implication here, is more likely or it can be a cause of making you overweight or maybe even making you obese. So the entire narrative here, from the title all the way through every paragraph, is look, breakfast prevents obesity."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "So the implication here is that breakfast makes you more active. And then this last sentence right over here, they say, over time, researchers found teens who regularly ate breakfast tended to gain less weight and had a lower body mass index than breakfast skippers. So they're telling us that breakfast skipping, this is the implication here, is more likely or it can be a cause of making you overweight or maybe even making you obese. So the entire narrative here, from the title all the way through every paragraph, is look, breakfast prevents obesity. Breakfast makes you active. Breakfast skipping will make you obese. So you just say, boy, I have to eat breakfast."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "So the entire narrative here, from the title all the way through every paragraph, is look, breakfast prevents obesity. Breakfast makes you active. Breakfast skipping will make you obese. So you just say, boy, I have to eat breakfast. And you should always think about the motivations and the industries around things like breakfast. But the more interesting question is, does this research really tell us that eating breakfast can prevent obesity? Does it really tell us that eating breakfast will cause someone to become more active?"}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "So you just say, boy, I have to eat breakfast. And you should always think about the motivations and the industries around things like breakfast. But the more interesting question is, does this research really tell us that eating breakfast can prevent obesity? Does it really tell us that eating breakfast will cause someone to become more active? Does it really tell us that breakfast skipping can make you overweight or make you obese? Or, it is more likely, are they just showing that these two things tend to go together? And this is a really important difference."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Does it really tell us that eating breakfast will cause someone to become more active? Does it really tell us that breakfast skipping can make you overweight or make you obese? Or, it is more likely, are they just showing that these two things tend to go together? And this is a really important difference. And let me kind of state slightly technical words here. And they sound fancy, but they really aren't that fancy. Are they pointing out causality?"}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "And this is a really important difference. And let me kind of state slightly technical words here. And they sound fancy, but they really aren't that fancy. Are they pointing out causality? Are they pointing out causality, which is what it seems like they're implying? Eating breakfast causes you to not be obese. Breakfast causes you to be active."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Are they pointing out causality? Are they pointing out causality, which is what it seems like they're implying? Eating breakfast causes you to not be obese. Breakfast causes you to be active. Breakfast skipping causes you to be obese. So it looks like they're kind of implying causality. They're implying cause and effect."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Breakfast causes you to be active. Breakfast skipping causes you to be obese. So it looks like they're kind of implying causality. They're implying cause and effect. But really, what the study looked at is correlation. So the whole point of this is to understand the difference between causality and correlation, because they're saying very different things. Causality versus correlation."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "They're implying cause and effect. But really, what the study looked at is correlation. So the whole point of this is to understand the difference between causality and correlation, because they're saying very different things. Causality versus correlation. And as I said, causality says A causes B. Well, correlation just says A and B tend to be observed at the same time. Whenever I see B happening, it looks like A is happening at the same time."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Causality versus correlation. And as I said, causality says A causes B. Well, correlation just says A and B tend to be observed at the same time. Whenever I see B happening, it looks like A is happening at the same time. Whenever A is happening, it looks like it also tends to happen with B. And the reason why it's super important to notice the distinction between these is you can come to very, very, very, very different conclusions. So the one thing that this research does do, assuming that it was performed well, is it does show a correlation."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Whenever I see B happening, it looks like A is happening at the same time. Whenever A is happening, it looks like it also tends to happen with B. And the reason why it's super important to notice the distinction between these is you can come to very, very, very, very different conclusions. So the one thing that this research does do, assuming that it was performed well, is it does show a correlation. So this study does show a correlation. It does show, if we believe all of their data, that breakfast skipping correlates with obesity, and obesity correlates with breakfast skipping. We're seeing it at the same time."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "So the one thing that this research does do, assuming that it was performed well, is it does show a correlation. So this study does show a correlation. It does show, if we believe all of their data, that breakfast skipping correlates with obesity, and obesity correlates with breakfast skipping. We're seeing it at the same time. Activity correlates with breakfast, and breakfast correlates with activity. All of these correlate. Well, they don't say, and there's no data here that lets me know one way or the other, what is causing what, or maybe you have some underlying cause that is causing both."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "We're seeing it at the same time. Activity correlates with breakfast, and breakfast correlates with activity. All of these correlate. Well, they don't say, and there's no data here that lets me know one way or the other, what is causing what, or maybe you have some underlying cause that is causing both. So for example, they're saying breakfast causes activity, or they're implying breakfast causes activity. They're not saying it explicitly. But maybe activity causes breakfast."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Well, they don't say, and there's no data here that lets me know one way or the other, what is causing what, or maybe you have some underlying cause that is causing both. So for example, they're saying breakfast causes activity, or they're implying breakfast causes activity. They're not saying it explicitly. But maybe activity causes breakfast. Maybe. They didn't write the study that people who are active, maybe they're more likely to be hungry in the morning. Activity causes breakfast."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "But maybe activity causes breakfast. Maybe. They didn't write the study that people who are active, maybe they're more likely to be hungry in the morning. Activity causes breakfast. And then you start having a different takeaway. Then you don't say, wait, maybe if you're active and you skip breakfast, and I'm not telling you that you should, I have no data one or the other, maybe you'll lose even more weight. Maybe it's even a healthier thing to do."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Activity causes breakfast. And then you start having a different takeaway. Then you don't say, wait, maybe if you're active and you skip breakfast, and I'm not telling you that you should, I have no data one or the other, maybe you'll lose even more weight. Maybe it's even a healthier thing to do. We're not sure. So they're trying to say, look, if you have breakfast, it's going to make you active, which is a very positive outcome. But maybe you can have the positive outcome without breakfast."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Maybe it's even a healthier thing to do. We're not sure. So they're trying to say, look, if you have breakfast, it's going to make you active, which is a very positive outcome. But maybe you can have the positive outcome without breakfast. Who knows? Likewise, they say breakfast skipping, or they're implying breakfast skipping can cause obesity. But maybe it's the other way around."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "But maybe you can have the positive outcome without breakfast. Who knows? Likewise, they say breakfast skipping, or they're implying breakfast skipping can cause obesity. But maybe it's the other way around. Maybe people who have high body fat, maybe for whatever reason, they're less likely to get hungry in the morning. So maybe it goes this way. Maybe there's a causality there."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "But maybe it's the other way around. Maybe people who have high body fat, maybe for whatever reason, they're less likely to get hungry in the morning. So maybe it goes this way. Maybe there's a causality there. Or even more likely, maybe there's some underlying cause that causes both of these things to happen. And you could think of a bunch of different examples of that. One could be the physical activity."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Maybe there's a causality there. Or even more likely, maybe there's some underlying cause that causes both of these things to happen. And you could think of a bunch of different examples of that. One could be the physical activity. So physical activity, and these are all just theories. I have no proof for it. But I just want to give you different ways of thinking about the same data, and maybe not just coming to the same conclusion that this article seems like it's trying to lead us to conclude that we should eat breakfast if we don't want to become obese."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "One could be the physical activity. So physical activity, and these are all just theories. I have no proof for it. But I just want to give you different ways of thinking about the same data, and maybe not just coming to the same conclusion that this article seems like it's trying to lead us to conclude that we should eat breakfast if we don't want to become obese. So maybe if you're physically active, that leads to you being hungry in the morning. So you're more likely to eat breakfast. And obviously, being physically active also makes it so that you've burned calories, you have more muscle, so that you're not obese."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "But I just want to give you different ways of thinking about the same data, and maybe not just coming to the same conclusion that this article seems like it's trying to lead us to conclude that we should eat breakfast if we don't want to become obese. So maybe if you're physically active, that leads to you being hungry in the morning. So you're more likely to eat breakfast. And obviously, being physically active also makes it so that you've burned calories, you have more muscle, so that you're not obese. So notice, if you view things this way, if you say physical activity is causing both of these, then all of a sudden you lose this connection between breakfast and obesity. Now you can't make the claim that somehow breakfast is the magic formula for someone to not be obese. So let's say that there is an obese person."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "And obviously, being physically active also makes it so that you've burned calories, you have more muscle, so that you're not obese. So notice, if you view things this way, if you say physical activity is causing both of these, then all of a sudden you lose this connection between breakfast and obesity. Now you can't make the claim that somehow breakfast is the magic formula for someone to not be obese. So let's say that there is an obese person. Let's say this is the reality, that physical activity is causing both of these things. And let's say that there is an obese person. What will you tell them to do?"}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "So let's say that there is an obese person. Let's say this is the reality, that physical activity is causing both of these things. And let's say that there is an obese person. What will you tell them to do? Will you tell them, eat breakfast and you won't become obese anymore? Well, that might not work, especially if they're not physically active. I mean, what's going to happen if you have an obese person who's not physically active?"}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "What will you tell them to do? Will you tell them, eat breakfast and you won't become obese anymore? Well, that might not work, especially if they're not physically active. I mean, what's going to happen if you have an obese person who's not physically active? And then you tell them to eat breakfast. Maybe that'll make things worse. And based on that, the advice or the implication from the article is the wrong thing."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "I mean, what's going to happen if you have an obese person who's not physically active? And then you tell them to eat breakfast. Maybe that'll make things worse. And based on that, the advice or the implication from the article is the wrong thing. Physical activity maybe is the thing that should be focused on. Maybe it's something other than physical activity. Maybe you have sleep."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "And based on that, the advice or the implication from the article is the wrong thing. Physical activity maybe is the thing that should be focused on. Maybe it's something other than physical activity. Maybe you have sleep. Maybe people who sleep late, and they're not getting enough sleep, maybe someone who's not getting enough sleep, maybe that leads to obesity. And obviously, because they're not getting enough sleep, they wake up as late as possible and they have to run to the next appointment, or they have to run to school in the case of students. And maybe that's why they skip breakfast."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Maybe you have sleep. Maybe people who sleep late, and they're not getting enough sleep, maybe someone who's not getting enough sleep, maybe that leads to obesity. And obviously, because they're not getting enough sleep, they wake up as late as possible and they have to run to the next appointment, or they have to run to school in the case of students. And maybe that's why they skip breakfast. So once again, if you find someone's obese, maybe the rule here isn't to force a breakfast down your throat. Maybe it'll become even worse, because maybe it is the lack of sleep that's causing your metabolism to slow down or whatever. So it's very, very important when you're looking at any of these studies to try to say, is this a correlation, or is this causality?"}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "And maybe that's why they skip breakfast. So once again, if you find someone's obese, maybe the rule here isn't to force a breakfast down your throat. Maybe it'll become even worse, because maybe it is the lack of sleep that's causing your metabolism to slow down or whatever. So it's very, very important when you're looking at any of these studies to try to say, is this a correlation, or is this causality? If it's correlation, you cannot make the judgment that, hey, eating breakfast is necessarily going to make someone less obese. All that tells you is that these things move together. A better study would be one that is able to prove causality."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "So it's very, very important when you're looking at any of these studies to try to say, is this a correlation, or is this causality? If it's correlation, you cannot make the judgment that, hey, eating breakfast is necessarily going to make someone less obese. All that tells you is that these things move together. A better study would be one that is able to prove causality. And then we could think of other underlying causes that would kind of break down the narrative that this piece is trying to say. I'm not saying it's wrong. Maybe it's absolutely true that eating breakfast will fight obesity."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "A better study would be one that is able to prove causality. And then we could think of other underlying causes that would kind of break down the narrative that this piece is trying to say. I'm not saying it's wrong. Maybe it's absolutely true that eating breakfast will fight obesity. But I think it's equally or more important to think about what the other causes are, not to just make a blanket statement like that. So for example, maybe poverty causes you to skip breakfast for multiple reasons. Maybe both of your parents are working."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Maybe it's absolutely true that eating breakfast will fight obesity. But I think it's equally or more important to think about what the other causes are, not to just make a blanket statement like that. So for example, maybe poverty causes you to skip breakfast for multiple reasons. Maybe both of your parents are working. There's no one there to give you breakfast. Maybe there's more stress in the family. Who knows what it might be?"}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Maybe both of your parents are working. There's no one there to give you breakfast. Maybe there's more stress in the family. Who knows what it might be? And so when you have poverty, maybe you're more likely to skip breakfast. And maybe when there's poverty, and maybe you have two, both your parents are working, and the kids have to make their own dinner and whatever else, maybe they also eat less healthy. So eat less healthy at all times of day, and then that leads to obesity."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Who knows what it might be? And so when you have poverty, maybe you're more likely to skip breakfast. And maybe when there's poverty, and maybe you have two, both your parents are working, and the kids have to make their own dinner and whatever else, maybe they also eat less healthy. So eat less healthy at all times of day, and then that leads to obesity. So once again, in this situation, if this is the reality of things, just telling someone to also eat breakfast, regardless of what that breakfast is, even if it's Froot Loops or syrup, that's probably not going to help the situation. Maybe it's just eating unhealthy dinners. Maybe eating unhealthy dinners is the underlying cause."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "So eat less healthy at all times of day, and then that leads to obesity. So once again, in this situation, if this is the reality of things, just telling someone to also eat breakfast, regardless of what that breakfast is, even if it's Froot Loops or syrup, that's probably not going to help the situation. Maybe it's just eating unhealthy dinners. Maybe eating unhealthy dinners is the underlying cause. And if you eat an unhealthy dinner, maybe by breakfast time, you're not hungry still, because you binge so much on breakfast, so you skip breakfast. And this also leads to obesity. But once again, if this is the actual reality, doing the advice that that article's saying might actually be a bad thing."}, {"video_title": "Correlation and causality Statistical studies Probability and Statistics Khan Academy (2).mp3", "Sentence": "Maybe eating unhealthy dinners is the underlying cause. And if you eat an unhealthy dinner, maybe by breakfast time, you're not hungry still, because you binge so much on breakfast, so you skip breakfast. And this also leads to obesity. But once again, if this is the actual reality, doing the advice that that article's saying might actually be a bad thing. If you eat an unhealthy dinner and then force yourself to eat a breakfast when you're not hungry, that might make the obesity even worse. So the whole point of this video isn't to say that the implications from that article are necessarily wrong. The important thing is to just realize that it might be wrong, and that just because you saw this correlation with the data, it doesn't mean that eating breakfast is going to somehow magically fight obesity."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And so these are the ages of everyone in the restaurant at that moment. And so you're interested in somehow presenting this, somehow visualizing the distribution of the ages, because you want to just say, well, are there more young people, are there more teenagers, are there more middle-aged people, are there more seniors here? And so when you just look at these numbers, it really doesn't give you a good sense of it. It's just a bunch of numbers. And so how could you do that? Well, one way to think about it is is to put these ages into different buckets, and then to think about how many people are there in each of those buckets, or sometimes someone might say, how many in each of those bins? So let's do that."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It's just a bunch of numbers. And so how could you do that? Well, one way to think about it is is to put these ages into different buckets, and then to think about how many people are there in each of those buckets, or sometimes someone might say, how many in each of those bins? So let's do that. So let's do buckets or categories. So I like, sometimes it's called a bin. So the bucket, I like to think of it more as a bucket."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So let's do that. So let's do buckets or categories. So I like, sometimes it's called a bin. So the bucket, I like to think of it more as a bucket. The bucket, and then the number in the bucket. The number in the bucket. Number, I'll just write the number, whoops."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So the bucket, I like to think of it more as a bucket. The bucket, and then the number in the bucket. The number in the bucket. Number, I'll just write the number, whoops. It's the, whoops. It's the number. It's the number in the bucket."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Number, I'll just write the number, whoops. It's the, whoops. It's the number. It's the number in the bucket. All right, so let's just make buckets, let's make them 10-year ranges. So let's say the first one is ages zero to nine. So how many people, and we just define all of the buckets here."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It's the number in the bucket. All right, so let's just make buckets, let's make them 10-year ranges. So let's say the first one is ages zero to nine. So how many people, and we just define all of the buckets here. So the next one is ages 10 to 19, then 20 to 29, then 30 to 39, and 40 to 49, 50 to 59. Make sure you can read that properly. Then you have 60 to 69."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So how many people, and we just define all of the buckets here. So the next one is ages 10 to 19, then 20 to 29, then 30 to 39, and 40 to 49, 50 to 59. Make sure you can read that properly. Then you have 60 to 69. I think that covers everyone. I don't see anyone 70 years old or older here. So then how many people fall into the zero to nine-year-old bucket?"}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Then you have 60 to 69. I think that covers everyone. I don't see anyone 70 years old or older here. So then how many people fall into the zero to nine-year-old bucket? Well, it's gonna be one, two, three, four, five, six people fall into that bucket. How many people fall into the, how many people fall into the 10 to 19-year-old bucket? Well, let's see."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So then how many people fall into the zero to nine-year-old bucket? Well, it's gonna be one, two, three, four, five, six people fall into that bucket. How many people fall into the, how many people fall into the 10 to 19-year-old bucket? Well, let's see. One, two, three. Three people. And I think you see where this is going."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, let's see. One, two, three. Three people. And I think you see where this is going. What about 20 to 29? So it's one, two, three, four, five people. Five people fall into that bucket."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And I think you see where this is going. What about 20 to 29? So it's one, two, three, four, five people. Five people fall into that bucket. All right, what about 30 to 39? We have one, and that's it. Only one person in that 30 to 39 bin or bucket or category."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Five people fall into that bucket. All right, what about 30 to 39? We have one, and that's it. Only one person in that 30 to 39 bin or bucket or category. All right, what about 40 to 49? We have one, two people. Two people are in that bucket."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Only one person in that 30 to 39 bin or bucket or category. All right, what about 40 to 49? We have one, two people. Two people are in that bucket. And then 50 to 59. So you have one, two people. Two people."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Two people are in that bucket. And then 50 to 59. So you have one, two people. Two people. And then finally, finally, ages 60 to 69, let me do it in a different color, 60 to 69, there is one person right over there. So this is one way of thinking about how the ages are distributed, but let's actually make a visualization of this. And the visualization that we're gonna create, this is called a histogram."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Two people. And then finally, finally, ages 60 to 69, let me do it in a different color, 60 to 69, there is one person right over there. So this is one way of thinking about how the ages are distributed, but let's actually make a visualization of this. And the visualization that we're gonna create, this is called a histogram. Histogram. Histogram. We're taking data that can take on a whole bunch of different values."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And the visualization that we're gonna create, this is called a histogram. Histogram. Histogram. We're taking data that can take on a whole bunch of different values. We're putting them into categories, and we're gonna plot how many folks are in each category. How big are each of those categories? And actually, I wrote histogram."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We're taking data that can take on a whole bunch of different values. We're putting them into categories, and we're gonna plot how many folks are in each category. How big are each of those categories? And actually, I wrote histogram. I wrote histograph, I should have written histogram. So a histogram. So let's do this."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And actually, I wrote histogram. I wrote histograph, I should have written histogram. So a histogram. So let's do this. All right, so on this axis, let's see, the largest category has six. So this is the number. Number of folks, and it's gonna go one, two, three, four, five, six."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So let's do this. All right, so on this axis, let's see, the largest category has six. So this is the number. Number of folks, and it's gonna go one, two, three, four, five, six. One, two, three, four, five, six. This is the number, and on this axis, I'm gonna make the buckets. The buckets, and let me scroll up a little bit now that I have my data here, I don't have to look at my data set again."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Number of folks, and it's gonna go one, two, three, four, five, six. One, two, three, four, five, six. This is the number, and on this axis, I'm gonna make the buckets. The buckets, and let me scroll up a little bit now that I have my data here, I don't have to look at my data set again. So I have one bucket. This is going to be the zero to nine bucket, right over here. Zero to nine."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "The buckets, and let me scroll up a little bit now that I have my data here, I don't have to look at my data set again. So I have one bucket. This is going to be the zero to nine bucket, right over here. Zero to nine. Then I'm going to have the three, actually, let me just plot them, since I have my pen that color. So zero to nine, there are six people. Zero to nine, there are six people."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Zero to nine. Then I'm going to have the three, actually, let me just plot them, since I have my pen that color. So zero to nine, there are six people. Zero to nine, there are six people. So I'll just plot it like that. And then we have the 10 to 19. There are three people."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Zero to nine, there are six people. So I'll just plot it like that. And then we have the 10 to 19. There are three people. So 10 to 19, there are three people. So I'll do a bar like this. Then 20 to 29, I have five people."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There are three people. So 10 to 19, there are three people. So I'll do a bar like this. Then 20 to 29, I have five people. 20 to 29, which is gonna be this one, which is getting, I'm writing too big. So 20 to 29, this is gonna be this bar. There's five people."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Then 20 to 29, I have five people. 20 to 29, which is gonna be this one, which is getting, I'm writing too big. So 20 to 29, this is gonna be this bar. There's five people. Five people there. So it'll look like this. I should have made the bars wide enough so I could write below them, but I've already, that train has already left."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There's five people. Five people there. So it'll look like this. I should have made the bars wide enough so I could write below them, but I've already, that train has already left. All right, then 30 to 39. I'll try to write smaller. 30 to 39, that's gonna be this bar right over here."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "I should have made the bars wide enough so I could write below them, but I've already, that train has already left. All right, then 30 to 39. I'll try to write smaller. 30 to 39, that's gonna be this bar right over here. We have one person. One person. And then we have 40 to 49."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "30 to 39, that's gonna be this bar right over here. We have one person. One person. And then we have 40 to 49. We have two people. 40 to 49, two people. So it looks like this."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then we have 40 to 49. We have two people. 40 to 49, two people. So it looks like this. 40 to 49, two people. Almost there, 50 to 59. We have two people."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So it looks like this. 40 to 49, two people. Almost there, 50 to 59. We have two people. 50 to 59, we also have two people. So that's that right over there. That's this category."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We have two people. 50 to 59, we also have two people. So that's that right over there. That's this category. And then finally, 60 to 69, we have one person. 60 to 69, we have one person. We have one person."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "That's this category. And then finally, 60 to 69, we have one person. 60 to 69, we have one person. We have one person. And what I have just constructed, I took our data, I put it into buckets that are kind of representative of the categories I care about, zero to nine is kind of young kids, 10 to 19, I guess you could call them adolescents or roughly teenagers, although obviously if you're 10 you're not quite a teenager yet. And then all the different age groups. And then when I counted the number in each bucket and I plotted it, now I can visually get a sense of how are the ages distributed in this restaurant."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We have one person. And what I have just constructed, I took our data, I put it into buckets that are kind of representative of the categories I care about, zero to nine is kind of young kids, 10 to 19, I guess you could call them adolescents or roughly teenagers, although obviously if you're 10 you're not quite a teenager yet. And then all the different age groups. And then when I counted the number in each bucket and I plotted it, now I can visually get a sense of how are the ages distributed in this restaurant. This must be some type of a restaurant that gives away toys or something because there's a lot of younger people. Maybe it's very family friendly. So every adult that comes in, maybe there's a lot of young adults with kids or maybe grandparents up here, and they just bring a lot of kids to this restaurant."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then when I counted the number in each bucket and I plotted it, now I can visually get a sense of how are the ages distributed in this restaurant. This must be some type of a restaurant that gives away toys or something because there's a lot of younger people. Maybe it's very family friendly. So every adult that comes in, maybe there's a lot of young adults with kids or maybe grandparents up here, and they just bring a lot of kids to this restaurant. So it gives you a view of what's going on here. Just a lot of kids here, a lot fewer senior citizens. So once again, this is just a way of visualizing things."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So every adult that comes in, maybe there's a lot of young adults with kids or maybe grandparents up here, and they just bring a lot of kids to this restaurant. So it gives you a view of what's going on here. Just a lot of kids here, a lot fewer senior citizens. So once again, this is just a way of visualizing things. We took a lot of data that can take multiple data points. Instead of plotting each data point like we might do in a dot plot, instead of saying how many one year olds are there, well there's only one one year old, how many three year olds are there? There's only one three year old."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So once again, this is just a way of visualizing things. We took a lot of data that can take multiple data points. Instead of plotting each data point like we might do in a dot plot, instead of saying how many one year olds are there, well there's only one one year old, how many three year olds are there? There's only one three year old. That wouldn't give us much information. We would just have like these single dots if we were doing a dot plot. But as a histogram, we're able to put them into buckets."}, {"video_title": "How to create a histogram Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There's only one three year old. That wouldn't give us much information. We would just have like these single dots if we were doing a dot plot. But as a histogram, we're able to put them into buckets. And we're just like, hey, generally between the ages zero and nine, we have six people. And so you see that plotted out just like that. And obviously this doesn't apply just to ages of people in a restaurant."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So for example, in this one here, in the horizontal axis, we might have something like age, and then here could be accident frequency. Accident frequency, and I'm just making this up. And I could just show these data points, maybe for some kind of statistical survey, that when the age is this, whatever number this is, maybe this is 20 years old, this is the accident frequency, and it could be a number of accidents per hundred. And that when the age is 21 years old, this is the frequency. And so these data scientists or statisticians went and plotted all of these in this scatter plot. This is often known as bivariate data, which is a very fancy way of saying, hey, you're plotting things that take two variables into consideration, and you're trying to see whether there's a pattern with how they relate. And what we're going to do in this video is think about, well, can we try to fit a line?"}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And that when the age is 21 years old, this is the frequency. And so these data scientists or statisticians went and plotted all of these in this scatter plot. This is often known as bivariate data, which is a very fancy way of saying, hey, you're plotting things that take two variables into consideration, and you're trying to see whether there's a pattern with how they relate. And what we're going to do in this video is think about, well, can we try to fit a line? Does it look like there's a linear or nonlinear relationship between the variables on the different axes? How strong is that variable? Is it a positive, is it a negative relationship?"}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And what we're going to do in this video is think about, well, can we try to fit a line? Does it look like there's a linear or nonlinear relationship between the variables on the different axes? How strong is that variable? Is it a positive, is it a negative relationship? And then we'll think about this idea of outliers. So let's just first think about whether there's a linear or nonlinear relationship. And I'll get my little ruler tool out here."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Is it a positive, is it a negative relationship? And then we'll think about this idea of outliers. So let's just first think about whether there's a linear or nonlinear relationship. And I'll get my little ruler tool out here. So this data right over here, it looks like I could put a line through it that gets pretty close through the data. You're not gonna, it's very unlikely you're gonna be able to go through all of the data points, but you can try to get a line, and I'm just doing this. There's more numerical, more precise ways of doing this, but I'm just eyeballing it right over here."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And I'll get my little ruler tool out here. So this data right over here, it looks like I could put a line through it that gets pretty close through the data. You're not gonna, it's very unlikely you're gonna be able to go through all of the data points, but you can try to get a line, and I'm just doing this. There's more numerical, more precise ways of doing this, but I'm just eyeballing it right over here. And it looks like I could plot a line that looks something like that that goes roughly through the data. So this looks pretty linear. So I would call this a linear relationship."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "There's more numerical, more precise ways of doing this, but I'm just eyeballing it right over here. And it looks like I could plot a line that looks something like that that goes roughly through the data. So this looks pretty linear. So I would call this a linear relationship. And since as we increase one variable, it looks like the other variable decreases. This is a downward-sloping line. I would say this is a negative, this is a negative linear relationship."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So I would call this a linear relationship. And since as we increase one variable, it looks like the other variable decreases. This is a downward-sloping line. I would say this is a negative, this is a negative linear relationship. But this one looks pretty strong. So because the dots aren't that far from my line, this one gets a little bit further, but it's not, you know, there's not some dots way out there. So most of them are pretty close to the line."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "I would say this is a negative, this is a negative linear relationship. But this one looks pretty strong. So because the dots aren't that far from my line, this one gets a little bit further, but it's not, you know, there's not some dots way out there. So most of them are pretty close to the line. So I'll call this a negative, reasonably strong linear relationship. Negative, strong. I call reasonably, I ought to say strong, but reasonably strong linear, linear relationship between these two variables."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So most of them are pretty close to the line. So I'll call this a negative, reasonably strong linear relationship. Negative, strong. I call reasonably, I ought to say strong, but reasonably strong linear, linear relationship between these two variables. Now let's look at this one, and pause this video and think about what this one would be for you. Well, let's see. I'll get my ruler tool out again."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "I call reasonably, I ought to say strong, but reasonably strong linear, linear relationship between these two variables. Now let's look at this one, and pause this video and think about what this one would be for you. Well, let's see. I'll get my ruler tool out again. It looks like I can try to put a line. It looks like generally speaking, as one variable increases, the other variable increases as well. So something like this goes through the data and approximates the direction."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "I'll get my ruler tool out again. It looks like I can try to put a line. It looks like generally speaking, as one variable increases, the other variable increases as well. So something like this goes through the data and approximates the direction. And this looks positive. As one variable increases, the other variable increases, roughly. So this is a positive relationship."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So something like this goes through the data and approximates the direction. And this looks positive. As one variable increases, the other variable increases, roughly. So this is a positive relationship. But this is weak. A lot of the data is off, well off of the line. So positive, weak."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So this is a positive relationship. But this is weak. A lot of the data is off, well off of the line. So positive, weak. But I'd say this is still linear. It seems that as we increase one, the other one increases at roughly the same rate, although these data points are all over the place. So I would still call this linear."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So positive, weak. But I'd say this is still linear. It seems that as we increase one, the other one increases at roughly the same rate, although these data points are all over the place. So I would still call this linear. Now there's also this notion of outliers. If I said, hey, this line is trying to describe the data, well, we have some data that is fairly off the line. So for example, even though we're saying it's a positive, weak, linear relationship, this one over here is reasonably high on the vertical variable, but it's low on the horizontal variable."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So I would still call this linear. Now there's also this notion of outliers. If I said, hey, this line is trying to describe the data, well, we have some data that is fairly off the line. So for example, even though we're saying it's a positive, weak, linear relationship, this one over here is reasonably high on the vertical variable, but it's low on the horizontal variable. And so this one right over here is an outlier. It's quite far away from the line. You could view that as an outlier."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So for example, even though we're saying it's a positive, weak, linear relationship, this one over here is reasonably high on the vertical variable, but it's low on the horizontal variable. And so this one right over here is an outlier. It's quite far away from the line. You could view that as an outlier. And this is a little bit subjective. Outliers, well, what looks pretty far from the rest of the data? This could also be an outlier."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "You could view that as an outlier. And this is a little bit subjective. Outliers, well, what looks pretty far from the rest of the data? This could also be an outlier. Let me label these. Outlier. Now pause the video and see if you can think about this one."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "This could also be an outlier. Let me label these. Outlier. Now pause the video and see if you can think about this one. Is this positive or negative? Is it linear, non-linear? Is it strong or weak?"}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Now pause the video and see if you can think about this one. Is this positive or negative? Is it linear, non-linear? Is it strong or weak? I'll get my ruler tool out here. So this goes here. It seems like I can fit a line pretty well to this."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Is it strong or weak? I'll get my ruler tool out here. So this goes here. It seems like I can fit a line pretty well to this. So I could fit, maybe I'll do the line in purple. I could fit a line that looks like that. And so this one looks like it's positive."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "It seems like I can fit a line pretty well to this. So I could fit, maybe I'll do the line in purple. I could fit a line that looks like that. And so this one looks like it's positive. As one variable increases, the other one does for these data points. So it's a positive. I'd say this is pretty strong."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And so this one looks like it's positive. As one variable increases, the other one does for these data points. So it's a positive. I'd say this is pretty strong. The dots are pretty close to the line. It really does look like a little bit of a fat line if you just look at the dots. So positive, strong, linear, linear relationship."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "I'd say this is pretty strong. The dots are pretty close to the line. It really does look like a little bit of a fat line if you just look at the dots. So positive, strong, linear, linear relationship. And none of these data points are really strong outliers. This one's a little bit further out. But they're all pretty close to the line and seem to describe that trend roughly."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So positive, strong, linear, linear relationship. And none of these data points are really strong outliers. This one's a little bit further out. But they're all pretty close to the line and seem to describe that trend roughly. All right, now let's look at this data right over here. So let me get my line tool out again. So it looks like I can fit a line."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "But they're all pretty close to the line and seem to describe that trend roughly. All right, now let's look at this data right over here. So let me get my line tool out again. So it looks like I can fit a line. So it looks like it's a positive relationship. The line would be upward sloping. It would look something like this."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So it looks like I can fit a line. So it looks like it's a positive relationship. The line would be upward sloping. It would look something like this. And once again, I'm eyeballing it. You can use computers and other methods to actually find a more precise line that minimizes the collective distance to all of the points. But it looks like there is a positive."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "It would look something like this. And once again, I'm eyeballing it. You can use computers and other methods to actually find a more precise line that minimizes the collective distance to all of the points. But it looks like there is a positive. But I would say this one is a weak linear relationship because you have a lot of points that are far off the line. So not so strong. So I would call this a positive, weak, linear relationship."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "But it looks like there is a positive. But I would say this one is a weak linear relationship because you have a lot of points that are far off the line. So not so strong. So I would call this a positive, weak, linear relationship. And there's a lot of outliers here. You know, this one over here is pretty far out. Now let's look at this one."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So I would call this a positive, weak, linear relationship. And there's a lot of outliers here. You know, this one over here is pretty far out. Now let's look at this one. Pause this video and think about is it positive, negative? Is it strong or weak? Is this linear, nonlinear?"}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Now let's look at this one. Pause this video and think about is it positive, negative? Is it strong or weak? Is this linear, nonlinear? Well, the first thing we wanna do is just think about it in linear, nonlinear. I could try to put a line on it. But if I try to put a line on it, it's actually quite difficult."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Is this linear, nonlinear? Well, the first thing we wanna do is just think about it in linear, nonlinear. I could try to put a line on it. But if I try to put a line on it, it's actually quite difficult. If I try to do a line like this, notice everything is kind of bending away from the line. It looks like generally as one variable increases, the other variable decreases. But they're not doing it in a linear fashion."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "But if I try to put a line on it, it's actually quite difficult. If I try to do a line like this, notice everything is kind of bending away from the line. It looks like generally as one variable increases, the other variable decreases. But they're not doing it in a linear fashion. It looks like there's some other type of curve at play. So I could try to do a fancier curve that looks something like this. And this seems to fit the data a lot better."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "But they're not doing it in a linear fashion. It looks like there's some other type of curve at play. So I could try to do a fancier curve that looks something like this. And this seems to fit the data a lot better. So this one I would describe as nonlinear. And it is a negative relationship. As one variable increases, the other variable decreases."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And this seems to fit the data a lot better. So this one I would describe as nonlinear. And it is a negative relationship. As one variable increases, the other variable decreases. So this is a negative, I would say reasonably strong nonlinear relationship. Pretty strong. Pretty strong."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "As one variable increases, the other variable decreases. So this is a negative, I would say reasonably strong nonlinear relationship. Pretty strong. Pretty strong. Once again, this is subjective. So I'll say negative, reasonably strong, nonlinear relationship. And maybe you could call this one an outlier, but it's not that far."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Pretty strong. Once again, this is subjective. So I'll say negative, reasonably strong, nonlinear relationship. And maybe you could call this one an outlier, but it's not that far. And I might even be able to fit a curve that gets a little bit closer to that. Once again, I'm eyeballing this. Now let's do this last one."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "And maybe you could call this one an outlier, but it's not that far. And I might even be able to fit a curve that gets a little bit closer to that. Once again, I'm eyeballing this. Now let's do this last one. So this one looks like a negative linear relationship to me. A fairly strong negative linear relationship, although there's some outliers. So let me draw this line."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "Now let's do this last one. So this one looks like a negative linear relationship to me. A fairly strong negative linear relationship, although there's some outliers. So let me draw this line. So that seems to fit the data pretty good. So this is a negative, reasonably strong, reasonably strong linear relationship. But these are very clear outliers."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So let me draw this line. So that seems to fit the data pretty good. So this is a negative, reasonably strong, reasonably strong linear relationship. But these are very clear outliers. These are well away from the data or from the cluster of where most of the points are. So with some significant, with at least these two significant outliers here. So hopefully this makes you a little bit familiar with some of this terminology."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "But these are very clear outliers. These are well away from the data or from the cluster of where most of the points are. So with some significant, with at least these two significant outliers here. So hopefully this makes you a little bit familiar with some of this terminology. And it's important to keep in mind, this is a little bit subjective. There'll be some cases that are more obvious than others. So for, and oftentimes you wanna make a comparison."}, {"video_title": "Bivariate relationship linearity, strength and direction AP Statistics Khan Academy.mp3", "Sentence": "So hopefully this makes you a little bit familiar with some of this terminology. And it's important to keep in mind, this is a little bit subjective. There'll be some cases that are more obvious than others. So for, and oftentimes you wanna make a comparison. That this is a stronger linear, positive linear relationship than this one is, right over here, because you can see most of the data is closer to the line. This one is for sure, this is more nonlinear than linear. It depends how you wanna describe, oftentimes making a comparison or making a subjective call on how to describe the data."}, {"video_title": "Expected payoff example protection plan Probability & combinatorics Khan Academy.mp3", "Sentence": "The customer pays $80 for the plan, and if their television is damaged or stops working, the store will replace it for no additional charge. The store knows that 2% of customers who buy this plan end up needing a replacement that costs the store $1,200 each. Here is a table that summarizes the possible outcomes from the store's perspective. Let X represent the store's net gain from one of these plans. Calculate the expected net gain. So pause this video and see if you can have a go at that before we work through this together. So we have the two scenarios here."}, {"video_title": "Expected payoff example protection plan Probability & combinatorics Khan Academy.mp3", "Sentence": "Let X represent the store's net gain from one of these plans. Calculate the expected net gain. So pause this video and see if you can have a go at that before we work through this together. So we have the two scenarios here. The first scenario is that the store does need to replace the TV because something happens, and so it's going to cost $1,200 to the store, but remember, they got $80 for the protection plan, so you have a net gain of negative $1,120 from the store's perspective. There's the other scenario, which is more favorable for the store, which is the customer does not need a replacement TV, so that has no cost, and so their net gain is just the $80 for the plan. So to figure out the expected net gain, we just have to figure out the probabilities of each of these and take the weighted average of them."}, {"video_title": "Expected payoff example protection plan Probability & combinatorics Khan Academy.mp3", "Sentence": "So we have the two scenarios here. The first scenario is that the store does need to replace the TV because something happens, and so it's going to cost $1,200 to the store, but remember, they got $80 for the protection plan, so you have a net gain of negative $1,120 from the store's perspective. There's the other scenario, which is more favorable for the store, which is the customer does not need a replacement TV, so that has no cost, and so their net gain is just the $80 for the plan. So to figure out the expected net gain, we just have to figure out the probabilities of each of these and take the weighted average of them. So what's the probability that they will have to replace the TV? Well, we know 2% of customers who buy this plan end up needing a replacement, so we could say this is 2 over 100, or maybe I'll write it as 0.02. This is the probability of X, and then the probability of not needing a replacement, 0.98, and so our expected net gain is going to be equal to the probability of needing a replacement times the net gain of a replacement, so it's going to be times negative $1,120, and then we're going to have plus the probability of not needing a replacement, which is 0.98, times the net gain there, so that is $80."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So I took some screen captures from the Khan Academy exercise on correlation coefficient intuition, and they've given us some correlation coefficients, and we need to match them to the various scatter plots. On that exercise, there's a little interface where we can drag these around in a table to match them to the different scatter plots. And the point isn't to figure out how exactly to calculate these. We'll do that in the future, but really to get an intuition of what we're trying to measure. And the main idea is that correlation coefficients are trying to measure how well a linear model can describe the relationship between two variables. So for example, if I have, let me draw, let me do some coordinate axes here. So let's say that's one variable."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "We'll do that in the future, but really to get an intuition of what we're trying to measure. And the main idea is that correlation coefficients are trying to measure how well a linear model can describe the relationship between two variables. So for example, if I have, let me draw, let me do some coordinate axes here. So let's say that's one variable. Say that's my y variable. And let's say that is my x variable. And so let's say when x is low, y is low."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So let's say that's one variable. Say that's my y variable. And let's say that is my x variable. And so let's say when x is low, y is low. When x is a little higher, y is a little higher. When x is a little bit higher, y is higher. When x is really high, y is even higher."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "And so let's say when x is low, y is low. When x is a little higher, y is a little higher. When x is a little bit higher, y is higher. When x is really high, y is even higher. This one, a linear model would describe it very, very, very well. It's quite easy to draw a line that goes through, that essentially goes through those points. So something like this would have an r of one."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "When x is really high, y is even higher. This one, a linear model would describe it very, very, very well. It's quite easy to draw a line that goes through, that essentially goes through those points. So something like this would have an r of one. r is equal to one. A linear model perfectly describes it, and it's a positive correlation. When one increases, when one variable gets larger, then the other variable is larger."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So something like this would have an r of one. r is equal to one. A linear model perfectly describes it, and it's a positive correlation. When one increases, when one variable gets larger, then the other variable is larger. When one variable is smaller, then the other variable is smaller, and vice versa. Now what would an r of negative one look like? Well, that would once again be a situation where a linear model works really well, but when one variable moves up, the other one moves down, and vice versa."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "When one increases, when one variable gets larger, then the other variable is larger. When one variable is smaller, then the other variable is smaller, and vice versa. Now what would an r of negative one look like? Well, that would once again be a situation where a linear model works really well, but when one variable moves up, the other one moves down, and vice versa. So let me draw my coordinates, my coordinate axes again. So I'm gonna try to draw a data set where the r would be negative one. So maybe when y is high, x is very low."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "Well, that would once again be a situation where a linear model works really well, but when one variable moves up, the other one moves down, and vice versa. So let me draw my coordinates, my coordinate axes again. So I'm gonna try to draw a data set where the r would be negative one. So maybe when y is high, x is very low. When y becomes lower, x becomes higher. When y becomes a good bit lower, x becomes a good bit higher. So once again, when y decreases, x increases, or as x increases, y decreases."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So maybe when y is high, x is very low. When y becomes lower, x becomes higher. When y becomes a good bit lower, x becomes a good bit higher. So once again, when y decreases, x increases, or as x increases, y decreases. So they're moving in opposite directions. But you can fit a line very easily to this. So the line would look something like this."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So once again, when y decreases, x increases, or as x increases, y decreases. So they're moving in opposite directions. But you can fit a line very easily to this. So the line would look something like this. So this would have an r of negative one. And an r of zero, r is equal to zero, would be a data set where a line doesn't really fit very well at all. So I'll do that one really small, since I don't have much space here."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So the line would look something like this. So this would have an r of negative one. And an r of zero, r is equal to zero, would be a data set where a line doesn't really fit very well at all. So I'll do that one really small, since I don't have much space here. So an r of zero might look something like this. Maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have one there, there, there, there, there. And it wouldn't necessarily be this well organized, but this gives you a sense of things."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So I'll do that one really small, since I don't have much space here. So an r of zero might look something like this. Maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have a data point here, maybe I have one there, there, there, there, there. And it wouldn't necessarily be this well organized, but this gives you a sense of things. Where would you actually, how would you actually try to fit a line here? You could equally justify a line that looks like that, or a line that looks like that, or a line that looks like that. So there really isn't, a linear model really does not describe the relationship between the two variables that well right over here."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "And it wouldn't necessarily be this well organized, but this gives you a sense of things. Where would you actually, how would you actually try to fit a line here? You could equally justify a line that looks like that, or a line that looks like that, or a line that looks like that. So there really isn't, a linear model really does not describe the relationship between the two variables that well right over here. So with that as a primer, let's see if we can tackle these scatter plots. And the way I'm gonna do it is I'm just gonna try to eyeball what a linear model might look like. And there's different methods of trying to fit a linear model to a dataset, an imperfect dataset."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So there really isn't, a linear model really does not describe the relationship between the two variables that well right over here. So with that as a primer, let's see if we can tackle these scatter plots. And the way I'm gonna do it is I'm just gonna try to eyeball what a linear model might look like. And there's different methods of trying to fit a linear model to a dataset, an imperfect dataset. I drew very perfect ones, at least for the r equals negative one and r equals one. But these are what the real world actually looks like. Nothing, very few times will things perfectly sit on a line."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "And there's different methods of trying to fit a linear model to a dataset, an imperfect dataset. I drew very perfect ones, at least for the r equals negative one and r equals one. But these are what the real world actually looks like. Nothing, very few times will things perfectly sit on a line. So for scatter plot A, if I were to try to fit a line, it would look something like, it would look something like that, if I were to try to minimize distances from these points to the line. I do see a general trend that when y is, you know, if we look at these data points over here, when y is high, x is low, and when x is high, when x is larger, y is smaller. So it looks like r is going to be less than zero, in a reasonable bit less than zero."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "Nothing, very few times will things perfectly sit on a line. So for scatter plot A, if I were to try to fit a line, it would look something like, it would look something like that, if I were to try to minimize distances from these points to the line. I do see a general trend that when y is, you know, if we look at these data points over here, when y is high, x is low, and when x is high, when x is larger, y is smaller. So it looks like r is going to be less than zero, in a reasonable bit less than zero. It's going to approach this thing here. And if we look at our choices, so it wouldn't be r equals 0.65, these are positive, so I wouldn't use that one or that one. And this one is almost no correlation."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So it looks like r is going to be less than zero, in a reasonable bit less than zero. It's going to approach this thing here. And if we look at our choices, so it wouldn't be r equals 0.65, these are positive, so I wouldn't use that one or that one. And this one is almost no correlation. r equals negative 0.02, this is pretty close to zero. So I feel good with r is equal to negative 0.72. r is equal to negative 0.72. Now I want to be clear, if I didn't have these choices here, I wouldn't just be able to say, just looking at these data points, without being able to do a calculation, that r is equal to negative 0.72."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "And this one is almost no correlation. r equals negative 0.02, this is pretty close to zero. So I feel good with r is equal to negative 0.72. r is equal to negative 0.72. Now I want to be clear, if I didn't have these choices here, I wouldn't just be able to say, just looking at these data points, without being able to do a calculation, that r is equal to negative 0.72. I'm just basing it on the intuition that it is a negative correlation. It seems pretty strong, you know, the pattern kind of jumps out at you that when y is large, y, x is small. When x is large, y is small."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "Now I want to be clear, if I didn't have these choices here, I wouldn't just be able to say, just looking at these data points, without being able to do a calculation, that r is equal to negative 0.72. I'm just basing it on the intuition that it is a negative correlation. It seems pretty strong, you know, the pattern kind of jumps out at you that when y is large, y, x is small. When x is large, y is small. And so I like something that's approaching r equals negative one. So I've used this one up already. Now scatter plot B, if I were to just try to eyeball it, once again, this is going to be imperfect, but the trend, if I were to try to fit a line, it looks something like that."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "When x is large, y is small. And so I like something that's approaching r equals negative one. So I've used this one up already. Now scatter plot B, if I were to just try to eyeball it, once again, this is going to be imperfect, but the trend, if I were to try to fit a line, it looks something like that. So it looks like a line fits in reasonably well. There are some points that would still be hard to fit, and they're still pretty far from the line. And it looks like it's a positive correlation."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "Now scatter plot B, if I were to just try to eyeball it, once again, this is going to be imperfect, but the trend, if I were to try to fit a line, it looks something like that. So it looks like a line fits in reasonably well. There are some points that would still be hard to fit, and they're still pretty far from the line. And it looks like it's a positive correlation. When x is small, when y is small, x is relatively small, and vice versa. And as x grows, y grows, and when y grows, x grows. So this one's going to be positive, and it looks like it would be reasonably positive."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "And it looks like it's a positive correlation. When x is small, when y is small, x is relatively small, and vice versa. And as x grows, y grows, and when y grows, x grows. So this one's going to be positive, and it looks like it would be reasonably positive. And I have two choices here. So I don't know which of these it's going to be. So it's either going to be r is equal to 0.65, or r is equal to 0.84."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So this one's going to be positive, and it looks like it would be reasonably positive. And I have two choices here. So I don't know which of these it's going to be. So it's either going to be r is equal to 0.65, or r is equal to 0.84. I also get scatter plot C. Now this one's all over the place. It kind of looks like what we did over here. You know, I could, you know, what does a line look like?"}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So it's either going to be r is equal to 0.65, or r is equal to 0.84. I also get scatter plot C. Now this one's all over the place. It kind of looks like what we did over here. You know, I could, you know, what does a line look like? You could almost imagine anything. Does it look like that? Does it look like that?"}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "You know, I could, you know, what does a line look like? You could almost imagine anything. Does it look like that? Does it look like that? Does a line look like that? These things really aren't, don't seem to, there's not a direction that you could say, well, as x increases, maybe y increases or decreases, there's no rhyme or reason here. So this looks very non-correlated."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "Does it look like that? Does a line look like that? These things really aren't, don't seem to, there's not a direction that you could say, well, as x increases, maybe y increases or decreases, there's no rhyme or reason here. So this looks very non-correlated. And so this one is pretty close to 0. So I feel pretty good that this is the r is equal to negative 0.02. In fact, you know, if we tried, probably the best line that could be fit would be one with a slight negative slope."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So this looks very non-correlated. And so this one is pretty close to 0. So I feel pretty good that this is the r is equal to negative 0.02. In fact, you know, if we tried, probably the best line that could be fit would be one with a slight negative slope. So it might look something like this. It might look something like this. And notice even when we try to fit a line, there's all sorts of points that are way off the line."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "In fact, you know, if we tried, probably the best line that could be fit would be one with a slight negative slope. So it might look something like this. It might look something like this. And notice even when we try to fit a line, there's all sorts of points that are way off the line. So the linear model did not fit it that well. So r is equal to negative 0.02. So we use that one."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "And notice even when we try to fit a line, there's all sorts of points that are way off the line. So the linear model did not fit it that well. So r is equal to negative 0.02. So we use that one. And so now we have scatterplot D. So that's going to use one of the other positive correlations. And it does look like, you know, there is a positive correlation when y is low, x is low, and when x is high, y is high, and vice versa. And so we could try to fit something that looks something like that."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "So we use that one. And so now we have scatterplot D. So that's going to use one of the other positive correlations. And it does look like, you know, there is a positive correlation when y is low, x is low, and when x is high, y is high, and vice versa. And so we could try to fit something that looks something like that. But it's still not as good as that one. You could see the points that we're trying to fit, there's several points that are still pretty far away from our model. So the model is not fitting it that well."}, {"video_title": "Example Correlation coefficient intuition Mathematics I High School Math Khan Academy.mp3", "Sentence": "And so we could try to fit something that looks something like that. But it's still not as good as that one. You could see the points that we're trying to fit, there's several points that are still pretty far away from our model. So the model is not fitting it that well. So I would say scatterplot B is a better fit. A linear model works better for scatterplot B than it works for scatterplot D. So I would give the higher r to scatterplot B, and the lower r, r equals 0.65 to scatterplot D. r is equal to 0.65. And once again, that's because with a linear model, it looks like there's a trend, but there's several data points that really, more data points are way off the line in scatterplot D than in the case of scatterplot B."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "She conducted a poll by calling 100 people whose names were randomly sampled from the phone book. Note that mobile phones and unlisted numbers are not in phone books. The senator's office called those numbers until they got a response from all 100 people chosen. The poll showed that 42% of respondents were very concerned about internet privacy. What is the most concerning source of bias in this scenario? And we should also think about, well, what kind of bias is that likely to introduce? Is this likely to be an overestimate or an underestimate of the number of respondents?"}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "The poll showed that 42% of respondents were very concerned about internet privacy. What is the most concerning source of bias in this scenario? And we should also think about, well, what kind of bias is that likely to introduce? Is this likely to be an overestimate or an underestimate of the number of respondents? And maybe there is no bias here. But our choices, and no bias is not one of the choices, so you can imagine, it's going to be one of these three. So I encourage you to pause this video and think about what we just said."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Is this likely to be an overestimate or an underestimate of the number of respondents? And maybe there is no bias here. But our choices, and no bias is not one of the choices, so you can imagine, it's going to be one of these three. So I encourage you to pause this video and think about what we just said. We're a senator, we're trying to figure out what percentage of our constituents are very concerned about internet privacy. And we go to the phone book, we sample 100 people, we keep calling them until they answer, and we get that 42% are very concerned. So what's the source of bias?"}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So I encourage you to pause this video and think about what we just said. We're a senator, we're trying to figure out what percentage of our constituents are very concerned about internet privacy. And we go to the phone book, we sample 100 people, we keep calling them until they answer, and we get that 42% are very concerned. So what's the source of bias? All right, now let's work through this together. So non-response would have been the case if we selected these 100 people, and let's say only 50 people answered the phone and we didn't keep calling them. Then we'd say, well, 50 of the people who we sampled to answer our survey didn't even respond."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So what's the source of bias? All right, now let's work through this together. So non-response would have been the case if we selected these 100 people, and let's say only 50 people answered the phone and we didn't keep calling them. Then we'd say, well, 50 of the people who we sampled to answer our survey didn't even respond. There was a non-response there. What was there about those 50 people? Maybe it was something that would have skewed the survey, or actually, if we had, we'd gotten them, it would have gotten maybe get better data."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Then we'd say, well, 50 of the people who we sampled to answer our survey didn't even respond. There was a non-response there. What was there about those 50 people? Maybe it was something that would have skewed the survey, or actually, if we had, we'd gotten them, it would have gotten maybe get better data. But in this case, they tell us. The senator's office called those numbers until they got a response from all 100 people chosen. So the 100 people that they chose, they made sure they got a response."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Maybe it was something that would have skewed the survey, or actually, if we had, we'd gotten them, it would have gotten maybe get better data. But in this case, they tell us. The senator's office called those numbers until they got a response from all 100 people chosen. So the 100 people that they chose, they made sure they got a response. So non-response is not going to be an issue here. All right, next choice, undercoverage. Well, undercoverage is where you're not able to sample from part of the population."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "So the 100 people that they chose, they made sure they got a response. So non-response is not going to be an issue here. All right, next choice, undercoverage. Well, undercoverage is where you're not able to sample from part of the population. A part of the population that actually might, because you didn't sample it, it might introduce bias. Let's think about what happened in this situation. We are a senator."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Well, undercoverage is where you're not able to sample from part of the population. A part of the population that actually might, because you didn't sample it, it might introduce bias. Let's think about what happened in this situation. We are a senator. We want to sample all of our constituents, but we choose, instead, we sample from the constituents who happen to be listed in the phone book. So these are the people who happen to be, who happen to be listed in the phone book. And so we're not sampling from people who are not in the phone book, who maybe have landlines and they're unlisted, and we're not sampling from people who don't have landlines, who only have mobile phones."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "We are a senator. We want to sample all of our constituents, but we choose, instead, we sample from the constituents who happen to be listed in the phone book. So these are the people who happen to be, who happen to be listed in the phone book. And so we're not sampling from people who are not in the phone book, who maybe have landlines and they're unlisted, and we're not sampling from people who don't have landlines, who only have mobile phones. And you might say, well, why is that important? Well, think about it. People who decide not to list in the phone book, or people who don't even have a landline, some of those people might be a little bit more concerned about privacy than everyone else."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "And so we're not sampling from people who are not in the phone book, who maybe have landlines and they're unlisted, and we're not sampling from people who don't have landlines, who only have mobile phones. And you might say, well, why is that important? Well, think about it. People who decide not to list in the phone book, or people who don't even have a landline, some of those people might be a little bit more concerned about privacy than everyone else. They explicitly chose not to be listed. So undercoverage is definitely a very concerning source of bias over here. We are sampling from only a subset of our entire population we care about."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "People who decide not to list in the phone book, or people who don't even have a landline, some of those people might be a little bit more concerned about privacy than everyone else. They explicitly chose not to be listed. So undercoverage is definitely a very concerning source of bias over here. We are sampling from only a subset of our entire population we care about. In particular, we're missing out on people who might care about privacy. And so I would say, because of undercoverage, 42% is likely to be an underestimate of the people concerned about internet privacy. Probably a higher proportion of the people out here care about privacy, because they're unlisted or they don't even have a landline."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "We are sampling from only a subset of our entire population we care about. In particular, we're missing out on people who might care about privacy. And so I would say, because of undercoverage, 42% is likely to be an underestimate of the people concerned about internet privacy. Probably a higher proportion of the people out here care about privacy, because they're unlisted or they don't even have a landline. So undercoverage, it probably introduced bias, and it implies that 42% is an under, underestimate of the percentage of the senator's constituents who care about internet privacy. Now the last question, volunteer response sampling. Well, this would be the case where you, you know, the senator, I don't know, put a billboard out or just told someone, told a bunch of people, maybe on her website, hey, vote for this, or give us your information on how much you care about internet privacy."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Probably a higher proportion of the people out here care about privacy, because they're unlisted or they don't even have a landline. So undercoverage, it probably introduced bias, and it implies that 42% is an under, underestimate of the percentage of the senator's constituents who care about internet privacy. Now the last question, volunteer response sampling. Well, this would be the case where you, you know, the senator, I don't know, put a billboard out or just told someone, told a bunch of people, maybe on her website, hey, vote for this, or give us your information on how much you care about internet privacy. And that would have been, the source of bias there is, well, who shows up on that website? Once again, if you did, hey, come to my website and fill it out, you're filling, you're only getting information from a subset of your population who are choosing, who are volunteering. That is not the situation that she did over here."}, {"video_title": "Example of under coverage introducing bias Study design AP Statistics Khan Academy.mp3", "Sentence": "Well, this would be the case where you, you know, the senator, I don't know, put a billboard out or just told someone, told a bunch of people, maybe on her website, hey, vote for this, or give us your information on how much you care about internet privacy. And that would have been, the source of bias there is, well, who shows up on that website? Once again, if you did, hey, come to my website and fill it out, you're filling, you're only getting information from a subset of your population who are choosing, who are volunteering. That is not the situation that she did over here. She didn't ask 100 people to volunteer. Her team went out and got them from the phone book. So this was definitely a case of undercoverage."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "This is the distribution for Portland, for example they get eight days between one and four degrees Celsius, they get 12 days between four and seven degrees Celsius, so forth and so on, and then this is the distribution for Minneapolis. Now when we make these comparisons, what we're going to focus on is the center of the distributions to compare that, and also the spread. Sometimes people will talk about the variability of the distributions, and so these are the things that we're going to compare. And in making the comparison, we're actually just going to try to eyeball it. We're not gonna try to pick a measure of central tendency, say the mean or the median, and then calculate precisely what those numbers are for these. We might wanna do those if they're close, but if we can eyeball it, that would be even better. Similar for the spread and variability."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "And in making the comparison, we're actually just going to try to eyeball it. We're not gonna try to pick a measure of central tendency, say the mean or the median, and then calculate precisely what those numbers are for these. We might wanna do those if they're close, but if we can eyeball it, that would be even better. Similar for the spread and variability. In either of these cases, there are multiple measures in our statistical toolkit. Center, mean, median is, mean, median is valuable for the center. For spread variability, the range, the interquartile range, the mean absolute deviation, the standard deviation, these are all measures."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "Similar for the spread and variability. In either of these cases, there are multiple measures in our statistical toolkit. Center, mean, median is, mean, median is valuable for the center. For spread variability, the range, the interquartile range, the mean absolute deviation, the standard deviation, these are all measures. But sometimes you can just kind of gauge it by looking. So in this first comparison, which distribution has a higher center, or are they comparable? Well, if you look at the distribution for Portland, the center of this distribution, let's say if we were to just think about the mean, although I think the mean and the median would be reasonably close right over here, it seems like it would be around, it would be around seven, or maybe a little bit lower than seven, so it would be kind of in that range, maybe between five and seven, would be our central tendency, would be either our mean or our median."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "For spread variability, the range, the interquartile range, the mean absolute deviation, the standard deviation, these are all measures. But sometimes you can just kind of gauge it by looking. So in this first comparison, which distribution has a higher center, or are they comparable? Well, if you look at the distribution for Portland, the center of this distribution, let's say if we were to just think about the mean, although I think the mean and the median would be reasonably close right over here, it seems like it would be around, it would be around seven, or maybe a little bit lower than seven, so it would be kind of in that range, maybe between five and seven, would be our central tendency, would be either our mean or our median. While for Minneapolis, it looks like our center is much closer to maybe negative two or negative three degrees Celsius. So here, even though we don't know precisely what the mean or the median is of each of these distributions, you can say that Portland, Portland distribution has a higher center, has higher center, however you wanna measure it, either mean or median. Now what about the spread or variability?"}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "Well, if you look at the distribution for Portland, the center of this distribution, let's say if we were to just think about the mean, although I think the mean and the median would be reasonably close right over here, it seems like it would be around, it would be around seven, or maybe a little bit lower than seven, so it would be kind of in that range, maybe between five and seven, would be our central tendency, would be either our mean or our median. While for Minneapolis, it looks like our center is much closer to maybe negative two or negative three degrees Celsius. So here, even though we don't know precisely what the mean or the median is of each of these distributions, you can say that Portland, Portland distribution has a higher center, has higher center, however you wanna measure it, either mean or median. Now what about the spread or variability? Well, if you just superficially thought about range, you see here that there's nothing below one degree Celsius and nothing above 13, so you have about a 13-degree range at most right over here. In fact, what might be contributing to this first column might be a bunch of things at three degrees or even 3.9 degrees, and similarly, what's contributing to this last column might be a bunch of things at 10.1 degrees, but at most, you have a 12-degree range right over here, while over here, it looks like you have, well, it looks like it's approaching a 27-degree range. So based on that, and even if you just eyeball it, this is just, we're using the same scales for our horizontal axes here, the temperature axes, and this is just a much wider distribution than what you see over here, and so you would say that the Minneapolis distribution has more spread or higher spread or more variability, so higher spread right over here."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "Now what about the spread or variability? Well, if you just superficially thought about range, you see here that there's nothing below one degree Celsius and nothing above 13, so you have about a 13-degree range at most right over here. In fact, what might be contributing to this first column might be a bunch of things at three degrees or even 3.9 degrees, and similarly, what's contributing to this last column might be a bunch of things at 10.1 degrees, but at most, you have a 12-degree range right over here, while over here, it looks like you have, well, it looks like it's approaching a 27-degree range. So based on that, and even if you just eyeball it, this is just, we're using the same scales for our horizontal axes here, the temperature axes, and this is just a much wider distribution than what you see over here, and so you would say that the Minneapolis distribution has more spread or higher spread or more variability, so higher spread right over here. Let's do another example, and we'll use a different representation for the data here. So we're told at the Olympic Games, many events have several rounds of competition. One of these events is the men's 100-meter backstroke."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "So based on that, and even if you just eyeball it, this is just, we're using the same scales for our horizontal axes here, the temperature axes, and this is just a much wider distribution than what you see over here, and so you would say that the Minneapolis distribution has more spread or higher spread or more variability, so higher spread right over here. Let's do another example, and we'll use a different representation for the data here. So we're told at the Olympic Games, many events have several rounds of competition. One of these events is the men's 100-meter backstroke. The upper dot plot shows the times in seconds of the top eight finishers in the final round of the 2012 Olympics, so that's in green right over here, the final round. The lower dot plot shows the times of the same eight swimmers but in the semifinal round. So given these distributions, which one has a higher center?"}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "One of these events is the men's 100-meter backstroke. The upper dot plot shows the times in seconds of the top eight finishers in the final round of the 2012 Olympics, so that's in green right over here, the final round. The lower dot plot shows the times of the same eight swimmers but in the semifinal round. So given these distributions, which one has a higher center? Well, once again, I mean, and here, you can actually, it's a little bit easier to eyeball even what the median might be. The mean, I would probably have to do a little bit more mathematics, but let's say the median, let's say there's one, two, three, four, five, six, seven, eight data points, so the median is gonna sit between the lower four and the upper four, so the central tendency right over here is for the final round, is looks like it's around 57.1 seconds, while the, especially if we think about the median, while the central tendency for the semifinal round, let's see, one, two, three, four, five, six, seven, eight, looks like it is right about there, so this is about 57, more than 57.3 seconds, so the semifinal round seems to have a higher central tendency, which is a little bit counterintuitive. You would expect the finalists to be running faster on average than the semifinalists, but that's not what this data is showing, so the semifinal round has higher center, higher, higher center, and I just eyeballed the median, and I suspect that the mean would also be higher in this second distribution, and now what about variability?"}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "So given these distributions, which one has a higher center? Well, once again, I mean, and here, you can actually, it's a little bit easier to eyeball even what the median might be. The mean, I would probably have to do a little bit more mathematics, but let's say the median, let's say there's one, two, three, four, five, six, seven, eight data points, so the median is gonna sit between the lower four and the upper four, so the central tendency right over here is for the final round, is looks like it's around 57.1 seconds, while the, especially if we think about the median, while the central tendency for the semifinal round, let's see, one, two, three, four, five, six, seven, eight, looks like it is right about there, so this is about 57, more than 57.3 seconds, so the semifinal round seems to have a higher central tendency, which is a little bit counterintuitive. You would expect the finalists to be running faster on average than the semifinalists, but that's not what this data is showing, so the semifinal round has higher center, higher, higher center, and I just eyeballed the median, and I suspect that the mean would also be higher in this second distribution, and now what about variability? Well, once again, if you just looked at range, and these are both at the same scale, if you just visually look, the variability here, the range for the final round, is larger than the range for the semifinal round, so you would say that the final round has higher variability, variability, it has a higher range. Eyeballing it, it looks like it has a higher spread, and there's, of course, times where one distribution could have a higher range, but then it might have a lower standard deviation. For example, you could have data that's like, you know, two data points that are really far apart, but then all of the other data just sits right, it's really, really closely packed, so for example, a distribution like this, and I'll draw the horizontal axis here, just so you can imagine it as a distribution."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "You would expect the finalists to be running faster on average than the semifinalists, but that's not what this data is showing, so the semifinal round has higher center, higher, higher center, and I just eyeballed the median, and I suspect that the mean would also be higher in this second distribution, and now what about variability? Well, once again, if you just looked at range, and these are both at the same scale, if you just visually look, the variability here, the range for the final round, is larger than the range for the semifinal round, so you would say that the final round has higher variability, variability, it has a higher range. Eyeballing it, it looks like it has a higher spread, and there's, of course, times where one distribution could have a higher range, but then it might have a lower standard deviation. For example, you could have data that's like, you know, two data points that are really far apart, but then all of the other data just sits right, it's really, really closely packed, so for example, a distribution like this, and I'll draw the horizontal axis here, just so you can imagine it as a distribution. A distribution like this might have a higher range, but lower standard deviation than a distribution like this. Let me just, I'm just drawing a very rough example. A distribution like this has a lower range, but actually might have a higher standard deviation, might have a higher standard deviation than the one above it."}, {"video_title": "Example Comparing distributions AP Statistics Khan Academy.mp3", "Sentence": "For example, you could have data that's like, you know, two data points that are really far apart, but then all of the other data just sits right, it's really, really closely packed, so for example, a distribution like this, and I'll draw the horizontal axis here, just so you can imagine it as a distribution. A distribution like this might have a higher range, but lower standard deviation than a distribution like this. Let me just, I'm just drawing a very rough example. A distribution like this has a lower range, but actually might have a higher standard deviation, might have a higher standard deviation than the one above it. In fact, I can make that even better. A distribution like this would have a lower range, but it would also have a higher standard deviation. So you can't just look at, it's not always the case that just by looking at one of these measures, the range or the standard deviation, you'll know for sure, but in cases like this, it's safe to say when you're looking at it by inspection that look, this green, the final round data does seem to have a higher range, higher variability, and so I'd feel pretty good at this."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And what I want to explore in this video is different ways of representing this data and then see if we can answer questions about the data. So the first way we can think about it is as a frequency table. Frequency table. Frequency table. And what we're going to do is we're going to look at each, for each age, for each possible age that we've measured here to see how many students in the class are of that age. So we could say the age is one column. And then the number."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Frequency table. And what we're going to do is we're going to look at each, for each age, for each possible age that we've measured here to see how many students in the class are of that age. So we could say the age is one column. And then the number. The number of students of that age. Or we could even say the frequency. Frequency."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then the number. The number of students of that age. Or we could even say the frequency. Frequency. When people say, how frequent do you do something, they're saying, how often does it happen? How often do you do that thing? Frequency."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Frequency. When people say, how frequent do you do something, they're saying, how often does it happen? How often do you do that thing? Frequency. Or we could also say, actually, let me just write number. I'm always a fan of the simpler number at age, which we could also consider the frequency at that age. Frequency of students."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Frequency. Or we could also say, actually, let me just write number. I'm always a fan of the simpler number at age, which we could also consider the frequency at that age. Frequency of students. All right. So what's the lowest age that we have here? Well, the lowest age is five."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Frequency of students. All right. So what's the lowest age that we have here? Well, the lowest age is five. So I'll start with five. And how many students in the class are age five? How frequent is the number five?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, the lowest age is five. So I'll start with five. And how many students in the class are age five? How frequent is the number five? Let's see. There is one, two. Let me keep scanning."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "How frequent is the number five? Let's see. There is one, two. Let me keep scanning. Looks like there's only two fives. So I could write a two here. There are two fives."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let me keep scanning. Looks like there's only two fives. So I could write a two here. There are two fives. And now let's go to six. How many sixes are there? Let's see."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There are two fives. And now let's go to six. How many sixes are there? Let's see. There is one sixth. There's only one six-year-old in the class. All right."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Let's see. There is one sixth. There's only one six-year-old in the class. All right. Seven-year-olds. See, there's one, two, three, four seven-year-olds. Now what about eight-year-olds?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "All right. Seven-year-olds. See, there's one, two, three, four seven-year-olds. Now what about eight-year-olds? I'm going to do this in a color that I have not used yet. Eight-year-olds, we have no eight-year-olds. Zero eight-year-olds."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Now what about eight-year-olds? I'm going to do this in a color that I have not used yet. Eight-year-olds, we have no eight-year-olds. Zero eight-year-olds. And then we have nine-year-olds. Let's see. Nine-year-olds, we have one, two, three, four nine-year-olds."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Zero eight-year-olds. And then we have nine-year-olds. Let's see. Nine-year-olds, we have one, two, three, four nine-year-olds. 10-year-olds, what do we have? We have one 10-year-old right over there. And then 11-year-olds."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Nine-year-olds, we have one, two, three, four nine-year-olds. 10-year-olds, what do we have? We have one 10-year-old right over there. And then 11-year-olds. There are no 11-year-olds. And then let me scroll up a little bit. And then finally 12-year-olds."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then 11-year-olds. There are no 11-year-olds. And then let me scroll up a little bit. And then finally 12-year-olds. 12-year-olds, there are one, two 12-year-olds. So what we have just constructed is a frequency table. It's a frequency table."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And then finally 12-year-olds. 12-year-olds, there are one, two 12-year-olds. So what we have just constructed is a frequency table. It's a frequency table. You can see for each age how many students are at that age. So it's giving you the same information as we have up here. You could take this table and construct what we have up here."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "It's a frequency table. You can see for each age how many students are at that age. So it's giving you the same information as we have up here. You could take this table and construct what we have up here. You would just write down two fives, one sixth, four sevens, no eights, four nines, one 10, no 11s, and two 12s. And then you would just have this list of numbers. Now a way to visually look at a frequency table is a dot plot."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "You could take this table and construct what we have up here. You would just write down two fives, one sixth, four sevens, no eights, four nines, one 10, no 11s, and two 12s. And then you would just have this list of numbers. Now a way to visually look at a frequency table is a dot plot. So let me draw a dot plot right over here. So a dot plot. And a dot plot, we essentially just take the same information and even think about it the same way, but we just show it visually."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Now a way to visually look at a frequency table is a dot plot. So let me draw a dot plot right over here. So a dot plot. And a dot plot, we essentially just take the same information and even think about it the same way, but we just show it visually. So in a dot plot, what we would have of, actually let me just not draw an even arrow there. We have the different age groups. So five, six, seven, eight, nine, 10, 11, and 12."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And a dot plot, we essentially just take the same information and even think about it the same way, but we just show it visually. So in a dot plot, what we would have of, actually let me just not draw an even arrow there. We have the different age groups. So five, six, seven, eight, nine, 10, 11, and 12. And we have a dot to represent, or we use a dot for each student at that age. So there's two five-year-olds, so I'll do two dots. One and two."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So five, six, seven, eight, nine, 10, 11, and 12. And we have a dot to represent, or we use a dot for each student at that age. So there's two five-year-olds, so I'll do two dots. One and two. There's one six-year-old, so that's going to be one dot right over here. There's four seven-year-olds, so one, two, three, four dots. There's no eight-year-olds."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "One and two. There's one six-year-old, so that's going to be one dot right over here. There's four seven-year-olds, so one, two, three, four dots. There's no eight-year-olds. There's four nine-year-olds, so one, two, three, and four. There's one 10-year-old, so let's put a dot, one dot, right over there for that one 10-year-old. There's no 11-year-olds, so I'm not going to put any dots there."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There's no eight-year-olds. There's four nine-year-olds, so one, two, three, and four. There's one 10-year-old, so let's put a dot, one dot, right over there for that one 10-year-old. There's no 11-year-olds, so I'm not going to put any dots there. And then there's two 12-year-olds. So one 12-year-old and another 12-year-old. So there you go."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "There's no 11-year-olds, so I'm not going to put any dots there. And then there's two 12-year-olds. So one 12-year-old and another 12-year-old. So there you go. We have frequency table, dot plot, list of numbers. These are all showing the same data, just in different ways. And once you have it represented in any of these ways, we can start to ask questions about it."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So there you go. We have frequency table, dot plot, list of numbers. These are all showing the same data, just in different ways. And once you have it represented in any of these ways, we can start to ask questions about it. So we could say, what is the most frequent age? Well, the most frequent age, when you look at it visually, or the easiest thing might be just look at the dot plot, because you see it visually, the most frequent age are the two highest stacks. So there's actually seven and nine are tied for the most frequent age."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And once you have it represented in any of these ways, we can start to ask questions about it. So we could say, what is the most frequent age? Well, the most frequent age, when you look at it visually, or the easiest thing might be just look at the dot plot, because you see it visually, the most frequent age are the two highest stacks. So there's actually seven and nine are tied for the most frequent age. You would have also seen it here, where seven and nine are tied at four. And if you just had this data, you'd have to count all of them to kind of come up with this again and say, OK, there's four sevens, four nines. That's the largest number."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So there's actually seven and nine are tied for the most frequent age. You would have also seen it here, where seven and nine are tied at four. And if you just had this data, you'd have to count all of them to kind of come up with this again and say, OK, there's four sevens, four nines. That's the largest number. So if you're looking for what's the most frequent age, when you just visually inspect here, it probably pops out at you the fastest. But there's other questions we can ask ourselves. We can ask ourselves, what is the range of ages in the classroom?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "That's the largest number. So if you're looking for what's the most frequent age, when you just visually inspect here, it probably pops out at you the fastest. But there's other questions we can ask ourselves. We can ask ourselves, what is the range of ages in the classroom? And this is, once again, where maybe the dot plot is the most, it jumps out at you the most, because the range is just the maximum age in your, or the maximum data point minus the minimum data point. So what's the maximum age here? Well, the maximum age here, we see it from the dot plot, is 12."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We can ask ourselves, what is the range of ages in the classroom? And this is, once again, where maybe the dot plot is the most, it jumps out at you the most, because the range is just the maximum age in your, or the maximum data point minus the minimum data point. So what's the maximum age here? Well, the maximum age here, we see it from the dot plot, is 12. And the minimum age here, you see, is five. So there's a range of seven. The difference between the maximum and the minimum is seven."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Well, the maximum age here, we see it from the dot plot, is 12. And the minimum age here, you see, is five. So there's a range of seven. The difference between the maximum and the minimum is seven. But you could have also done that over here. You could say, hey, the maximum age here is 12. Minimum age here is five."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "The difference between the maximum and the minimum is seven. But you could have also done that over here. You could say, hey, the maximum age here is 12. Minimum age here is five. And so you find the difference between 12 and five, which is seven. Here, you still could have done it. You could say, OK, what's the lowest?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Minimum age here is five. And so you find the difference between 12 and five, which is seven. Here, you still could have done it. You could say, OK, what's the lowest? Let's look at five. Are there any fours here? Nope, there's no fours."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "You could say, OK, what's the lowest? Let's look at five. Are there any fours here? Nope, there's no fours. So five's the minimum age. And what's the largest? Is it seven?"}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Nope, there's no fours. So five's the minimum age. And what's the largest? Is it seven? No, is it nine? Nine, not even 10? Oh, 12."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Is it seven? No, is it nine? Nine, not even 10? Oh, 12. 12. Are there any 13s? No, 12 is the maximum."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Oh, 12. 12. Are there any 13s? No, 12 is the maximum. So you say 12 minus five is seven to get the range. But then we could ask ourselves other questions. We could say, how many older than nine is a question we could ask ourselves."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "No, 12 is the maximum. So you say 12 minus five is seven to get the range. But then we could ask ourselves other questions. We could say, how many older than nine is a question we could ask ourselves. And then if we were to look at the dot plot, we say, OK, this is nine. And we care about how many are older than nine. So that would be this one, two, and three."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "We could say, how many older than nine is a question we could ask ourselves. And then if we were to look at the dot plot, we say, OK, this is nine. And we care about how many are older than nine. So that would be this one, two, and three. Or you could look over here. How many are older than nine? Well, it's the one person who's 10, and then the two who are 12."}, {"video_title": "Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So that would be this one, two, and three. Or you could look over here. How many are older than nine? Well, it's the one person who's 10, and then the two who are 12. So there are three. And over here, if you said how many are older than nine, well, then you would just have to go through the list and say, OK, no, no, no, no, no, no, no, no. OK, here, one, two, three."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "We're told a certain lottery ticket costs $2, and the back of the ticket says, the overall odds of winning a prize with this ticket are one to 50, and the expected return for this ticket is 95 cents. Which interpretations of the expected value are correct? Choose all answers that apply. Pause this video, have a go at that. All right, now let's go through each of these choices. So choice A says the probability that one of these tickets wins a prize is 0.95 on average. Well, I see where they're getting that 0.95."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "Pause this video, have a go at that. All right, now let's go through each of these choices. So choice A says the probability that one of these tickets wins a prize is 0.95 on average. Well, I see where they're getting that 0.95. They're getting it from right over here, but that's not the probability that you're winning. That's the expected return. The probability that you win is much lower."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "Well, I see where they're getting that 0.95. They're getting it from right over here, but that's not the probability that you're winning. That's the expected return. The probability that you win is much lower. If the odds are one to 50, that means that the probability of winning is one to 51. So it's a much lower probability than this right over here. So definitely rule that out."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "The probability that you win is much lower. If the odds are one to 50, that means that the probability of winning is one to 51. So it's a much lower probability than this right over here. So definitely rule that out. Someone who buys this ticket is most likely to win 95 cents. That is not necessarily the case either. We don't know what the different outcomes are for the prize."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "So definitely rule that out. Someone who buys this ticket is most likely to win 95 cents. That is not necessarily the case either. We don't know what the different outcomes are for the prize. It's very likely that there's no outcome for that prize where you win exactly 95 cents. Instead, there's likely to be outcomes that are much larger than that with very low probabilities, and then when you take the weighted average of all of the outcomes, then you get an expected return of 95 cents. So it's actually maybe even impossible to win exactly 95 cents, so I would rule that out."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "We don't know what the different outcomes are for the prize. It's very likely that there's no outcome for that prize where you win exactly 95 cents. Instead, there's likely to be outcomes that are much larger than that with very low probabilities, and then when you take the weighted average of all of the outcomes, then you get an expected return of 95 cents. So it's actually maybe even impossible to win exactly 95 cents, so I would rule that out. If we looked at many of these tickets, the average return would be about 95 cents per ticket. That one feels pretty interesting because we're looking at many of these tickets, and so across many of them, you would expect to, on average, get the expected return as your return. And so this is what we are saying here."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "So it's actually maybe even impossible to win exactly 95 cents, so I would rule that out. If we looked at many of these tickets, the average return would be about 95 cents per ticket. That one feels pretty interesting because we're looking at many of these tickets, and so across many of them, you would expect to, on average, get the expected return as your return. And so this is what we are saying here. The average return would be about that, would be approximately that. So I like that choice. That is a good interpretation of expected value."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "And so this is what we are saying here. The average return would be about that, would be approximately that. So I like that choice. That is a good interpretation of expected value. And then choice D, if 1,000 people each bought one of these tickets, they'd expect a net gain of about $950 in total. This one is tempting. Instead of net gain, if it just said return, this would make a lot of sense."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "That is a good interpretation of expected value. And then choice D, if 1,000 people each bought one of these tickets, they'd expect a net gain of about $950 in total. This one is tempting. Instead of net gain, if it just said return, this would make a lot of sense. In fact, it would be completely consistent with choice C. If you have 1,000 people, that would be many tickets, and if on average, if their average return is about 95 cents per ticket, then their total return would be about $950. But they didn't write return here. They wrote net gain."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "Instead of net gain, if it just said return, this would make a lot of sense. In fact, it would be completely consistent with choice C. If you have 1,000 people, that would be many tickets, and if on average, if their average return is about 95 cents per ticket, then their total return would be about $950. But they didn't write return here. They wrote net gain. Net gain would be how much you get minus how much you paid. And 1,000 people would have to pay, if they each got a ticket, would pay $2,000. So they would pay 2,000."}, {"video_title": "Interpreting expected value Probability & combinatorics Khan Academy.mp3", "Sentence": "They wrote net gain. Net gain would be how much you get minus how much you paid. And 1,000 people would have to pay, if they each got a ticket, would pay $2,000. So they would pay 2,000. They would expect a return of $950. Their net gain would actually be negative $1,050. So we would rule that one out as well."}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "What is the probability that Kyra is chosen for the conference? Pause this video and see if you can have a go at this before we work through this together. All right, now let's work through this together. So we wanna figure out this probability. And so one way to think about it is, what are the number of ways that Kyra can be on a team or the number of possible teams, teams with Kyra, and then over the total number of possible teams, total number of possible teams. And if this little hint gets you even more inspired, if you weren't able to do it the first time, I encourage you to try to pause it again and then work through it. All right, now I will continue to continue."}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "So we wanna figure out this probability. And so one way to think about it is, what are the number of ways that Kyra can be on a team or the number of possible teams, teams with Kyra, and then over the total number of possible teams, total number of possible teams. And if this little hint gets you even more inspired, if you weren't able to do it the first time, I encourage you to try to pause it again and then work through it. All right, now I will continue to continue. So first let me do the denominator here. What are the total possible number of teams? Some of y'all might've found that a little bit easier to figure out."}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "All right, now I will continue to continue. So first let me do the denominator here. What are the total possible number of teams? Some of y'all might've found that a little bit easier to figure out. Well, we know that we're choosing from 13 people and we're picking three of them. And we don't care about order. It's not like we're saying someone's going to be president of the team, someone's going to be vice president, and someone's going to be treasurer."}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "Some of y'all might've found that a little bit easier to figure out. Well, we know that we're choosing from 13 people and we're picking three of them. And we don't care about order. It's not like we're saying someone's going to be president of the team, someone's going to be vice president, and someone's going to be treasurer. We just say there are three people in the team. And so this is a situation where out of 13, we are choosing three people. Now, what are the total number of teams, possible teams that could have Kyra in it?"}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "It's not like we're saying someone's going to be president of the team, someone's going to be vice president, and someone's going to be treasurer. We just say there are three people in the team. And so this is a situation where out of 13, we are choosing three people. Now, what are the total number of teams, possible teams that could have Kyra in it? Well, one way to think about it is if we know that Kyra's on a team, then the possibilities are who's going to be the other two people on the team and who are the possible candidates for the other two people? Well, if Kyra's already on the team, then there's a possible 12 people to pick from. So there's 12 people to choose from for those other two slots."}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "Now, what are the total number of teams, possible teams that could have Kyra in it? Well, one way to think about it is if we know that Kyra's on a team, then the possibilities are who's going to be the other two people on the team and who are the possible candidates for the other two people? Well, if Kyra's already on the team, then there's a possible 12 people to pick from. So there's 12 people to choose from for those other two slots. And so we're going to choose two. And once again, we don't care about the order with which we are choosing them. So once again, it is going to be a combination."}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "So there's 12 people to choose from for those other two slots. And so we're going to choose two. And once again, we don't care about the order with which we are choosing them. So once again, it is going to be a combination. And then we can just go ahead and calculate each of these combinations here. What is 12 choose two? Well, there's 12 possible people for that first non-Kyra seat."}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "So once again, it is going to be a combination. And then we can just go ahead and calculate each of these combinations here. What is 12 choose two? Well, there's 12 possible people for that first non-Kyra seat. And then there would be 11 people there for that other non-Kyra spot. And of course, it's a combination. We don't care what order we picked it in."}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "Well, there's 12 possible people for that first non-Kyra seat. And then there would be 11 people there for that other non-Kyra spot. And of course, it's a combination. We don't care what order we picked it in. And so there are two ways to get these two people. We could say two factorial, but that's just the same thing as two or two times one. And then the denominator here, for that first spot, there's 13 people to pick from."}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "We don't care what order we picked it in. And so there are two ways to get these two people. We could say two factorial, but that's just the same thing as two or two times one. And then the denominator here, for that first spot, there's 13 people to pick from. Then in that second spot, there are 12. Then in that third spot, there are 11. And then once again, we don't care about order."}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "And then the denominator here, for that first spot, there's 13 people to pick from. Then in that second spot, there are 12. Then in that third spot, there are 11. And then once again, we don't care about order. Three factorial ways to arrange three people. So I could write three times two. And for kicks, I could write one right over here."}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "And then once again, we don't care about order. Three factorial ways to arrange three people. So I could write three times two. And for kicks, I could write one right over here. And then we can, let's go down here. This is going to be equal to, my numerator over here is going to be six times 11. And then my denominator is going to be 12 divided by six right over here is two."}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "And for kicks, I could write one right over here. And then we can, let's go down here. This is going to be equal to, my numerator over here is going to be six times 11. And then my denominator is going to be 12 divided by six right over here is two. So it's going to be 13 times 11 times two. Just to be clear, I divided both the denominator and this numerator over here by six to get two right over there. Now this cancels with that."}, {"video_title": "Probability with combinations example choosing groups Probability & combinatorics.mp3", "Sentence": "And then my denominator is going to be 12 divided by six right over here is two. So it's going to be 13 times 11 times two. Just to be clear, I divided both the denominator and this numerator over here by six to get two right over there. Now this cancels with that. And then if we divide the numerator and denominators by two, this is going to be three here. This is going to be one. And so we are left with a probability of 313ths that Kyra is chosen for the conference."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "I'll assume it's a quarter or something. Let's see. So this is a quarter. Let me draw my best attempt at a profile of George Washington. Well, that'll do. So it's a fair coin, and we're going to flip it a bunch of times and figure out the different probabilities. So let's start with a straightforward one."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "Let me draw my best attempt at a profile of George Washington. Well, that'll do. So it's a fair coin, and we're going to flip it a bunch of times and figure out the different probabilities. So let's start with a straightforward one. Let's just flip it once. So with one flip of the coin, what's the probability of getting heads? Well, there's two equally likely possibilities, and the one with heads is one of those two equally likely possibilities, so there's a 1 half chance."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So let's start with a straightforward one. Let's just flip it once. So with one flip of the coin, what's the probability of getting heads? Well, there's two equally likely possibilities, and the one with heads is one of those two equally likely possibilities, so there's a 1 half chance. Same thing if we were to ask what is the probability of getting tails. There are two equally likely possibilities, and one of those gives us tails, so 1 half. And this is one thing to realize."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "Well, there's two equally likely possibilities, and the one with heads is one of those two equally likely possibilities, so there's a 1 half chance. Same thing if we were to ask what is the probability of getting tails. There are two equally likely possibilities, and one of those gives us tails, so 1 half. And this is one thing to realize. If you take the probabilities of heads plus the probabilities of tails, you get 1 half plus 1 half, which is 1. And this is generally 2. The sum of the probabilities of all of the possible events should be equal to 1, and that makes sense because you're adding up all of these fractions, and the numerator will then add up to all of the possible events."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "And this is one thing to realize. If you take the probabilities of heads plus the probabilities of tails, you get 1 half plus 1 half, which is 1. And this is generally 2. The sum of the probabilities of all of the possible events should be equal to 1, and that makes sense because you're adding up all of these fractions, and the numerator will then add up to all of the possible events. The denominator is always all of the possible events, so you have all of the possible events over all of the possible events when you add all of these things up. Now let's take it up a notch. Let's figure out the probability of..."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "The sum of the probabilities of all of the possible events should be equal to 1, and that makes sense because you're adding up all of these fractions, and the numerator will then add up to all of the possible events. The denominator is always all of the possible events, so you have all of the possible events over all of the possible events when you add all of these things up. Now let's take it up a notch. Let's figure out the probability of... I'm going to take this coin, and I'm going to flip it twice. The probability of getting heads and then getting another heads. The probability of getting a head and then another head."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "Let's figure out the probability of... I'm going to take this coin, and I'm going to flip it twice. The probability of getting heads and then getting another heads. The probability of getting a head and then another head. So there's two ways to think about it. One way is to just think about all of the different possibilities. I could get a head on the first flip and a head on the second flip, head on the first flip, tail on the second flip."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "The probability of getting a head and then another head. So there's two ways to think about it. One way is to just think about all of the different possibilities. I could get a head on the first flip and a head on the second flip, head on the first flip, tail on the second flip. I could get tails on the first flip, heads on the second flip, or I could get tails on both flips. So there's four distinct, equally likely possibilities. Four distinct, equally likely outcomes here."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "I could get a head on the first flip and a head on the second flip, head on the first flip, tail on the second flip. I could get tails on the first flip, heads on the second flip, or I could get tails on both flips. So there's four distinct, equally likely possibilities. Four distinct, equally likely outcomes here. One way to think about it is on the first flip, I have two possibilities. On the second flip, I have another two possibilities. I could have heads or tails, heads or tails."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "Four distinct, equally likely outcomes here. One way to think about it is on the first flip, I have two possibilities. On the second flip, I have another two possibilities. I could have heads or tails, heads or tails. So I have four possibilities. For each of these possibilities, for each of these two, I have two possibilities here. So either way, I have four equally likely possibilities."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "I could have heads or tails, heads or tails. So I have four possibilities. For each of these possibilities, for each of these two, I have two possibilities here. So either way, I have four equally likely possibilities. How many of those meet our constraints? Well, we have it right over here. This one right over here, having two heads meets our constraints."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So either way, I have four equally likely possibilities. How many of those meet our constraints? Well, we have it right over here. This one right over here, having two heads meets our constraints. So this is, and there's only one of those possibilities. I've only circled one of the four scenarios. So there's a 1 4th chance of that happening."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "This one right over here, having two heads meets our constraints. So this is, and there's only one of those possibilities. I've only circled one of the four scenarios. So there's a 1 4th chance of that happening. Another way you could think about this, and this is because these are independent events, and this is a very important idea to understand in probability, and we'll also study scenarios that are not independent, but these are independent events. What happens in the first flip in no way affects what happens in the second flip. This is actually one thing that many people don't realize."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So there's a 1 4th chance of that happening. Another way you could think about this, and this is because these are independent events, and this is a very important idea to understand in probability, and we'll also study scenarios that are not independent, but these are independent events. What happens in the first flip in no way affects what happens in the second flip. This is actually one thing that many people don't realize. There's something called the gambler's fallacy, where someone thinks if I got a bunch of heads in a row, then all of a sudden it becomes more likely on the next flip to get a tail. That is not the case. Every flip is an independent event."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "This is actually one thing that many people don't realize. There's something called the gambler's fallacy, where someone thinks if I got a bunch of heads in a row, then all of a sudden it becomes more likely on the next flip to get a tail. That is not the case. Every flip is an independent event. What happened in the past in these flips does not affect the probabilities going forward. So the probability of getting heads on the first flip in no way, or the fact that you got heads on the first flip, in no way affects that you got heads on the second flip. So if you can make that assumption, you can say that the probability of getting heads and heads, or heads and then heads, is going to be the same thing as the probability of getting heads on the first flip times the probability of getting heads on the second flip."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "Every flip is an independent event. What happened in the past in these flips does not affect the probabilities going forward. So the probability of getting heads on the first flip in no way, or the fact that you got heads on the first flip, in no way affects that you got heads on the second flip. So if you can make that assumption, you can say that the probability of getting heads and heads, or heads and then heads, is going to be the same thing as the probability of getting heads on the first flip times the probability of getting heads on the second flip. And we know the probability of getting heads on the first flip is one half. And the probability of getting heads on the second flip is one half. And so we have 1 half times 1 half, which is equal to 1 fourth, which is exactly what we got when we tried out all of the different scenarios, all of the equally likely possibilities."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So if you can make that assumption, you can say that the probability of getting heads and heads, or heads and then heads, is going to be the same thing as the probability of getting heads on the first flip times the probability of getting heads on the second flip. And we know the probability of getting heads on the first flip is one half. And the probability of getting heads on the second flip is one half. And so we have 1 half times 1 half, which is equal to 1 fourth, which is exactly what we got when we tried out all of the different scenarios, all of the equally likely possibilities. Let's take it up another notch. Let's figure out the probability, and we've kind of been ignoring tails, so let's pay some attention to tails. The probability of getting tails and then heads and then tails."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "And so we have 1 half times 1 half, which is equal to 1 fourth, which is exactly what we got when we tried out all of the different scenarios, all of the equally likely possibilities. Let's take it up another notch. Let's figure out the probability, and we've kind of been ignoring tails, so let's pay some attention to tails. The probability of getting tails and then heads and then tails. So this exact series of events. So I'm not saying in any order, 2 tails and a head. I'm saying this exact order."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "The probability of getting tails and then heads and then tails. So this exact series of events. So I'm not saying in any order, 2 tails and a head. I'm saying this exact order. The first flip is a tails, second flip is a heads, and then third flip is a tail. So once again, these are all independent events. The fact that I get tails on the first flip in no way affects the probability of getting a heads on the second flip."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "I'm saying this exact order. The first flip is a tails, second flip is a heads, and then third flip is a tail. So once again, these are all independent events. The fact that I get tails on the first flip in no way affects the probability of getting a heads on the second flip. And that in no way affects the probability of getting a tails on the third flip. So because these are independent events, we can say this is the same thing as the probability of getting tails on the first flip times the probability of getting heads on the second flip times the probability of getting tails on the third flip. And we know these are all independent events, so this right over here is 1 half times 1 half times 1 half."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "The fact that I get tails on the first flip in no way affects the probability of getting a heads on the second flip. And that in no way affects the probability of getting a tails on the third flip. So because these are independent events, we can say this is the same thing as the probability of getting tails on the first flip times the probability of getting heads on the second flip times the probability of getting tails on the third flip. And we know these are all independent events, so this right over here is 1 half times 1 half times 1 half. 1 half times 1 half is 1 fourth. 1 fourth times 1 half is equal to 1 eighth. So this is equal to 1 eighth."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "And we know these are all independent events, so this right over here is 1 half times 1 half times 1 half. 1 half times 1 half is 1 fourth. 1 fourth times 1 half is equal to 1 eighth. So this is equal to 1 eighth. And we can verify it. Let's try it all of the different scenarios again. So you could get heads, heads, heads."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So this is equal to 1 eighth. And we can verify it. Let's try it all of the different scenarios again. So you could get heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "So you could get heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. You can get tails, heads, heads. This is a little tricky sometimes. You want to make sure you're being exhaustive in all of the different possibilities here."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "You could get heads, tails, tails. You can get tails, heads, heads. This is a little tricky sometimes. You want to make sure you're being exhaustive in all of the different possibilities here. You could get tails, heads, tails. You could get tails, tails, heads. Or you could get tails, tails, tails."}, {"video_title": "Compound probability of independent events Probability and Statistics Khan Academy.mp3", "Sentence": "You want to make sure you're being exhaustive in all of the different possibilities here. You could get tails, heads, tails. You could get tails, tails, heads. Or you could get tails, tails, tails. And what we see here is that we got exactly 8 equally likely possibilities. And the tail, heads, tails is exactly one of them. It is this possibility right over here."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And I'm using theirs because it's open source. It's actually quite a good book. The problems are, I think, good practice for us. So let's see, number three. You could go to their site, and I think you can download the book. Assume that the mean weight of one-year-old girls in the US is normally distributed with a mean of about 9.5 grams. That's got to be kilograms."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So let's see, number three. You could go to their site, and I think you can download the book. Assume that the mean weight of one-year-old girls in the US is normally distributed with a mean of about 9.5 grams. That's got to be kilograms. I have a 10-month-old son, and he weighs about 20 pounds, which is about 9 kilograms. Not 9.5. 9.5 grams is nothing."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "That's got to be kilograms. I have a 10-month-old son, and he weighs about 20 pounds, which is about 9 kilograms. Not 9.5. 9.5 grams is nothing. This would be, you know, we're talking about like mice or something. This has got to be kilograms. But anyway, it's about 9.5 kilograms with a standard deviation of approximately 1.1 grams."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "9.5 grams is nothing. This would be, you know, we're talking about like mice or something. This has got to be kilograms. But anyway, it's about 9.5 kilograms with a standard deviation of approximately 1.1 grams. So the mean is equal to 9.5 kilograms, I'm assuming. And the standard deviation is equal to 1.1 grams. Without using a calculator."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "But anyway, it's about 9.5 kilograms with a standard deviation of approximately 1.1 grams. So the mean is equal to 9.5 kilograms, I'm assuming. And the standard deviation is equal to 1.1 grams. Without using a calculator. So that's an interesting clue. Estimate the percentage of one-year-old girls in the US that meet the following conditions. So when they say that without a calculator estimate, that's a big clue or a big giveaway that we're supposed to use the empirical rule."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Without using a calculator. So that's an interesting clue. Estimate the percentage of one-year-old girls in the US that meet the following conditions. So when they say that without a calculator estimate, that's a big clue or a big giveaway that we're supposed to use the empirical rule. The empirical rule. Sometimes called the 68-95-99.7 rule. And this is actually, if you remember, this is the name of the rule."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So when they say that without a calculator estimate, that's a big clue or a big giveaway that we're supposed to use the empirical rule. The empirical rule. Sometimes called the 68-95-99.7 rule. And this is actually, if you remember, this is the name of the rule. You've essentially remembered the rule. What that tells us is if we have a normal distribution, I'll do a bit of a review here before we jump into this problem. If we have a normal distribution, let me draw a normal distribution."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And this is actually, if you remember, this is the name of the rule. You've essentially remembered the rule. What that tells us is if we have a normal distribution, I'll do a bit of a review here before we jump into this problem. If we have a normal distribution, let me draw a normal distribution. Say it looks like that. That's my normal distribution. I didn't draw it perfectly, but you get the idea."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "If we have a normal distribution, let me draw a normal distribution. Say it looks like that. That's my normal distribution. I didn't draw it perfectly, but you get the idea. It should be symmetrical. This is our mean right there. That's our mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "I didn't draw it perfectly, but you get the idea. It should be symmetrical. This is our mean right there. That's our mean. If we go one standard deviation above the mean and one standard deviation below the mean. So this is our mean plus one standard deviation. This is our mean minus one standard deviation."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "That's our mean. If we go one standard deviation above the mean and one standard deviation below the mean. So this is our mean plus one standard deviation. This is our mean minus one standard deviation. The probability of finding a result, if we're dealing with a perfect normal distribution, that's between one standard deviation below the mean and one standard deviation above the mean, that would be this area. And it would be, you could guess, 68%. 68% chance you're going to get something within one standard deviation of the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "This is our mean minus one standard deviation. The probability of finding a result, if we're dealing with a perfect normal distribution, that's between one standard deviation below the mean and one standard deviation above the mean, that would be this area. And it would be, you could guess, 68%. 68% chance you're going to get something within one standard deviation of the mean. Either a standard deviation below or above or anywhere in between. Now, if we're talking about two standard deviations around the mean, so if we go down another standard deviation, so we go down another standard deviation in that direction and another standard deviation above the mean, and we were to ask ourselves, what's the probability of finding something within those two or within that range, then it's, you could guess it, 95%. And that includes this middle area right here."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "68% chance you're going to get something within one standard deviation of the mean. Either a standard deviation below or above or anywhere in between. Now, if we're talking about two standard deviations around the mean, so if we go down another standard deviation, so we go down another standard deviation in that direction and another standard deviation above the mean, and we were to ask ourselves, what's the probability of finding something within those two or within that range, then it's, you could guess it, 95%. And that includes this middle area right here. So the 68% is a subset of that 95%. And I think you know where this is going. If we go three standard deviations below the mean and above the mean, the empirical rule, or the 68, 95, 99.7 rule tells us that there is a 99.7% chance of finding a result in a normal distribution that is within three standard deviations of the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And that includes this middle area right here. So the 68% is a subset of that 95%. And I think you know where this is going. If we go three standard deviations below the mean and above the mean, the empirical rule, or the 68, 95, 99.7 rule tells us that there is a 99.7% chance of finding a result in a normal distribution that is within three standard deviations of the mean. So above three standard deviations below the mean and below three standard deviations above the mean. That's what the empirical rule tells us. Now let's see if we can apply it to this problem."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "If we go three standard deviations below the mean and above the mean, the empirical rule, or the 68, 95, 99.7 rule tells us that there is a 99.7% chance of finding a result in a normal distribution that is within three standard deviations of the mean. So above three standard deviations below the mean and below three standard deviations above the mean. That's what the empirical rule tells us. Now let's see if we can apply it to this problem. So they gave us the mean and the standard deviation. Let me draw that out. Let me draw my axis first as best as I can."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Now let's see if we can apply it to this problem. So they gave us the mean and the standard deviation. Let me draw that out. Let me draw my axis first as best as I can. That's my axis. Let me draw my bell curve. That's about as good as a bell curve is you can expect a freehand drawer to do."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Let me draw my axis first as best as I can. That's my axis. Let me draw my bell curve. That's about as good as a bell curve is you can expect a freehand drawer to do. And the mean here is 9 point, and this should be symmetric. This height should be the same as that height there. I think you get the idea."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "That's about as good as a bell curve is you can expect a freehand drawer to do. And the mean here is 9 point, and this should be symmetric. This height should be the same as that height there. I think you get the idea. I'm not a computer. 9.5 is the mean. I won't write the units."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "I think you get the idea. I'm not a computer. 9.5 is the mean. I won't write the units. It's all in kilograms. One standard deviation above the mean. So one standard deviation above the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "I won't write the units. It's all in kilograms. One standard deviation above the mean. So one standard deviation above the mean. We should add 1.1 to that. They told us the standard deviation is 1.1. That's going to be 10.6."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So one standard deviation above the mean. We should add 1.1 to that. They told us the standard deviation is 1.1. That's going to be 10.6. If we go, let me just draw a little dotted line there. One standard deviation below the mean. One standard deviation below the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "That's going to be 10.6. If we go, let me just draw a little dotted line there. One standard deviation below the mean. One standard deviation below the mean. We're going to subtract 1.1 from 9.5. And so that would be 8.4. If we go two standard deviations above the mean, we would add another standard deviation here."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "One standard deviation below the mean. We're going to subtract 1.1 from 9.5. And so that would be 8.4. If we go two standard deviations above the mean, we would add another standard deviation here. We went one standard deviation, two standard deviations. That would get us to 11.7. And if we were to go three standard deviations, we'd add 1.1 again."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "If we go two standard deviations above the mean, we would add another standard deviation here. We went one standard deviation, two standard deviations. That would get us to 11.7. And if we were to go three standard deviations, we'd add 1.1 again. That would get us to 12.8. Doing it on the other side. One standard deviation below the mean is 8.4."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And if we were to go three standard deviations, we'd add 1.1 again. That would get us to 12.8. Doing it on the other side. One standard deviation below the mean is 8.4. Two standard deviations below the mean. Subtract 1.1 again would be 7.3. And then three standard deviations below the mean, maybe right there, would be 6.2 kilograms."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "One standard deviation below the mean is 8.4. Two standard deviations below the mean. Subtract 1.1 again would be 7.3. And then three standard deviations below the mean, maybe right there, would be 6.2 kilograms. So that's our setup for the problem. So what's the probability that we would find a one-year-old girl in the US that weighs less than 8.4 kilograms? Or maybe I should say whose mass is less than 8.4 kilograms."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And then three standard deviations below the mean, maybe right there, would be 6.2 kilograms. So that's our setup for the problem. So what's the probability that we would find a one-year-old girl in the US that weighs less than 8.4 kilograms? Or maybe I should say whose mass is less than 8.4 kilograms. So if we look here, the probability of finding a baby, or a female baby, that's one year old, with a mass or a weight of less than 8.4 kilograms, that's this area right here. I said mass because kilograms is actually a unit of mass. But most people use it as weight as well."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Or maybe I should say whose mass is less than 8.4 kilograms. So if we look here, the probability of finding a baby, or a female baby, that's one year old, with a mass or a weight of less than 8.4 kilograms, that's this area right here. I said mass because kilograms is actually a unit of mass. But most people use it as weight as well. So that's that area right there. So how can we figure out that area under this normal distribution using the empirical rule? Well, we know what this area is."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "But most people use it as weight as well. So that's that area right there. So how can we figure out that area under this normal distribution using the empirical rule? Well, we know what this area is. We know what this area between minus one standard deviation and plus one standard deviation is. We know that that is 68%. And if that's 68%, then that means in the parts that aren't in that middle region, you have 32%."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Well, we know what this area is. We know what this area between minus one standard deviation and plus one standard deviation is. We know that that is 68%. And if that's 68%, then that means in the parts that aren't in that middle region, you have 32%. Because the area under the entire normal distribution is 100% or 100%, or 1%, depending on how you want to think about it, because you can't have all of the possibilities combined, it can only add up to 1. You can't have it more than 100% there. So if you add up this leg and this leg, so this plus that leg, is going to be the remainder."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And if that's 68%, then that means in the parts that aren't in that middle region, you have 32%. Because the area under the entire normal distribution is 100% or 100%, or 1%, depending on how you want to think about it, because you can't have all of the possibilities combined, it can only add up to 1. You can't have it more than 100% there. So if you add up this leg and this leg, so this plus that leg, is going to be the remainder. So 100 minus 68, that's 32%. 32% is if you add up this left leg and this right leg over here. And this is a perfect normal distribution."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So if you add up this leg and this leg, so this plus that leg, is going to be the remainder. So 100 minus 68, that's 32%. 32% is if you add up this left leg and this right leg over here. And this is a perfect normal distribution. They told us it's normally distributed, so it's going to be perfectly symmetrical. So if this side and that side add up to 32, but they're both symmetrical, meaning they have the exact same area, then this side right here, do it in pink, this side right here, I ended up looking more like purple, would be 16%. And this side right here would be 16%."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And this is a perfect normal distribution. They told us it's normally distributed, so it's going to be perfectly symmetrical. So if this side and that side add up to 32, but they're both symmetrical, meaning they have the exact same area, then this side right here, do it in pink, this side right here, I ended up looking more like purple, would be 16%. And this side right here would be 16%. So your probability of getting a result more than one standard deviation above the mean, so this right-hand side would be 16%, or the probability of having a result less than one standard deviation below the mean, that's this right here, 16%. So they want to know the probability of having a baby, or at one year's old, less than 8.4 kilograms. Less than 8.4 kilograms is this area right here, and that's 16%."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And this side right here would be 16%. So your probability of getting a result more than one standard deviation above the mean, so this right-hand side would be 16%, or the probability of having a result less than one standard deviation below the mean, that's this right here, 16%. So they want to know the probability of having a baby, or at one year's old, less than 8.4 kilograms. Less than 8.4 kilograms is this area right here, and that's 16%. So that's 16% for part A. Let's do part B. Between 7.3 and 11.7 kilograms."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Less than 8.4 kilograms is this area right here, and that's 16%. So that's 16% for part A. Let's do part B. Between 7.3 and 11.7 kilograms. So between 7.3, that's right there, that's two standard deviations below the mean, and 11.7. It's one, two standard deviations above the mean. So they're essentially asking us, what's the probability of getting a result within two standard deviations of the mean?"}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Between 7.3 and 11.7 kilograms. So between 7.3, that's right there, that's two standard deviations below the mean, and 11.7. It's one, two standard deviations above the mean. So they're essentially asking us, what's the probability of getting a result within two standard deviations of the mean? This is the mean right here. This is two standard deviations below. This is two standard deviations above."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So they're essentially asking us, what's the probability of getting a result within two standard deviations of the mean? This is the mean right here. This is two standard deviations below. This is two standard deviations above. Well, that's pretty straightforward. The empirical rule tells us, between two standard deviations, you have a 95% chance of getting that result. Or a 95% chance of getting a result that is within two standard deviations."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "This is two standard deviations above. Well, that's pretty straightforward. The empirical rule tells us, between two standard deviations, you have a 95% chance of getting that result. Or a 95% chance of getting a result that is within two standard deviations. So the empirical rule just gives us that answer. And then finally, part C. The probability of having a one-year-old US baby girl more than 12.8 kilograms. So 12.8 kilograms is three standard deviations above the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Or a 95% chance of getting a result that is within two standard deviations. So the empirical rule just gives us that answer. And then finally, part C. The probability of having a one-year-old US baby girl more than 12.8 kilograms. So 12.8 kilograms is three standard deviations above the mean. So we want to know the probability of having more than three standard deviations above the mean. So that is this area way out there. I drew an orange."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So 12.8 kilograms is three standard deviations above the mean. So we want to know the probability of having more than three standard deviations above the mean. So that is this area way out there. I drew an orange. Maybe I should do it in a different color to really contrast it. So it's this long tail out here. This little small area."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "I drew an orange. Maybe I should do it in a different color to really contrast it. So it's this long tail out here. This little small area. So what is that probability? So let's turn back to our empirical rule. Well, we know the probability."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "This little small area. So what is that probability? So let's turn back to our empirical rule. Well, we know the probability. We know this area. We know the area between minus three standard deviations and plus three standard deviations. We know this."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Well, we know the probability. We know this area. We know the area between minus three standard deviations and plus three standard deviations. We know this. I can, since this is the last problem, I can color the whole thing in. We know this area right here. Between minus three and plus three, that is 99.7%."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "We know this. I can, since this is the last problem, I can color the whole thing in. We know this area right here. Between minus three and plus three, that is 99.7%. The bulk of the results fall under there. I mean, almost all of them. So what do we have left over for the two tails?"}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Between minus three and plus three, that is 99.7%. The bulk of the results fall under there. I mean, almost all of them. So what do we have left over for the two tails? Remember, there are two tails. This is one of them. And then you have the results that are less than three standard deviations below the mean."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "So what do we have left over for the two tails? Remember, there are two tails. This is one of them. And then you have the results that are less than three standard deviations below the mean. This tail right there. So that tells us that less than three standard deviations below the mean and more than three standard deviations above the mean combined have to be the rest. Well, the rest, it's only 0.3% for the rest."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "And then you have the results that are less than three standard deviations below the mean. This tail right there. So that tells us that less than three standard deviations below the mean and more than three standard deviations above the mean combined have to be the rest. Well, the rest, it's only 0.3% for the rest. And these two things are symmetrical. They're going to be equal. So this right here has to be half of this, or 0.15%."}, {"video_title": "ck12.org normal distribution problems Empirical rule Probability and Statistics Khan Academy.mp3", "Sentence": "Well, the rest, it's only 0.3% for the rest. And these two things are symmetrical. They're going to be equal. So this right here has to be half of this, or 0.15%. And this right here is going to be 0.15%. So the probability of having a one-year-old baby girl in the US that is more than 12.8 kilograms, if you assume a perfect normal distribution, is the area under this curve, the area that is more than three standard deviations above the mean. And that is 0.15%."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "The specific test they use has a false positive rate of 2% and a false negative rate of 1%. Suppose that 5% of all their applicants are actually using illegal drugs and we randomly select an applicant. Given the applicant tests positive, what is the probability that they are actually on drugs? So let's work through this together. So first, let's just make sure we understand what they're telling us. So there is this drug test for the job applicants and then the test has a false positive rate of 2%. What does that mean?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "So let's work through this together. So first, let's just make sure we understand what they're telling us. So there is this drug test for the job applicants and then the test has a false positive rate of 2%. What does that mean? That means that in 2% of the cases when it should have read negative, that the person didn't do the drugs, it actually read positive. It is a false positive. It should have read negative, but it read positive."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "What does that mean? That means that in 2% of the cases when it should have read negative, that the person didn't do the drugs, it actually read positive. It is a false positive. It should have read negative, but it read positive. Another way to think about it, if someone did not do drugs and you take this test, there's a 2% chance saying that you did do the illegal drugs. They also say that there is a false negative rate of 1%. What does that mean?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "It should have read negative, but it read positive. Another way to think about it, if someone did not do drugs and you take this test, there's a 2% chance saying that you did do the illegal drugs. They also say that there is a false negative rate of 1%. What does that mean? That means that 1% of the time, if someone did actually take the illegal drugs, it'll say that they didn't. It is falsely giving a negative result when it should have given a positive one. And then they say that 5% of all their applicants are actually using illegal drugs."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "What does that mean? That means that 1% of the time, if someone did actually take the illegal drugs, it'll say that they didn't. It is falsely giving a negative result when it should have given a positive one. And then they say that 5% of all their applicants are actually using illegal drugs. So there's several ways that we can think about it. One of the easiest ways to conceptualize, let's just make up a large number of applicants and I'll use a number where it's fairly straightforward to do the mathematics. So let's say that we start off with 10,000 applicants."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And then they say that 5% of all their applicants are actually using illegal drugs. So there's several ways that we can think about it. One of the easiest ways to conceptualize, let's just make up a large number of applicants and I'll use a number where it's fairly straightforward to do the mathematics. So let's say that we start off with 10,000 applicants. And so I will both talk in absolute numbers, and I just made this number up. It could have been 1,000, it could have been 100,000, but I like this number because it's easy to do the math. It's better than saying 9,785."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "So let's say that we start off with 10,000 applicants. And so I will both talk in absolute numbers, and I just made this number up. It could have been 1,000, it could have been 100,000, but I like this number because it's easy to do the math. It's better than saying 9,785. And so this is also going to be 100% of the applicants. Now they give us some crucial information here. They tell us that 5% of all their applicants are actually using illegal drugs."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "It's better than saying 9,785. And so this is also going to be 100% of the applicants. Now they give us some crucial information here. They tell us that 5% of all their applicants are actually using illegal drugs. So we can immediately break this 10,000 group into the ones that are doing the drugs and the ones that are not. So 5% are actually on the drugs. 95% are not on the drugs."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "They tell us that 5% of all their applicants are actually using illegal drugs. So we can immediately break this 10,000 group into the ones that are doing the drugs and the ones that are not. So 5% are actually on the drugs. 95% are not on the drugs. So what's 5% of 10,000? So that would be 500. So 500 on drugs."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "95% are not on the drugs. So what's 5% of 10,000? So that would be 500. So 500 on drugs. On drugs. And so once again, this is 5% of our original population. And then how many are not on drugs?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "So 500 on drugs. On drugs. And so once again, this is 5% of our original population. And then how many are not on drugs? Well, 9,500 not. Not on drugs. And once again, this is 95% of our group of applicants."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And then how many are not on drugs? Well, 9,500 not. Not on drugs. And once again, this is 95% of our group of applicants. So now let's administer the test. So what is going to happen when we administer the test to the people who are on drugs? Well, the test ideally would give a positive result."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And once again, this is 95% of our group of applicants. So now let's administer the test. So what is going to happen when we administer the test to the people who are on drugs? Well, the test ideally would give a positive result. It would say positive for all of them, but we know that it's not a perfect test. It's going to give negative for some of them. It will falsely give a negative result for some of them."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Well, the test ideally would give a positive result. It would say positive for all of them, but we know that it's not a perfect test. It's going to give negative for some of them. It will falsely give a negative result for some of them. And we know that because it has a false negative rate of 1%. And so of these 500, 99% is going to get the correct result in that they're going to test positive. So what is 99% of 500?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "It will falsely give a negative result for some of them. And we know that because it has a false negative rate of 1%. And so of these 500, 99% is going to get the correct result in that they're going to test positive. So what is 99% of 500? Well, let's see, that would be 495. 495 are going to test positive. I will just use a positive right over there."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "So what is 99% of 500? Well, let's see, that would be 495. 495 are going to test positive. I will just use a positive right over there. And then we're going to have 1%, 1%, which is five, are going to test negative. They are going to falsely test negative. This is the false negative rate."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "I will just use a positive right over there. And then we're going to have 1%, 1%, which is five, are going to test negative. They are going to falsely test negative. This is the false negative rate. And so if we say what percent of our original applicant pool is on drugs and tests positive? Well, 495 over 10,000, this is 4.95%. What percent is of the original applicant pool that is on drugs but tests negative for drugs?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "This is the false negative rate. And so if we say what percent of our original applicant pool is on drugs and tests positive? Well, 495 over 10,000, this is 4.95%. What percent is of the original applicant pool that is on drugs but tests negative for drugs? The test says that, hey, they're not taking drugs. Well, this is going to be five out of 10,000, which is 0.05%. Another way that you can get these percentages, if you take 5% and multiply by 1%, you're going to get 0.05%, 5 hundredths of a percent."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "What percent is of the original applicant pool that is on drugs but tests negative for drugs? The test says that, hey, they're not taking drugs. Well, this is going to be five out of 10,000, which is 0.05%. Another way that you can get these percentages, if you take 5% and multiply by 1%, you're going to get 0.05%, 5 hundredths of a percent. If you take 5% and multiply by 99%, you're going to get 4.95%. Now let's keep going. Now let's go to the folks who aren't taking the drugs."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Another way that you can get these percentages, if you take 5% and multiply by 1%, you're going to get 0.05%, 5 hundredths of a percent. If you take 5% and multiply by 99%, you're going to get 4.95%. Now let's keep going. Now let's go to the folks who aren't taking the drugs. And this is where the false positive rate is going to come into effect. So we have a false positive rate of 2%. So 2% are going to test positive."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Now let's go to the folks who aren't taking the drugs. And this is where the false positive rate is going to come into effect. So we have a false positive rate of 2%. So 2% are going to test positive. What's 2% of 9,500? It's 190 would test positive, even though they're not on drugs. This is the false positive rate."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "So 2% are going to test positive. What's 2% of 9,500? It's 190 would test positive, even though they're not on drugs. This is the false positive rate. So they are testing positive. And then the other 98% will correctly come out negative. And so the other 98%, so 9,500 minus 190, that's going to be 9,310 will correctly test negative."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "This is the false positive rate. So they are testing positive. And then the other 98% will correctly come out negative. And so the other 98%, so 9,500 minus 190, that's going to be 9,310 will correctly test negative. Now what percent of the original applicant pool is this? Well, 190 is 1.9%. And we could calculate it by 190 over 10,000, or you could just say 2% of 95% is 1.9%."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And so the other 98%, so 9,500 minus 190, that's going to be 9,310 will correctly test negative. Now what percent of the original applicant pool is this? Well, 190 is 1.9%. And we could calculate it by 190 over 10,000, or you could just say 2% of 95% is 1.9%. Once again, multiply the path along the tree. What percent is 9,310? Well, that is going to be 93.10%."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And we could calculate it by 190 over 10,000, or you could just say 2% of 95% is 1.9%. Once again, multiply the path along the tree. What percent is 9,310? Well, that is going to be 93.10%. You could say this is 9,310 over 10,000, or you can multiply by the path on our probability tree here. 95% times 98% gets us to 93.10%. But now I think we are ready to answer the question."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Well, that is going to be 93.10%. You could say this is 9,310 over 10,000, or you can multiply by the path on our probability tree here. 95% times 98% gets us to 93.10%. But now I think we are ready to answer the question. Given that the applicant tests positive, what is the probability that they are actually on drugs? So let's look at the first part. Given the applicant tests positive."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "But now I think we are ready to answer the question. Given that the applicant tests positive, what is the probability that they are actually on drugs? So let's look at the first part. Given the applicant tests positive. So which applicants actually tested positive? You have these 495 here tested positive, correctly tested positive, and then you have these 190 right over here incorrectly tested positive, but they did test positive. So how many tested positive?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Given the applicant tests positive. So which applicants actually tested positive? You have these 495 here tested positive, correctly tested positive, and then you have these 190 right over here incorrectly tested positive, but they did test positive. So how many tested positive? Well, we have 495 plus 190 tested positive. That's the total number that tested positive. And then which of them were actually on the drugs?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "So how many tested positive? Well, we have 495 plus 190 tested positive. That's the total number that tested positive. And then which of them were actually on the drugs? Well, of the ones that tested positive, 495 were actually on the drugs. We have 495 divided by 495 plus 190 is equal to 0.7226, so we could say approximately 72%. Approximately 72%."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And then which of them were actually on the drugs? Well, of the ones that tested positive, 495 were actually on the drugs. We have 495 divided by 495 plus 190 is equal to 0.7226, so we could say approximately 72%. Approximately 72%. Now this is really interesting. Given the applicant tests positive, what is the probability that they are actually on drugs? When you look at these false positive and false negative rates, they seem quite low."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Approximately 72%. Now this is really interesting. Given the applicant tests positive, what is the probability that they are actually on drugs? When you look at these false positive and false negative rates, they seem quite low. But now when we actually did the calculation, the probability that someone's actually on drugs, it's high, but it's not that high. It's not like if someone were to test positive that you'd say, oh, they are definitely taking the drugs. And you could also get to this result just by using the percentages."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "When you look at these false positive and false negative rates, they seem quite low. But now when we actually did the calculation, the probability that someone's actually on drugs, it's high, but it's not that high. It's not like if someone were to test positive that you'd say, oh, they are definitely taking the drugs. And you could also get to this result just by using the percentages. For example, you could think in terms of what percentage of the original applicants end up testing positive? Well, that's 4.95% plus 1.9%. 4.95, we'll just do it in terms of percent, plus 1.9%, and of them, what percentage were actually on the drugs?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "And you could also get to this result just by using the percentages. For example, you could think in terms of what percentage of the original applicants end up testing positive? Well, that's 4.95% plus 1.9%. 4.95, we'll just do it in terms of percent, plus 1.9%, and of them, what percentage were actually on the drugs? Well, that was the 4.95%. And notice, this would give you the exact same result. Now there's an interesting takeaway here."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "4.95, we'll just do it in terms of percent, plus 1.9%, and of them, what percentage were actually on the drugs? Well, that was the 4.95%. And notice, this would give you the exact same result. Now there's an interesting takeaway here. Because this is saying, of the people that test positive, 72% are actually on the drugs. You could think about it the other way around. Of the people who test positive, 4.95 plus 1.90, what percentage aren't on drugs?"}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Now there's an interesting takeaway here. Because this is saying, of the people that test positive, 72% are actually on the drugs. You could think about it the other way around. Of the people who test positive, 4.95 plus 1.90, what percentage aren't on drugs? Well, that was 1.90. And this comes out to be approximately 28%. 100% minus 72%."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Of the people who test positive, 4.95 plus 1.90, what percentage aren't on drugs? Well, that was 1.90. And this comes out to be approximately 28%. 100% minus 72%. And so, if we were in a court of law, and let's say the prosecuting attorney, let's say I got tested positive for drugs, and the prosecuting attorney says, look, this test is very good. It only has a false positive rate of 2%. Sal, and Sal tested positive, he is probably taking drugs."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "100% minus 72%. And so, if we were in a court of law, and let's say the prosecuting attorney, let's say I got tested positive for drugs, and the prosecuting attorney says, look, this test is very good. It only has a false positive rate of 2%. Sal, and Sal tested positive, he is probably taking drugs. A jury who doesn't really understand this well, or go through the trouble that we just did, might say, oh yeah, Sal probably took the drugs. But when we look at this, even if I test positive using this test, there's a 28% chance that I'm not taking drugs, that I was just in this false positive group. And the reason why this number is a good bit larger than this number is because when we looked at the original division between those who take drugs and don't take drugs, most don't take the illegal drugs."}, {"video_title": "Conditional probability tree diagram example Probability AP Statistics Khan Academy.mp3", "Sentence": "Sal, and Sal tested positive, he is probably taking drugs. A jury who doesn't really understand this well, or go through the trouble that we just did, might say, oh yeah, Sal probably took the drugs. But when we look at this, even if I test positive using this test, there's a 28% chance that I'm not taking drugs, that I was just in this false positive group. And the reason why this number is a good bit larger than this number is because when we looked at the original division between those who take drugs and don't take drugs, most don't take the illegal drugs. And so 2% of this larger group of the ones that don't take the drugs, well, this is actually a fairly large number relative to the percentage that do take the drugs and test positive. So I will leave you there. This is fascinating, not just for this particular case, but you will see analysis like this all the time when we're looking at whether a certain medication is effective or a certain procedure is effective."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "You roll a fair six-sided die three times. If the sum of the rolls is 10 or greater, you win. If it is less than 10, you lose. What is the probability of winning three rolls to 10? So there are several ways that you can approach this. The way we're going to tackle it in this video is we're going to try to come up with an experimental probability. We're going to do many experiments trying to win three rolls to 10 and figure out the proportion that we actually win."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "What is the probability of winning three rolls to 10? So there are several ways that you can approach this. The way we're going to tackle it in this video is we're going to try to come up with an experimental probability. We're going to do many experiments trying to win three rolls to 10 and figure out the proportion that we actually win. And the more experiments we try, the better, the more likely that we're gonna get a good approximation of the actual probability. So let's do that. And to help us, I'm going to have a computer generate a string of random digits from zero to nine."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "We're going to do many experiments trying to win three rolls to 10 and figure out the proportion that we actually win. And the more experiments we try, the better, the more likely that we're gonna get a good approximation of the actual probability. So let's do that. And to help us, I'm going to have a computer generate a string of random digits from zero to nine. And the way that we're going to use this is, remember, we're rolling a fair six-sided die. So the outcome could be one, two, three, four, five, or six for each roll. In this random number list that the computer has generated, I do get digits from one to six, but I also get the digits seven, eight, nine, and zero."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "And to help us, I'm going to have a computer generate a string of random digits from zero to nine. And the way that we're going to use this is, remember, we're rolling a fair six-sided die. So the outcome could be one, two, three, four, five, or six for each roll. In this random number list that the computer has generated, I do get digits from one to six, but I also get the digits seven, eight, nine, and zero. And so what I'm going to do for each experiment, I'm gonna start at the top left and I'm gonna consider each digit a roll. If it gives me an invalid result for a six-sided die, so if it's a zero, an eight, a seven, or a nine, I will just ignore that. I will just say, well, that wasn't a valid roll."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "In this random number list that the computer has generated, I do get digits from one to six, but I also get the digits seven, eight, nine, and zero. And so what I'm going to do for each experiment, I'm gonna start at the top left and I'm gonna consider each digit a roll. If it gives me an invalid result for a six-sided die, so if it's a zero, an eight, a seven, or a nine, I will just ignore that. I will just say, well, that wasn't a valid roll. It's like you roll the die and it fell off the table or something like that. So let's do that. Let's do multiple experiments of taking three rolls, sum them up, and we'll see how many we can do to figure out an experimental probability of winning Pascal's game."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "I will just say, well, that wasn't a valid roll. It's like you roll the die and it fell off the table or something like that. So let's do that. Let's do multiple experiments of taking three rolls, sum them up, and we'll see how many we can do to figure out an experimental probability of winning Pascal's game. So let me set up a little table here. So I want space to show the sum. So this is going to be the experiments."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Let's do multiple experiments of taking three rolls, sum them up, and we'll see how many we can do to figure out an experimental probability of winning Pascal's game. So let me set up a little table here. So I want space to show the sum. So this is going to be the experiments. Experiment. So let me write the sum. And over here, we're gonna say, did we win?"}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So this is going to be the experiments. Experiment. So let me write the sum. And over here, we're gonna say, did we win? All right, so let's start with experiment one. So our first roll, we got a one. Our second roll, we got a five."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "And over here, we're gonna say, did we win? All right, so let's start with experiment one. So our first roll, we got a one. Our second roll, we got a five. We're doing quite well. And then our third roll, we got a six. Did we win?"}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Our second roll, we got a five. We're doing quite well. And then our third roll, we got a six. Did we win? Well, one plus five plus six is 12. Yes, we won. Let's do another experiment."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Did we win? Well, one plus five plus six is 12. Yes, we won. Let's do another experiment. This is going to be experiment two. We can just keep going here. These are random digits."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Let's do another experiment. This is going to be experiment two. We can just keep going here. These are random digits. So we have a six in our first roll. We got a two in our second roll. We got a four in our third roll."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "These are random digits. So we have a six in our first roll. We got a two in our second roll. We got a four in our third roll. Did we win? Yes, once again, this summed up to 12. So we won."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "We got a four in our third roll. Did we win? Yes, once again, this summed up to 12. So we won. All right, let's do another experiment. So experiment number three. So this first thing is invalid."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So we won. All right, let's do another experiment. So experiment number three. So this first thing is invalid. So this is our first roll. We got a six. And then this is invalid."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So this first thing is invalid. So this is our first roll. We got a six. And then this is invalid. Our second roll, we get a three. This is invalid, that is invalid, that is invalid. And then in our third roll, we got a two."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "And then this is invalid. Our second roll, we get a three. This is invalid, that is invalid, that is invalid. And then in our third roll, we got a two. So we squeaked by, this adds up to 11. Yes, that looks like a win. All right, let's do our fourth experiment here."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "And then in our third roll, we got a two. So we squeaked by, this adds up to 11. Yes, that looks like a win. All right, let's do our fourth experiment here. So our first roll, we got a one. This is invalid. Second roll, we got a two."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "All right, let's do our fourth experiment here. So our first roll, we got a one. This is invalid. Second roll, we got a two. This is invalid. Third roll, we get a five. Did we win?"}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Second roll, we got a two. This is invalid. Third roll, we get a five. Did we win? One plus two plus five is eight. No, we did not win. So that was our first non-win."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Did we win? One plus two plus five is eight. No, we did not win. So that was our first non-win. So let's keep going. This is interesting. All right, this is invalid."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So that was our first non-win. So let's keep going. This is interesting. All right, this is invalid. So we're going to have, so this is trial five. We are going to have four plus three plus one. Four plus three plus one adds up to eight."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "All right, this is invalid. So we're going to have, so this is trial five. We are going to have four plus three plus one. Four plus three plus one adds up to eight. Did we win? No. Let's just keep going here."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Four plus three plus one adds up to eight. Did we win? No. Let's just keep going here. So I'm gonna keep going with my table where I have experiment. I'll do five more trials. Experiment, sum, and do we win?"}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Let's just keep going here. So I'm gonna keep going with my table where I have experiment. I'll do five more trials. Experiment, sum, and do we win? Let me make the table. This is just a continuation of the table we had before. I don't wanna go below the page because I wanna be able to look at our random numbers here."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Experiment, sum, and do we win? Let me make the table. This is just a continuation of the table we had before. I don't wanna go below the page because I wanna be able to look at our random numbers here. So we are on to experiment six. Experiment six, we are getting a three in the first roll, a three in the second roll. This isn't looking good."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "I don't wanna go below the page because I wanna be able to look at our random numbers here. So we are on to experiment six. Experiment six, we are getting a three in the first roll, a three in the second roll. This isn't looking good. And then a two in our third roll. Did we win? No, this is less than 10."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "This isn't looking good. And then a two in our third roll. Did we win? No, this is less than 10. Now we go to experiment seven. Experiment seven, we get a two in our first roll. This is invalid."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "No, this is less than 10. Now we go to experiment seven. Experiment seven, we get a two in our first roll. This is invalid. We get a three in our second roll, plus three. And we get a one in our third roll, so plus one. Once again, we did not win."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "This is invalid. We get a three in our second roll, plus three. And we get a one in our third roll, so plus one. Once again, we did not win. Now we go to experiment, we will go to experiment eight. We get a one in our first roll. We get a three in our second roll."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Once again, we did not win. Now we go to experiment, we will go to experiment eight. We get a one in our first roll. We get a three in our second roll. This is invalid. The die fell off the table. We can think of it that way."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "We get a three in our second roll. This is invalid. The die fell off the table. We can think of it that way. And then in our third roll, we get a five, plus five. Did we win? No, this adds up to nine."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "We can think of it that way. And then in our third roll, we get a five, plus five. Did we win? No, this adds up to nine. So we had a string of wins to begin with, but now we're getting a string of non-wins. All right, now let's go to experiment nine. So we get a six in our first roll."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "No, this adds up to nine. So we had a string of wins to begin with, but now we're getting a string of non-wins. All right, now let's go to experiment nine. So we get a six in our first roll. We get a four in our second roll. And then these are all invalid. And then we get a five in our third roll."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "So we get a six in our first roll. We get a four in our second roll. And then these are all invalid. And then we get a five in our third roll. Did we win here? Yes, we won over here. This is definitely going to be greater than 10."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "And then we get a five in our third roll. Did we win here? Yes, we won over here. This is definitely going to be greater than 10. This is 15 here. All right, last experiment, or at least for this video, last experiment. You could keep going."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "This is definitely going to be greater than 10. This is 15 here. All right, last experiment, or at least for this video, last experiment. You could keep going. In fact, I encourage you to after this to see if you can get a more accurate, a better approximation of the theoretical probability of winning by doing more experiments to calculate an experimental probability. So here, experiment 10. First roll, we get a five."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "You could keep going. In fact, I encourage you to after this to see if you can get a more accurate, a better approximation of the theoretical probability of winning by doing more experiments to calculate an experimental probability. So here, experiment 10. First roll, we get a five. Second roll, we get a two. This is invalid, invalid, invalid. Then we get a six."}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "First roll, we get a five. Second roll, we get a two. This is invalid, invalid, invalid. Then we get a six. Here, we definitely won. So with 10 trials, based on 10 trials, or 10 experiments, what is our experimental probability of winning this game? Well, out of the 10 experiments, how many did we win?"}, {"video_title": "Random numbers for experimental probability Probability AP Statistics Khan Academy.mp3", "Sentence": "Then we get a six. Here, we definitely won. So with 10 trials, based on 10 trials, or 10 experiments, what is our experimental probability of winning this game? Well, out of the 10 experiments, how many did we win? Well, it looks like we won one, two, three, four, five. So based on just these 10 experiments, we got a pretty clean 50%. So do you think the theoretical probability is actually 50%?"}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "We're told that Maya and Doug are finalists in a crafting competition. For the final round, each of them spin a wheel to determine what star material must be in their craft. Maya and Doug both want to get silk as their star material. Maya will spin first, followed by Doug. What is the probability that neither contestant gets silk? Pause this video and think through this on your own before we work through this together. All right, so first let's think about what they're asking."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "Maya will spin first, followed by Doug. What is the probability that neither contestant gets silk? Pause this video and think through this on your own before we work through this together. All right, so first let's think about what they're asking. They want to figure out the probability that neither gets silk. So I'm gonna write this in shorthand. So I'm gonna use MNS for Maya no silk."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "All right, so first let's think about what they're asking. They want to figure out the probability that neither gets silk. So I'm gonna write this in shorthand. So I'm gonna use MNS for Maya no silk. And we're also thinking about Doug not being able to pick silk. So Maya no silk and Doug no silk. So we know that this could be viewed as the probability that Maya doesn't get silk."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "So I'm gonna use MNS for Maya no silk. And we're also thinking about Doug not being able to pick silk. So Maya no silk and Doug no silk. So we know that this could be viewed as the probability that Maya doesn't get silk. She, after all, does get to spin this wheel first. And then we can multiply that by the probability that Doug doesn't get silk, Doug no silk, given that Maya did not get silk, Maya no silk. Now it's important to think about whether Doug's probability is independent or dependent on whether Maya got silk or not."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "So we know that this could be viewed as the probability that Maya doesn't get silk. She, after all, does get to spin this wheel first. And then we can multiply that by the probability that Doug doesn't get silk, Doug no silk, given that Maya did not get silk, Maya no silk. Now it's important to think about whether Doug's probability is independent or dependent on whether Maya got silk or not. So let's remember, Maya will spin first, but it's not like if she picks silk that somehow silk is taken out of the running. In fact, no matter what she picks, it's not taken out of the running. Doug will then spin it again."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "Now it's important to think about whether Doug's probability is independent or dependent on whether Maya got silk or not. So let's remember, Maya will spin first, but it's not like if she picks silk that somehow silk is taken out of the running. In fact, no matter what she picks, it's not taken out of the running. Doug will then spin it again. And so these are really two independent events. And so the probability that Doug doesn't get silk given that Maya doesn't get silk, this is going to be the same thing as the probability that just Doug doesn't get silk. It doesn't matter what happens to Maya."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "Doug will then spin it again. And so these are really two independent events. And so the probability that Doug doesn't get silk given that Maya doesn't get silk, this is going to be the same thing as the probability that just Doug doesn't get silk. It doesn't matter what happens to Maya. And so what are each of these? Well, this is all going to be equal to the probability that Maya does not get silk. There's six pieces or six options of this wheel right over here."}, {"video_title": "General multiplication rule example independent events Probability & combinatorics.mp3", "Sentence": "It doesn't matter what happens to Maya. And so what are each of these? Well, this is all going to be equal to the probability that Maya does not get silk. There's six pieces or six options of this wheel right over here. Five of them entail her not getting silk on her spin. So five over six. And then similarly, when Doug goes to spin this wheel, there are six possibilities."}, {"video_title": "Invalid conclusions from studies example Study design AP Statistics Khan Academy.mp3", "Sentence": "Jerry was reading about a study that looked at the connection between smartphone usage and happiness based on data from approximately 5,000 randomly selected teenagers. The study found that on average, the teens who spent more time on smartphones were significantly less happy than those who spent less time on smartphones. Jerry concluded that spending more time on smartphones makes teens less happy. All right, this is interesting. So what I want you to do is think about whether Jerry is making a valid conclusion or not and why or why don't you think he's making a valid conclusion. All right, now let's work on this together. This is really important to understand because you will see things like this in the popular media all the time that try to establish a causality when there might not be causality or at least where the study might not be able to show causality."}, {"video_title": "Invalid conclusions from studies example Study design AP Statistics Khan Academy.mp3", "Sentence": "All right, this is interesting. So what I want you to do is think about whether Jerry is making a valid conclusion or not and why or why don't you think he's making a valid conclusion. All right, now let's work on this together. This is really important to understand because you will see things like this in the popular media all the time that try to establish a causality when there might not be causality or at least where the study might not be able to show causality. So right now, Jerry is saying he's concluding that smartphone usage, smartphone usage, makes teens less happy. So he's assuming there's a causal connection. Smartphone usage causes teens to be less happy, less happy."}, {"video_title": "Invalid conclusions from studies example Study design AP Statistics Khan Academy.mp3", "Sentence": "This is really important to understand because you will see things like this in the popular media all the time that try to establish a causality when there might not be causality or at least where the study might not be able to show causality. So right now, Jerry is saying he's concluding that smartphone usage, smartphone usage, makes teens less happy. So he's assuming there's a causal connection. Smartphone usage causes teens to be less happy, less happy. Can he actually make that conclusion from this study based on how it was designed? Well, the first thing to ask ourselves is is this an experimental study that is designed to establish causality or is it an observational study where we really can just say there's an association but we really can't make a statement about causality? Well, an experimental study, he would have had to have a control group and then a treatment group, sometimes called a experimental group."}, {"video_title": "Invalid conclusions from studies example Study design AP Statistics Khan Academy.mp3", "Sentence": "Smartphone usage causes teens to be less happy, less happy. Can he actually make that conclusion from this study based on how it was designed? Well, the first thing to ask ourselves is is this an experimental study that is designed to establish causality or is it an observational study where we really can just say there's an association but we really can't make a statement about causality? Well, an experimental study, he would have had to have a control group and then a treatment group, sometimes called a experimental group. So I'll say that's control group, that's the treatment or the experimental group. And then you randomly assign folks to one of those two groups and then you would make that treatment group use the cell phone more and see if they are less happy. That's not what happened here."}, {"video_title": "Invalid conclusions from studies example Study design AP Statistics Khan Academy.mp3", "Sentence": "Well, an experimental study, he would have had to have a control group and then a treatment group, sometimes called a experimental group. So I'll say that's control group, that's the treatment or the experimental group. And then you randomly assign folks to one of those two groups and then you would make that treatment group use the cell phone more and see if they are less happy. That's not what happened here. What happened here was an observational study. In this study, we are looking at two variables. So you have your smartphone usage and then you have the teen happiness."}, {"video_title": "Invalid conclusions from studies example Study design AP Statistics Khan Academy.mp3", "Sentence": "That's not what happened here. What happened here was an observational study. In this study, we are looking at two variables. So you have your smartphone usage and then you have the teen happiness. And they took these 5,000 randomly selected teenagers and they figured out their smartphone usage and their happiness, maybe with a survey of some kind. And then you could plot those data points. You would have 5,000 data points."}, {"video_title": "Invalid conclusions from studies example Study design AP Statistics Khan Academy.mp3", "Sentence": "So you have your smartphone usage and then you have the teen happiness. And they took these 5,000 randomly selected teenagers and they figured out their smartphone usage and their happiness, maybe with a survey of some kind. And then you could plot those data points. You would have 5,000 data points. So this data point right over here would be a very happy teenager that doesn't use a smartphone much. This would be a not-so-happy teenager that uses a smartphone a lot. And so you would plot those data points and there might be a teenager who's unhappy and doesn't use a smartphone or one that is happy and that uses a smartphone a lot."}, {"video_title": "Invalid conclusions from studies example Study design AP Statistics Khan Academy.mp3", "Sentence": "You would have 5,000 data points. So this data point right over here would be a very happy teenager that doesn't use a smartphone much. This would be a not-so-happy teenager that uses a smartphone a lot. And so you would plot those data points and there might be a teenager who's unhappy and doesn't use a smartphone or one that is happy and that uses a smartphone a lot. But you could see there's a trend, there's an association that, in general, the teenagers who use the smartphones more seem to be less happy and the teenagers who use the smartphones less seem to be more happy. But it's important to realize that the causality could go the other way around. Maybe less happy teenagers use their smartphones more and maybe more happy teenagers don't find a need to use a smartphone."}, {"video_title": "Invalid conclusions from studies example Study design AP Statistics Khan Academy.mp3", "Sentence": "And so you would plot those data points and there might be a teenager who's unhappy and doesn't use a smartphone or one that is happy and that uses a smartphone a lot. But you could see there's a trend, there's an association that, in general, the teenagers who use the smartphones more seem to be less happy and the teenagers who use the smartphones less seem to be more happy. But it's important to realize that the causality could go the other way around. Maybe less happy teenagers use their smartphones more and maybe more happy teenagers don't find a need to use a smartphone. Or there could be some variable that's not even being observed in this study that has a causal relationship with both of these. So there could be some other variable that might cause someone to be less happy and use their smartphone more. So in an observational study, you can really just say there's an association."}, {"video_title": "Invalid conclusions from studies example Study design AP Statistics Khan Academy.mp3", "Sentence": "Maybe less happy teenagers use their smartphones more and maybe more happy teenagers don't find a need to use a smartphone. Or there could be some variable that's not even being observed in this study that has a causal relationship with both of these. So there could be some other variable that might cause someone to be less happy and use their smartphone more. So in an observational study, you can really just say there's an association. You wouldn't be able to say that there is causality. So Jerry is not making a valid conclusion. It's an observational study."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "The school nurse plans to provide additional screening to students whose resting pulse rates are in the top 30% of the students who were tested. What is the minimum resting pulse rate at that school for students who will receive additional screening, round to the nearest whole number? If you feel like you know how to tackle this, I encourage you to pause this video and try to work it out. All right, now let's work this out together. They're telling us that the distribution of resting pulse rates are approximately normal. So we could use a normal distribution, and they tell us several things about this normal distribution. They tell us that the mean is 80 beats per minute, so that is the mean right over there, and they tell us that the standard deviation is nine beats per minute."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "All right, now let's work this out together. They're telling us that the distribution of resting pulse rates are approximately normal. So we could use a normal distribution, and they tell us several things about this normal distribution. They tell us that the mean is 80 beats per minute, so that is the mean right over there, and they tell us that the standard deviation is nine beats per minute. So on this normal distribution, we have one standard deviation above the mean, two standard deviations above the mean, so this distance right over here is nine, so this would be 89. This one right over here would be 98, and you could also go standard deviations below the mean. This right over here would be 71."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "They tell us that the mean is 80 beats per minute, so that is the mean right over there, and they tell us that the standard deviation is nine beats per minute. So on this normal distribution, we have one standard deviation above the mean, two standard deviations above the mean, so this distance right over here is nine, so this would be 89. This one right over here would be 98, and you could also go standard deviations below the mean. This right over here would be 71. This would be 62, but what we're concerned about is the top 30% because that is who is going to be tested. So there's gonna be some value here, some threshold. Let's say it is right over here, that if you are in that, if you are at that score, you have reached the minimum threshold to get additional screening."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "This right over here would be 71. This would be 62, but what we're concerned about is the top 30% because that is who is going to be tested. So there's gonna be some value here, some threshold. Let's say it is right over here, that if you are in that, if you are at that score, you have reached the minimum threshold to get additional screening. You are in the top 30%. So that means that this area right over here is going to be 30%, or 0.3. So what we can do, we can use a z-table to say for what z-score is 70% of the distribution less than that, and then we can take that z-score and use the mean and the standard deviation to come up with an actual value."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "Let's say it is right over here, that if you are in that, if you are at that score, you have reached the minimum threshold to get additional screening. You are in the top 30%. So that means that this area right over here is going to be 30%, or 0.3. So what we can do, we can use a z-table to say for what z-score is 70% of the distribution less than that, and then we can take that z-score and use the mean and the standard deviation to come up with an actual value. In previous examples, we started with a z-score and we're looking for the percentage. This time, we're looking for the percentage. We want it to be at least 70% and then come up with the corresponding z-score."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "So what we can do, we can use a z-table to say for what z-score is 70% of the distribution less than that, and then we can take that z-score and use the mean and the standard deviation to come up with an actual value. In previous examples, we started with a z-score and we're looking for the percentage. This time, we're looking for the percentage. We want it to be at least 70% and then come up with the corresponding z-score. So let's see, immediately when we look at this, and we are to the right of the mean, and so we're gonna have a positive z-score, so we're starting at 50% here. We definitely wanna get, this is 67%, 68, 69. We're getting close, and on our table, this is the lowest z-score that gets us across that 70% threshold."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "We want it to be at least 70% and then come up with the corresponding z-score. So let's see, immediately when we look at this, and we are to the right of the mean, and so we're gonna have a positive z-score, so we're starting at 50% here. We definitely wanna get, this is 67%, 68, 69. We're getting close, and on our table, this is the lowest z-score that gets us across that 70% threshold. It's at 0.7019. So it definitely crosses the threshold, and so that is a z-score of 0.53. 0.52 is too little."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "We're getting close, and on our table, this is the lowest z-score that gets us across that 70% threshold. It's at 0.7019. So it definitely crosses the threshold, and so that is a z-score of 0.53. 0.52 is too little. So we need a z-score of 0.53. Let's write that down. 0.53, right over there, and we just now have to figure out what value gives us a z-score of 0.53."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "0.52 is too little. So we need a z-score of 0.53. Let's write that down. 0.53, right over there, and we just now have to figure out what value gives us a z-score of 0.53. Well, this just means 0.53 standard deviations above the mean. So to get the value, we would take our mean, and we would add 0.53 standard deviation, so 0.53 times nine, and this will get us 0.53 times nine is equal to 4.77, plus 80 is equal to 84.77. 84.77, and they want us to round to the nearest whole number."}, {"video_title": "Finding z-score for a percentile AP Statistics Khan Academy.mp3", "Sentence": "0.53, right over there, and we just now have to figure out what value gives us a z-score of 0.53. Well, this just means 0.53 standard deviations above the mean. So to get the value, we would take our mean, and we would add 0.53 standard deviation, so 0.53 times nine, and this will get us 0.53 times nine is equal to 4.77, plus 80 is equal to 84.77. 84.77, and they want us to round to the nearest whole number. So we will just round to 85 beats per minute. So that's the threshold. If you have that resting heartbeat, then the school nurse is going to give you some additional screening."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "A hand is chosen, a hand is a collection of 9 cards, which can be sorted however the player chooses. Fair enough. How many 9 card hands are possible? So let's think about it. There are 36 unique cards, and I won't worry about there's 9 numbers in each suit, and there are 4 suits, 4 times 9 is 36. But let's just think of the cards as being 1 through 36, and we're going to pick 9 of them. So at first we'll say, well look, I have 9 slots in my hand, 1, 2, 3, 4, 5, 6, 7, 8, 9."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So let's think about it. There are 36 unique cards, and I won't worry about there's 9 numbers in each suit, and there are 4 suits, 4 times 9 is 36. But let's just think of the cards as being 1 through 36, and we're going to pick 9 of them. So at first we'll say, well look, I have 9 slots in my hand, 1, 2, 3, 4, 5, 6, 7, 8, 9. I'm going to pick 9 cards for my hand. And so for the very first card, how many possible cards can I pick from? Well, there's 36 unique cards, so for that first slot, there's 36."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So at first we'll say, well look, I have 9 slots in my hand, 1, 2, 3, 4, 5, 6, 7, 8, 9. I'm going to pick 9 cards for my hand. And so for the very first card, how many possible cards can I pick from? Well, there's 36 unique cards, so for that first slot, there's 36. But then that's now part of my hand. Now for the second slot, how many will there be left to pick from? Well, I've already picked 1, so there'll only be 35 to pick from, and then for the third slot, 34, and then it just keeps going, then 33 to pick from, 32, 31, 30, 29, and 28."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Well, there's 36 unique cards, so for that first slot, there's 36. But then that's now part of my hand. Now for the second slot, how many will there be left to pick from? Well, I've already picked 1, so there'll only be 35 to pick from, and then for the third slot, 34, and then it just keeps going, then 33 to pick from, 32, 31, 30, 29, and 28. So you might want to say that there are 36 times 35 times 34 times 33 times 32 times 31 times 30 times 29 times 28 possible hands. Now, this would be true if order mattered. This would be true if maybe I have card 15 here."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Well, I've already picked 1, so there'll only be 35 to pick from, and then for the third slot, 34, and then it just keeps going, then 33 to pick from, 32, 31, 30, 29, and 28. So you might want to say that there are 36 times 35 times 34 times 33 times 32 times 31 times 30 times 29 times 28 possible hands. Now, this would be true if order mattered. This would be true if maybe I have card 15 here. Maybe I have a 9 of spades here, and then I have a bunch of cards. And maybe I have, and that's one hand, and then I have another, so then I have cards 1, 2, 3, 4, 5, 6, 7, 8. I have 8 other cards."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "This would be true if maybe I have card 15 here. Maybe I have a 9 of spades here, and then I have a bunch of cards. And maybe I have, and that's one hand, and then I have another, so then I have cards 1, 2, 3, 4, 5, 6, 7, 8. I have 8 other cards. Or maybe another hand is I have the 8 cards, 1, 2, 3, 4, 5, 6, 7, 8, and then I have the 9 of spades. If we were thinking of these as two different hands, because we have the exact same cards, but they're in different order, then what I just calculated would make a lot of sense, because we did it based on order. But they're telling us that the cards can be sorted however the player chooses."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "I have 8 other cards. Or maybe another hand is I have the 8 cards, 1, 2, 3, 4, 5, 6, 7, 8, and then I have the 9 of spades. If we were thinking of these as two different hands, because we have the exact same cards, but they're in different order, then what I just calculated would make a lot of sense, because we did it based on order. But they're telling us that the cards can be sorted however the player chooses. So order doesn't matter. So we're over-counting. We're counting all of the different ways that the same number of cards can be arranged."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "But they're telling us that the cards can be sorted however the player chooses. So order doesn't matter. So we're over-counting. We're counting all of the different ways that the same number of cards can be arranged. So in order to, I guess, not over-count, we have to divide this by the way 9 cards can be rearranged. So how many ways can 9 cards be rearranged? If I have 9 cards, and I'm going to pick one of 9 to be in the first slot, well, that means I have 9 ways to put something in the first slot."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "We're counting all of the different ways that the same number of cards can be arranged. So in order to, I guess, not over-count, we have to divide this by the way 9 cards can be rearranged. So how many ways can 9 cards be rearranged? If I have 9 cards, and I'm going to pick one of 9 to be in the first slot, well, that means I have 9 ways to put something in the first slot. Then in the second slot, I have 8 ways of putting a card in the second slot, because I took 1 to put in the first, so I have 8 left, then 7, then 6, then 5, then 4, then 3, then 2, then 1. That last slot, there's only going to be one card left to put in it. So this number right here, where you take 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1, or 9, you start with 9, and then you multiply it by every number less than 9, every, I guess we could say, natural number less than 9, this is called 9 factorial."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "If I have 9 cards, and I'm going to pick one of 9 to be in the first slot, well, that means I have 9 ways to put something in the first slot. Then in the second slot, I have 8 ways of putting a card in the second slot, because I took 1 to put in the first, so I have 8 left, then 7, then 6, then 5, then 4, then 3, then 2, then 1. That last slot, there's only going to be one card left to put in it. So this number right here, where you take 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1, or 9, you start with 9, and then you multiply it by every number less than 9, every, I guess we could say, natural number less than 9, this is called 9 factorial. And you express it as an exclamation mark. So if we want to think about all of the different ways that we can have all of the different combinations for hands, this is the number of hands, if we cared about the order, but then we want to divide by the number of ways we can order things so that we don't overcount. And this will be an answer."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So this number right here, where you take 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1, or 9, you start with 9, and then you multiply it by every number less than 9, every, I guess we could say, natural number less than 9, this is called 9 factorial. And you express it as an exclamation mark. So if we want to think about all of the different ways that we can have all of the different combinations for hands, this is the number of hands, if we cared about the order, but then we want to divide by the number of ways we can order things so that we don't overcount. And this will be an answer. And this will be the correct answer. Now, this is a super, super, duper large number. Let's figure out how large of a number this is."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And this will be an answer. And this will be the correct answer. Now, this is a super, super, duper large number. Let's figure out how large of a number this is. We have 36 times 35 times 34 times 33 times 32 times 31 times 30 times 29 times 28 divided by 9. Well, I could do it this way. I could put a parentheses."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Let's figure out how large of a number this is. We have 36 times 35 times 34 times 33 times 32 times 31 times 30 times 29 times 28 divided by 9. Well, I could do it this way. I could put a parentheses. Divided by parentheses 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. Now, hopefully the calculator can handle this. And it gave us this number."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "I could put a parentheses. Divided by parentheses 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. Now, hopefully the calculator can handle this. And it gave us this number. Was it 94,143,280? Let me put this on the side so I can read it. So this number right here gives us 94,143,280."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And it gave us this number. Was it 94,143,280? Let me put this on the side so I can read it. So this number right here gives us 94,143,280. So that's the answer for this problem. That there are 94,143,280 possible 9-card hands in this situation. Now, we kind of just worked through it."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So this number right here gives us 94,143,280. So that's the answer for this problem. That there are 94,143,280 possible 9-card hands in this situation. Now, we kind of just worked through it. We reasoned our way through it. There is a formula for this that does essentially the exact same thing. And the way that people denote this formula is they say, look, we have 36 things."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "Now, we kind of just worked through it. We reasoned our way through it. There is a formula for this that does essentially the exact same thing. And the way that people denote this formula is they say, look, we have 36 things. And we are going to choose 9 of them. We don't care about order. So sometimes it'll be written as n choose k. Let me write it this way."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And the way that people denote this formula is they say, look, we have 36 things. And we are going to choose 9 of them. We don't care about order. So sometimes it'll be written as n choose k. Let me write it this way. So what did we do here? We have 36 things. We chose 9."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So sometimes it'll be written as n choose k. Let me write it this way. So what did we do here? We have 36 things. We chose 9. So this numerator over here, this was 36 factorial. But 36 factorial would go all the way down to 27, 26, 25. It would just keep going."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "We chose 9. So this numerator over here, this was 36 factorial. But 36 factorial would go all the way down to 27, 26, 25. It would just keep going. But we stopped only 9 away from 36. So this is 36 factorial. So this part right here, that part right there, is not just 36 factorial."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "It would just keep going. But we stopped only 9 away from 36. So this is 36 factorial. So this part right here, that part right there, is not just 36 factorial. It's 36 factorial divided by 36 minus 9 factorial. What is 36 minus 9? It's 27."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So this part right here, that part right there, is not just 36 factorial. It's 36 factorial divided by 36 minus 9 factorial. What is 36 minus 9? It's 27. So 27 factorial. So let's think about this. 36 factorial, it'd be 36 times 35."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "It's 27. So 27 factorial. So let's think about this. 36 factorial, it'd be 36 times 35. You keep going. All the way times 28 times 27, all the way down to 1. That is 36 factorial."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "36 factorial, it'd be 36 times 35. You keep going. All the way times 28 times 27, all the way down to 1. That is 36 factorial. Now what is 36 minus 9 factorial? That's 27 factorial. So if you divide by 27 factorial, 27 factorial is 27 times 26, all the way down to 1."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "That is 36 factorial. Now what is 36 minus 9 factorial? That's 27 factorial. So if you divide by 27 factorial, 27 factorial is 27 times 26, all the way down to 1. Well, this and this are the exact same thing. This is 27 times 26. So that and that would cancel out."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So if you divide by 27 factorial, 27 factorial is 27 times 26, all the way down to 1. Well, this and this are the exact same thing. This is 27 times 26. So that and that would cancel out. So if you do 36 divided by 36 minus 9 factorial, you just get the first, I guess the largest 9 terms of 36 factorial, which is exactly what we have over there. So that is that. And then we divided it by 9 factorial."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "So that and that would cancel out. So if you do 36 divided by 36 minus 9 factorial, you just get the first, I guess the largest 9 terms of 36 factorial, which is exactly what we have over there. So that is that. And then we divided it by 9 factorial. And we divided it by 9 factorial. And this right here is called 36 choose 9. And sometimes you'll see this formula written like this."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And then we divided it by 9 factorial. And we divided it by 9 factorial. And this right here is called 36 choose 9. And sometimes you'll see this formula written like this. n choose k, and they'll write the formula as equal to n factorial over n minus k factorial. And also in the denominator, k factorial. And this is a general formula that if you have n things and you want to find out all of the possible ways you can pick k things from those n things and you don't care about the order."}, {"video_title": "Example 9 card hands Probability and combinatorics Precalculus Khan Academy.mp3", "Sentence": "And sometimes you'll see this formula written like this. n choose k, and they'll write the formula as equal to n factorial over n minus k factorial. And also in the denominator, k factorial. And this is a general formula that if you have n things and you want to find out all of the possible ways you can pick k things from those n things and you don't care about the order. All you care is about which k things you picked. You don't care about the order in which you picked those k things. So that's what we did here."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So for example, this one over here in the top left, it's made out of chocolate on the outside, but it doesn't have coconut on the inside. While this one right over here does, is chocolate on the outside, and has coconut on the inside. While this one, whoops, I didn't want to do that. While this one, while this one right over here does not have chocolate on the outside, but it does have coconut on the inside. And this one right over here has neither chocolate nor coconut. And what I want to think about is ways to represent this information that we are looking at. And one way to do it is using a Venn diagram."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "While this one, while this one right over here does not have chocolate on the outside, but it does have coconut on the inside. And this one right over here has neither chocolate nor coconut. And what I want to think about is ways to represent this information that we are looking at. And one way to do it is using a Venn diagram. So let me draw a Venn diagram. So Venn diagram is one way to represent it. And the way it's typically done, my intention is that you would make a rectangle to represent the universe that you care about."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And one way to do it is using a Venn diagram. So let me draw a Venn diagram. So Venn diagram is one way to represent it. And the way it's typically done, my intention is that you would make a rectangle to represent the universe that you care about. In this case, it would be all the chocolates. So all the numbers inside of this should add up to the number of chocolates I have. So it should add up to 12."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And the way it's typically done, my intention is that you would make a rectangle to represent the universe that you care about. In this case, it would be all the chocolates. So all the numbers inside of this should add up to the number of chocolates I have. So it should add up to 12. So that's our universe right over here. And then I'll draw circles to represent the sets that I care about. So say for this one, I care about the set of the things that have chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So it should add up to 12. So that's our universe right over here. And then I'll draw circles to represent the sets that I care about. So say for this one, I care about the set of the things that have chocolate. So I'll draw that with a circle. Oftentimes, you could draw them to scale, but I'm not going to draw them to scale. So that is my chocolate set."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So say for this one, I care about the set of the things that have chocolate. So I'll draw that with a circle. Oftentimes, you could draw them to scale, but I'm not going to draw them to scale. So that is my chocolate set. And then I'll have a coconut set. So coconut, once again, not drawn to scale. I drew them roughly the same size, but you can see the chocolate set is bigger than the coconut set in reality."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So that is my chocolate set. And then I'll have a coconut set. So coconut, once again, not drawn to scale. I drew them roughly the same size, but you can see the chocolate set is bigger than the coconut set in reality. Coconut set. And now we can fill in the different sections. So how many of these things have chocolate but no coconut?"}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "I drew them roughly the same size, but you can see the chocolate set is bigger than the coconut set in reality. Coconut set. And now we can fill in the different sections. So how many of these things have chocolate but no coconut? Let's see. We have one, two, three, four, five, six have chocolate but no coconut. So that's going to be the... Actually, let me do that in a different color because I think the colors are important."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So how many of these things have chocolate but no coconut? Let's see. We have one, two, three, four, five, six have chocolate but no coconut. So that's going to be the... Actually, let me do that in a different color because I think the colors are important. So let me do it in green. So one, two, three, four, five, and six. So this section right over here is six."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So that's going to be the... Actually, let me do that in a different color because I think the colors are important. So let me do it in green. So one, two, three, four, five, and six. So this section right over here is six. And once again, I'm not talking about the whole brown thing. I'm talking about just this area that I've shaded in green. Now how many have chocolate and coconut?"}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So this section right over here is six. And once again, I'm not talking about the whole brown thing. I'm talking about just this area that I've shaded in green. Now how many have chocolate and coconut? Well, that's going to be one, two, three. So three of them have chocolate and coconut. And notice that's this section here that's in the overlap between."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Now how many have chocolate and coconut? Well, that's going to be one, two, three. So three of them have chocolate and coconut. And notice that's this section here that's in the overlap between. Three of them go into both sets, both categories. These three have coconut and they have chocolate. How many total have chocolate?"}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And notice that's this section here that's in the overlap between. Three of them go into both sets, both categories. These three have coconut and they have chocolate. How many total have chocolate? Well, six plus three, nine. How many total have coconut? Well, we're going to have to figure that out in a second."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "How many total have chocolate? Well, six plus three, nine. How many total have coconut? Well, we're going to have to figure that out in a second. So how many have coconut but no chocolate? Well, there's only one with coconut and no chocolate. So that's that one right over there."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Well, we're going to have to figure that out in a second. So how many have coconut but no chocolate? Well, there's only one with coconut and no chocolate. So that's that one right over there. And that represents this area that I'm shading in in white. So how many total coconut are there? Well, one plus three or four."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So that's that one right over there. And that represents this area that I'm shading in in white. So how many total coconut are there? Well, one plus three or four. And you see that, one, two, three, four. And then the last thing we'd want to fill in, because notice six plus three plus one only adds up to ten. What about the other two?"}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Well, one plus three or four. And you see that, one, two, three, four. And then the last thing we'd want to fill in, because notice six plus three plus one only adds up to ten. What about the other two? Well, the other two are neither chocolate nor coconut. Actually, let me color this. So that's one, two."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "What about the other two? Well, the other two are neither chocolate nor coconut. Actually, let me color this. So that's one, two. These are neither chocolate nor coconut. And I could write these two right over here. These are neither chocolate nor coconut."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So that's one, two. These are neither chocolate nor coconut. And I could write these two right over here. These are neither chocolate nor coconut. So that's one way to represent the information of how many chocolates, how many coconuts, and how many chocolate and coconuts, and how many neither. But there's other ways that we could do it. Another way to do it would be with a two-way table."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "These are neither chocolate nor coconut. So that's one way to represent the information of how many chocolates, how many coconuts, and how many chocolate and coconuts, and how many neither. But there's other ways that we could do it. Another way to do it would be with a two-way table. A two-way table. And on one axis, say the vertical axis, we could say, let me write this. So has chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Another way to do it would be with a two-way table. A two-way table. And on one axis, say the vertical axis, we could say, let me write this. So has chocolate. Has chocolate. I'll write chalk for short. And then I'll write no chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So has chocolate. Has chocolate. I'll write chalk for short. And then I'll write no chocolate. No chocolate, chalk for short. And then over here, I could write coconut. I want to do that in white."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And then I'll write no chocolate. No chocolate, chalk for short. And then over here, I could write coconut. I want to do that in white. I got new tools, and sometimes the color changing isn't so easy. So this is coconut. And then over here, I'll write no coconut."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "I want to do that in white. I got new tools, and sometimes the color changing isn't so easy. So this is coconut. And then over here, I'll write no coconut. No coconut. And then let me make a little table. Let me make a table, make it clear what I'm doing here."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And then over here, I'll write no coconut. No coconut. And then let me make a little table. Let me make a table, make it clear what I'm doing here. So a line there and a line there. And then I'll add a line over here as well. And then I can just fill in the different things."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Let me make a table, make it clear what I'm doing here. So a line there and a line there. And then I'll add a line over here as well. And then I can just fill in the different things. So how many have this cell right over this square? This is going to represent the number that has coconut and chocolate. Coconut and chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And then I can just fill in the different things. So how many have this cell right over this square? This is going to represent the number that has coconut and chocolate. Coconut and chocolate. Well, we already looked into that. That's one, two, three. That's these three right over here."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Coconut and chocolate. Well, we already looked into that. That's one, two, three. That's these three right over here. So that's those three right over there. This one right over here is it has chocolate, but it doesn't have coconut. Well, that's this six right over here."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "That's these three right over here. So that's those three right over there. This one right over here is it has chocolate, but it doesn't have coconut. Well, that's this six right over here. It has chocolate, but it doesn't have coconut. So let me write this is that six right over there. And then this box would be it has coconut, but no chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Well, that's this six right over here. It has chocolate, but it doesn't have coconut. So let me write this is that six right over there. And then this box would be it has coconut, but no chocolate. Well, how many is that? Well, coconut, no chocolate. That's that one there."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "And then this box would be it has coconut, but no chocolate. Well, how many is that? Well, coconut, no chocolate. That's that one there. And this one is going to be no coconut and no chocolate. And we know what that's going to be. No coconut and no chocolate is going to be two."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "That's that one there. And this one is going to be no coconut and no chocolate. And we know what that's going to be. No coconut and no chocolate is going to be two. And if we wanted to, we could even throw in totals over here. We could write, actually let me just do that just for fun. I could write total."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "No coconut and no chocolate is going to be two. And if we wanted to, we could even throw in totals over here. We could write, actually let me just do that just for fun. I could write total. And if I total it vertically, so three plus one, this is four. Six plus two is eight. So this four is the total number that have coconut, both the has chocolate and doesn't have chocolate."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "I could write total. And if I total it vertically, so three plus one, this is four. Six plus two is eight. So this four is the total number that have coconut, both the has chocolate and doesn't have chocolate. And that's the three plus one. This eight is the total that does not have coconut. We're in no coconuts, the total of no coconut."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "So this four is the total number that have coconut, both the has chocolate and doesn't have chocolate. And that's the three plus one. This eight is the total that does not have coconut. We're in no coconuts, the total of no coconut. And that, of course, is going to be the six plus this two. And we could total horizontally. Three plus six is nine."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "We're in no coconuts, the total of no coconut. And that, of course, is going to be the six plus this two. And we could total horizontally. Three plus six is nine. One plus two is three. What's this nine? That's the total amount of chocolate, six plus three."}, {"video_title": "Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3", "Sentence": "Three plus six is nine. One plus two is three. What's this nine? That's the total amount of chocolate, six plus three. What's this three? This is the total amount no chocolate. That's this one plus two."}, {"video_title": "Using probability to make fair decisions.mp3", "Sentence": "We're told that Roberto and Jocelyn decide to roll a pair of fair six-sided dice to determine who has to dust their apartment. If the sum is seven, then Roberto will dust. If the sum is 10 or 11, then Jocelyn will dust. If the sum is anything else, they'll roll again. Is this a fair way to decide who dusts? Why or why not? So pause this video and see if you can figure this out before we do it together."}, {"video_title": "Using probability to make fair decisions.mp3", "Sentence": "If the sum is anything else, they'll roll again. Is this a fair way to decide who dusts? Why or why not? So pause this video and see if you can figure this out before we do it together. All right, now let's do this together. So what I wanna do is make a table that shows all of the different scenarios for rolling two fair six-sided dice. So let me make columns for roll one."}, {"video_title": "Using probability to make fair decisions.mp3", "Sentence": "So pause this video and see if you can figure this out before we do it together. All right, now let's do this together. So what I wanna do is make a table that shows all of the different scenarios for rolling two fair six-sided dice. So let me make columns for roll one. So that is, you get a one, this is when you get a two, this is when you get a three, this is when you get a four, this is when you get a five, and then this is when you get a six. And then here, let's do it the other die. So this is when you get a one, this is when you get a two, this is when you get a three, this is when you get a four, this is when you get a five, and then this is when you get a six."}, {"video_title": "Using probability to make fair decisions.mp3", "Sentence": "So let me make columns for roll one. So that is, you get a one, this is when you get a two, this is when you get a three, this is when you get a four, this is when you get a five, and then this is when you get a six. And then here, let's do it the other die. So this is when you get a one, this is when you get a two, this is when you get a three, this is when you get a four, this is when you get a five, and then this is when you get a six. So one way to think about it is this is roll one, or let me write it this way, die one and die two. This could be a one, a two, a three, a four, a five, or six. And this could be a one, a two, a three, a four, a five, or a six."}, {"video_title": "Using probability to make fair decisions.mp3", "Sentence": "So this is when you get a one, this is when you get a two, this is when you get a three, this is when you get a four, this is when you get a five, and then this is when you get a six. So one way to think about it is this is roll one, or let me write it this way, die one and die two. This could be a one, a two, a three, a four, a five, or six. And this could be a one, a two, a three, a four, a five, or a six. Now what we could do is fill in these 36 squares to figure out what the sum is. Actually, let me just do that and I'll try to do it really fast. One plus one is two, so it's three, four, five, six, seven."}, {"video_title": "Using probability to make fair decisions.mp3", "Sentence": "And this could be a one, a two, a three, a four, a five, or a six. Now what we could do is fill in these 36 squares to figure out what the sum is. Actually, let me just do that and I'll try to do it really fast. One plus one is two, so it's three, four, five, six, seven. This is three, four, five, six, seven, eight. This is four, five, six, seven, eight, nine. This is five, six, seven, eight, nine, 10."}, {"video_title": "Using probability to make fair decisions.mp3", "Sentence": "One plus one is two, so it's three, four, five, six, seven. This is three, four, five, six, seven, eight. This is four, five, six, seven, eight, nine. This is five, six, seven, eight, nine, 10. This is six, seven, eight, nine, 10, 11. Last but not least, seven, eight, nine, 10, 11, and 12. Took a little less time than I suspected."}, {"video_title": "Using probability to make fair decisions.mp3", "Sentence": "This is five, six, seven, eight, nine, 10. This is six, seven, eight, nine, 10, 11. Last but not least, seven, eight, nine, 10, 11, and 12. Took a little less time than I suspected. All right, let's think about this scenario. If the sum is seven, then Roberto will dust. So where is the sum seven?"}, {"video_title": "Using probability to make fair decisions.mp3", "Sentence": "Took a little less time than I suspected. All right, let's think about this scenario. If the sum is seven, then Roberto will dust. So where is the sum seven? So we have that once, twice, three times, four, five, six. So six out of, so six of these outcomes result in a sum of seven. And how many possible equally likely outcomes are there?"}, {"video_title": "Using probability to make fair decisions.mp3", "Sentence": "So where is the sum seven? So we have that once, twice, three times, four, five, six. So six out of, so six of these outcomes result in a sum of seven. And how many possible equally likely outcomes are there? Well, there are six times six equally possible outcomes, or 36. So six out of the 36, or this is another way of saying there's a 1 6th probability that Roberto will dust. And then let's think about the 10s or 11s."}, {"video_title": "Using probability to make fair decisions.mp3", "Sentence": "And how many possible equally likely outcomes are there? Well, there are six times six equally possible outcomes, or 36. So six out of the 36, or this is another way of saying there's a 1 6th probability that Roberto will dust. And then let's think about the 10s or 11s. If the sum is 10 or 11, then Jocelyn will dust. So 10 or 11, so we have one, two, three, four, five. So this is only happening five out of the 36 times."}, {"video_title": "Using probability to make fair decisions.mp3", "Sentence": "And then let's think about the 10s or 11s. If the sum is 10 or 11, then Jocelyn will dust. So 10 or 11, so we have one, two, three, four, five. So this is only happening five out of the 36 times. So in any given roll, it's a higher probability that Roberto will dust than Jocelyn will dust. And of course, if neither of these happen, they're going to roll again. But on that second roll, there's a higher probability that Roberto will dust than Jocelyn will dust."}, {"video_title": "Using probability to make fair decisions.mp3", "Sentence": "So this is only happening five out of the 36 times. So in any given roll, it's a higher probability that Roberto will dust than Jocelyn will dust. And of course, if neither of these happen, they're going to roll again. But on that second roll, there's a higher probability that Roberto will dust than Jocelyn will dust. So in general, this is not fair. There's a higher probability that Roberto dusts. So this is our choice."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics.mp3", "Sentence": "I guess the primary ingredient, and we can see it could be chard, spinach, romaine, lettuce, I'm guessing, cabbage, arugula, or kale. And so then they give us these different types of events, or at least the symbols for these different types of events, and then give us their meaning. So K sub one means the first contestant lands on kale. K sub two means the second contestant lands on kale. K sub one with this superscript C, which we could view as complement. So K sub one complement, the first contestant does not land on kale. So it's the complement of this one right over here."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics.mp3", "Sentence": "K sub two means the second contestant lands on kale. K sub one with this superscript C, which we could view as complement. So K sub one complement, the first contestant does not land on kale. So it's the complement of this one right over here. And then K sub two complement would be that the second contestant does not land on kale. So the not of K sub two right over here. Using the general multiplication rule, express symbolically the probability that neither contestant lands on kale."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics.mp3", "Sentence": "So it's the complement of this one right over here. And then K sub two complement would be that the second contestant does not land on kale. So the not of K sub two right over here. Using the general multiplication rule, express symbolically the probability that neither contestant lands on kale. So pause this video and see if you can have a go at this. All right. So the general multiplication rule is just saying this notion that the probability of two events, A and B, is going to be equal to the probability of, let's say A given B, times the probability of B."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics.mp3", "Sentence": "Using the general multiplication rule, express symbolically the probability that neither contestant lands on kale. So pause this video and see if you can have a go at this. All right. So the general multiplication rule is just saying this notion that the probability of two events, A and B, is going to be equal to the probability of, let's say A given B, times the probability of B. Now, if they're independent events, if the probability of A occurring does not depend in any way on whether B occurred or not, then this would simplify to this probability of A given B would just become the probability of A. And so if you have two independent events, you would just multiply their probabilities. So that's just all they're talking about, the general multiplication rule."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics.mp3", "Sentence": "So the general multiplication rule is just saying this notion that the probability of two events, A and B, is going to be equal to the probability of, let's say A given B, times the probability of B. Now, if they're independent events, if the probability of A occurring does not depend in any way on whether B occurred or not, then this would simplify to this probability of A given B would just become the probability of A. And so if you have two independent events, you would just multiply their probabilities. So that's just all they're talking about, the general multiplication rule. But let me express what they're actually asking us to express, the probability that neither contestant lands on kale. So that means that this is going to happen. The first contestant does not land on kale, and this is going to happen."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics.mp3", "Sentence": "So that's just all they're talking about, the general multiplication rule. But let me express what they're actually asking us to express, the probability that neither contestant lands on kale. So that means that this is going to happen. The first contestant does not land on kale, and this is going to happen. The second contestant does not land on kale. So I could write it this way. The probability that K sub one complement and K sub two complement, and I could write it this way."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics.mp3", "Sentence": "The first contestant does not land on kale, and this is going to happen. The second contestant does not land on kale. So I could write it this way. The probability that K sub one complement and K sub two complement, and I could write it this way. This is going to be equal to, we know that these are independent events because if the first contestant gets kale or whatever they get, it doesn't get taken out of the running for the second contestant. The second contestant still has an equal probability of getting or not getting kale, regardless of what happened for the first contestant. So that means we're just in the situation where we multiply these probabilities."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics.mp3", "Sentence": "The probability that K sub one complement and K sub two complement, and I could write it this way. This is going to be equal to, we know that these are independent events because if the first contestant gets kale or whatever they get, it doesn't get taken out of the running for the second contestant. The second contestant still has an equal probability of getting or not getting kale, regardless of what happened for the first contestant. So that means we're just in the situation where we multiply these probabilities. So that's going to be the probability of K sub one complement times the probability of K sub two complement. All right, now let's do part two. Interpret what each part of this probability statement represents."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics.mp3", "Sentence": "So that means we're just in the situation where we multiply these probabilities. So that's going to be the probability of K sub one complement times the probability of K sub two complement. All right, now let's do part two. Interpret what each part of this probability statement represents. So I encourage you, like always, pause this video and try to figure that out. All right, so first let's think about what is going on here. So this is saying the probability that this is K sub one complement."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics.mp3", "Sentence": "Interpret what each part of this probability statement represents. So I encourage you, like always, pause this video and try to figure that out. All right, so first let's think about what is going on here. So this is saying the probability that this is K sub one complement. So the first contestant does not land on kale. So first contestant does not get kale and, all right, and in caps, and second contestant does get kale. And second does get kale."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics.mp3", "Sentence": "So this is saying the probability that this is K sub one complement. So the first contestant does not land on kale. So first contestant does not get kale and, all right, and in caps, and second contestant does get kale. And second does get kale. So that's what this left hand is saying. And now they say that that is going to be equal to, so this part right over here, probability that the first contestant does not get kale, probability that first does not get kale times, right over here, and the second part right over here is the probability that the second contestant gets kale given that the first contestant does not get kale. So probability that the second gets kale given, that's what this vertical line right over here means."}, {"video_title": "Interpreting general multiplication rule Probability & combinatorics.mp3", "Sentence": "And second does get kale. So that's what this left hand is saying. And now they say that that is going to be equal to, so this part right over here, probability that the first contestant does not get kale, probability that first does not get kale times, right over here, and the second part right over here is the probability that the second contestant gets kale given that the first contestant does not get kale. So probability that the second gets kale given, that's what this vertical line right over here means. It means given, shorthand for given. Given, I wrote it up there too. Given that first does not get kale."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "Let's say that your school has a population of 80 students in it. Maybe it's not your whole school, maybe it's just your grade. So there's 80 students in your population, and you want to get an estimate of the average height in your population. And you think it's too hard for you to go and measure the height of all 80 students, so you decide to find a simple, or take a simple random sample. You think it's reasonable for you to measure the heights of 30 of these students. And so what you want to do is randomly sample 30 of the 80 students, and take their average height, and say, well, that's probably a pretty good estimate for the population parameter, for the average height of the entire population. So once you decide to do this, you say, well, how do I select those 30 students, and how do I select it so that I feel good that it is actually random?"}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And you think it's too hard for you to go and measure the height of all 80 students, so you decide to find a simple, or take a simple random sample. You think it's reasonable for you to measure the heights of 30 of these students. And so what you want to do is randomly sample 30 of the 80 students, and take their average height, and say, well, that's probably a pretty good estimate for the population parameter, for the average height of the entire population. So once you decide to do this, you say, well, how do I select those 30 students, and how do I select it so that I feel good that it is actually random? And there's several ways that you could approach this. One way to do it is associate every person in your school with a piece of paper, and put them all in a bowl, and then pick them out. So let's do that."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "So once you decide to do this, you say, well, how do I select those 30 students, and how do I select it so that I feel good that it is actually random? And there's several ways that you could approach this. One way to do it is associate every person in your school with a piece of paper, and put them all in a bowl, and then pick them out. So let's do that. So let's say this is alphabetically the first person in the school, they're on a slip of paper, then the next slip of paper gets the next person, and you're gonna go all the way down, so you're gonna have 80 pieces of paper. They all should be the same size. And then you throw them all, you throw them all into a bowl of some kind."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "So let's do that. So let's say this is alphabetically the first person in the school, they're on a slip of paper, then the next slip of paper gets the next person, and you're gonna go all the way down, so you're gonna have 80 pieces of paper. They all should be the same size. And then you throw them all, you throw them all into a bowl of some kind. And this seems like a very basic way of doing it, but it's actually a pretty effective way of getting a simple, of getting a simple random sample. So I'll try to draw a little, I don't know, it looks like a little fishbowl or something. All right, so that's our bowl."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And then you throw them all, you throw them all into a bowl of some kind. And this seems like a very basic way of doing it, but it's actually a pretty effective way of getting a simple, of getting a simple random sample. So I'll try to draw a little, I don't know, it looks like a little fishbowl or something. All right, so that's our bowl. And so all the pieces of paper go in there. And then you put a blindfold on someone, and they can't feel what names are there, and so they should pick out the first 30 without replacing them, because you obviously don't wanna pick the same, you don't wanna pick out the same name twice. And those 30 names that you pick, that would be your simple random sample, and then you could measure their heights to estimate the average height for the population."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "All right, so that's our bowl. And so all the pieces of paper go in there. And then you put a blindfold on someone, and they can't feel what names are there, and so they should pick out the first 30 without replacing them, because you obviously don't wanna pick the same, you don't wanna pick out the same name twice. And those 30 names that you pick, that would be your simple random sample, and then you could measure their heights to estimate the average height for the population. This would be a completely legitimate way of doing it. Other ways that you could do it, if you have a computer or a calculator, you could use a random number generator. And the random functions on computer programming languages or on your calculator, they tend to be something, you know, some place you'll see something like a math.rand, short for random."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And those 30 names that you pick, that would be your simple random sample, and then you could measure their heights to estimate the average height for the population. This would be a completely legitimate way of doing it. Other ways that you could do it, if you have a computer or a calculator, you could use a random number generator. And the random functions on computer programming languages or on your calculator, they tend to be something, you know, some place you'll see something like a math.rand, short for random. You might see something like random. You might see, you might see something like random. Without anything passed into it, it might give you a number between zero and one, or zero and 100."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And the random functions on computer programming languages or on your calculator, they tend to be something, you know, some place you'll see something like a math.rand, short for random. You might see something like random. You might see, you might see something like random. Without anything passed into it, it might give you a number between zero and one, or zero and 100. And you have to be very careful on how you use this to make sure that you have an even chance of picking certain numbers. But what you would do in this situation, if you had access to some random number generator, and it could even pick out a random number between one and 80, including one and 80, is you would maybe line up all the students' names alphabetically. And so the first student alphabetically, assign the number zero, one."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "Without anything passed into it, it might give you a number between zero and one, or zero and 100. And you have to be very careful on how you use this to make sure that you have an even chance of picking certain numbers. But what you would do in this situation, if you had access to some random number generator, and it could even pick out a random number between one and 80, including one and 80, is you would maybe line up all the students' names alphabetically. And so the first student alphabetically, assign the number zero, one. And you could just say one if you're using a random number generator, but I'll use two digits for it just because it'll be useful and consistent. And in a little bit, we'll use another technique where it's gonna be nice to be consistent with our number of digits. And so the next one, zero, two, and you go all the way to 79, and all the way to 80."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And so the first student alphabetically, assign the number zero, one. And you could just say one if you're using a random number generator, but I'll use two digits for it just because it'll be useful and consistent. And in a little bit, we'll use another technique where it's gonna be nice to be consistent with our number of digits. And so the next one, zero, two, and you go all the way to 79, and all the way to 80. And then you use your random number generator to keep generating numbers from one to 80. And as long as you don't get repeats, you pick the first 30 to be your actual random sample. Another related technique, which is a little bit more old school, but is definitely the way that it has been done in the past and even done now sometimes, is to use a random digit table."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And so the next one, zero, two, and you go all the way to 79, and all the way to 80. And then you use your random number generator to keep generating numbers from one to 80. And as long as you don't get repeats, you pick the first 30 to be your actual random sample. Another related technique, which is a little bit more old school, but is definitely the way that it has been done in the past and even done now sometimes, is to use a random digit table. You still start with these number associations with each student in the class, and then you use a randomly generated list of numbers. And so let's say that's our randomly generated list of numbers, and it keeps going well beyond this. And you start at the beginning."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "Another related technique, which is a little bit more old school, but is definitely the way that it has been done in the past and even done now sometimes, is to use a random digit table. You still start with these number associations with each student in the class, and then you use a randomly generated list of numbers. And so let's say that's our randomly generated list of numbers, and it keeps going well beyond this. And you start at the beginning. And you say, okay, we're interested in getting 30 two-digit numbers from one to 80, including one in 80. So one technique that you could use is you start it right at the beginning, and you could say, all right, this is a randomly generated list of numbers. So the first number here is 59."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "And you start at the beginning. And you say, okay, we're interested in getting 30 two-digit numbers from one to 80, including one in 80. So one technique that you could use is you start it right at the beginning, and you could say, all right, this is a randomly generated list of numbers. So the first number here is 59. Is 59 between one and 80? Sure is. As long as we, you know, if this was a zero one, that would have worked."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "So the first number here is 59. Is 59 between one and 80? Sure is. As long as we, you know, if this was a zero one, that would have worked. If this was an eight zero, that would have worked. If this was a zero zero, it wouldn't have worked. If this was an eight one, it wouldn't have worked."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "As long as we, you know, if this was a zero one, that would have worked. If this was an eight zero, that would have worked. If this was a zero zero, it wouldn't have worked. If this was an eight one, it wouldn't have worked. But this would be our, this right over here, that would be our first name that we, you could imagine the same as picking that first name out of the hat, whoever's associated with number 59. Now, you would move on. You get the next two digits."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "If this was an eight one, it wouldn't have worked. But this would be our, this right over here, that would be our first name that we, you could imagine the same as picking that first name out of the hat, whoever's associated with number 59. Now, you would move on. You get the next two digits. The next two digits are 83. They don't fall into our range from one to 80, so we're not going to use it. Then you look at the next two digits."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "You get the next two digits. The next two digits are 83. They don't fall into our range from one to 80, so we're not going to use it. Then you look at the next two digits. So we get a five and a nine. Well, that fits in our range, but we already picked 59. We already picked person 59, so we're not gonna pick 59 again."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "Then you look at the next two digits. So we get a five and a nine. Well, that fits in our range, but we already picked 59. We already picked person 59, so we're not gonna pick 59 again. So we keep moving on. Then we get a 37. Well, that's in our range."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "We already picked person 59, so we're not gonna pick 59 again. So we keep moving on. Then we get a 37. Well, that's in our range. We haven't picked that yet. We do that. Then we get a zero zero."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "Well, that's in our range. We haven't picked that yet. We do that. Then we get a zero zero. Once again, not in our range. I think you see where this is going. 91, not in our range."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "Then we get a zero zero. Once again, not in our range. I think you see where this is going. 91, not in our range. 23, it's in our range, and we haven't picked it yet. So we're gonna pick the 23. I think you see where this is going."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "91, not in our range. 23, it's in our range, and we haven't picked it yet. So we're gonna pick the 23. I think you see where this is going. We're gonna keep going down this list in the way that I've just described until we get 30 of these. We've just gotten three. We just have to keep on going."}, {"video_title": "Techniques for generating a simple random sample Study design AP Statistics Khan Academy.mp3", "Sentence": "I think you see where this is going. We're gonna keep going down this list in the way that I've just described until we get 30 of these. We've just gotten three. We just have to keep on going. And this isn't an exhaustive list of all of the different ways that you can get random numbers, but it starts to give you some techniques in your toolkit. And you might say, oh, well, why don't I just randomly come up with some numbers in my head? And I would really suggest that you don't do that because humans are famously bad at being truly random."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "In this video, we're going to get our bearings on the different types of studies you might statistically analyze or statistical studies. So first of all, it's worth differentiating between an experiment and an observational study. I encourage you, pause this video and think about what the difference is, at least in your head, between an experiment and an observational study. Well, you might already be familiar with experiments. You oftentimes have a hypothesis that if you do something to one group, that it might have some type of statistically significant impact on them relative to a group that you did not do it to, and you would be generally right. That is the flavor of what we're talking about when we're talking about an experiment. An experiment where actively putting people or things into a control versus treatment group."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "Well, you might already be familiar with experiments. You oftentimes have a hypothesis that if you do something to one group, that it might have some type of statistically significant impact on them relative to a group that you did not do it to, and you would be generally right. That is the flavor of what we're talking about when we're talking about an experiment. An experiment where actively putting people or things into a control versus treatment group. In the treatment group, you put the people, and you usually would want to randomly select people into the treatment group. Maybe it's a new type of medication, and maybe in the treatment group, they actually get the medication, while in the control group, which you would put people into randomly, whether they're in control or treatment, here, they might get a placebo, where they get a pill that looks just like the medication, but it really doesn't do anything. And then you wait some time, and you can see is there a statistically significant difference between the treatment group, on average, and the control group."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "An experiment where actively putting people or things into a control versus treatment group. In the treatment group, you put the people, and you usually would want to randomly select people into the treatment group. Maybe it's a new type of medication, and maybe in the treatment group, they actually get the medication, while in the control group, which you would put people into randomly, whether they're in control or treatment, here, they might get a placebo, where they get a pill that looks just like the medication, but it really doesn't do anything. And then you wait some time, and you can see is there a statistically significant difference between the treatment group, on average, and the control group. So that's what an experiment does. It's kind of this active sorting and figuring out whether some type of stimulus is able to show a difference. While an observational study, you don't actively put into groups."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "And then you wait some time, and you can see is there a statistically significant difference between the treatment group, on average, and the control group. So that's what an experiment does. It's kind of this active sorting and figuring out whether some type of stimulus is able to show a difference. While an observational study, you don't actively put into groups. Instead, you just collect data and see if you can have some insights from that data. If you can say, okay, the data, there's a population here. Can I come up with some statistics that are indicative of the population?"}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "While an observational study, you don't actively put into groups. Instead, you just collect data and see if you can have some insights from that data. If you can say, okay, the data, there's a population here. Can I come up with some statistics that are indicative of the population? I might just wanna look at averages, or I might wanna find some correlations between variables. But even when we're talking about an observational study, there are different types of it, depending on what type of data we're looking at, whether the data is backward-looking, forward-looking, or it's data that we are collecting right now based on what people think or say right now. So if we're thinking about an observational study that is looking at past data, and I could imagine doing something like this at Khan Academy, where we could look at maybe usage of Khan Academy over time."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "Can I come up with some statistics that are indicative of the population? I might just wanna look at averages, or I might wanna find some correlations between variables. But even when we're talking about an observational study, there are different types of it, depending on what type of data we're looking at, whether the data is backward-looking, forward-looking, or it's data that we are collecting right now based on what people think or say right now. So if we're thinking about an observational study that is looking at past data, and I could imagine doing something like this at Khan Academy, where we could look at maybe usage of Khan Academy over time. We have these things in our server logs, and we're able to do some analysis there. Maybe we're able to analyze and say, okay, on average, students are spending two hours per month on Khan Academy over in 2019. That would be past data, and that type of observational study would be called a retrospective study."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "So if we're thinking about an observational study that is looking at past data, and I could imagine doing something like this at Khan Academy, where we could look at maybe usage of Khan Academy over time. We have these things in our server logs, and we're able to do some analysis there. Maybe we're able to analyze and say, okay, on average, students are spending two hours per month on Khan Academy over in 2019. That would be past data, and that type of observational study would be called a retrospective study. Retro for backwards, and spective, looking. So a retrospective observational study would sample past data in order to come up with some insights. Now, you could imagine there might be the other side."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "That would be past data, and that type of observational study would be called a retrospective study. Retro for backwards, and spective, looking. So a retrospective observational study would sample past data in order to come up with some insights. Now, you could imagine there might be the other side. What if we are trying to observe things into the future? Well, here, you might take a sample of folks who you think are indicative of a population, and you might want to just track their data. So you could even consider that to be future data."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "Now, you could imagine there might be the other side. What if we are trying to observe things into the future? Well, here, you might take a sample of folks who you think are indicative of a population, and you might want to just track their data. So you could even consider that to be future data. So you pick the group, the sample, ahead of time, and then you track their data over time. I'm just gonna draw it as these little arrows that you're tracking their data. And then you see what happens."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "So you could even consider that to be future data. So you pick the group, the sample, ahead of time, and then you track their data over time. I'm just gonna draw it as these little arrows that you're tracking their data. And then you see what happens. For example, you might randomly select, hopefully a random sample of 100 women, and you wanna see in the coming year how many eggs do they eat on average per day. Well, what you would do is you selected those folks, and then you would track that data for each of them every day. And then once you have the data, you could actually do it while you're collecting it, but at the end of the study, you'll be able to see what those averages are, but you can also keep track of it while you're taking that data."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "And then you see what happens. For example, you might randomly select, hopefully a random sample of 100 women, and you wanna see in the coming year how many eggs do they eat on average per day. Well, what you would do is you selected those folks, and then you would track that data for each of them every day. And then once you have the data, you could actually do it while you're collecting it, but at the end of the study, you'll be able to see what those averages are, but you can also keep track of it while you're taking that data. And you could imagine what this was called. Instead of retrospective, we're now looking forward. So it is prospective, forward-looking observational study."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "And then once you have the data, you could actually do it while you're collecting it, but at the end of the study, you'll be able to see what those averages are, but you can also keep track of it while you're taking that data. And you could imagine what this was called. Instead of retrospective, we're now looking forward. So it is prospective, forward-looking observational study. Last but not least, some of y'all are probably thinking, what about if we're doing something now? If we go out there and we were to survey a bunch of people and say, how many eggs did you eat today? Or who are you going to vote for?"}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "So it is prospective, forward-looking observational study. Last but not least, some of y'all are probably thinking, what about if we're doing something now? If we go out there and we were to survey a bunch of people and say, how many eggs did you eat today? Or who are you going to vote for? What might we call that? Well, it's tempting to call it something with a prefix and then spective, so it all matches, but it turns out that the terminology that statisticians will typically use is a sample survey. Sample survey."}, {"video_title": "Types of studies AP Statistics Khan Academy.mp3", "Sentence": "Or who are you going to vote for? What might we call that? Well, it's tempting to call it something with a prefix and then spective, so it all matches, but it turns out that the terminology that statisticians will typically use is a sample survey. Sample survey. That right now, you're going to take a, hopefully random sample of individuals from the population that you care about, and you are just going to survey them right now and ask them, say, some questions or observe some data about them right now. So I'll leave you there. This video is to just give you a little bit of the vocabulary and a little bit of a taxonomy on the types of studies that you'll see in general, which is especially useful to know when you're exploring the world of statistics."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And they gave us a bunch of data points. And it says if it helps, you might drag the numbers around, which I will do, because that will be useful. And they say the order isn't checked, and that's because I'm doing this on Khan Academy exercises up here in the top right where you can't see there's actually a check answer. So I encourage you to use the exercises yourself. But let's just use this as an example. So the first thing, if I'm going to do a box and whiskers, I'm going to order these numbers. So let me order these numbers from least to greatest."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So I encourage you to use the exercises yourself. But let's just use this as an example. So the first thing, if I'm going to do a box and whiskers, I'm going to order these numbers. So let me order these numbers from least to greatest. So let's see, there is a 1 here. And we've got some 2's here. And some 3's."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So let me order these numbers from least to greatest. So let's see, there is a 1 here. And we've got some 2's here. And some 3's. I have one 4. Then 5's. I have a 6."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And some 3's. I have one 4. Then 5's. I have a 6. I have a 7. I have a couple of 8's. And I have a 10."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "I have a 6. I have a 7. I have a couple of 8's. And I have a 10. So there you go. I have ordered these numbers from least to greatest. And now, well, just like that I can plot the whiskers because I see the range."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And I have a 10. So there you go. I have ordered these numbers from least to greatest. And now, well, just like that I can plot the whiskers because I see the range. My lowest number is 1. So my lowest number is 1. My largest number is 10."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "And now, well, just like that I can plot the whiskers because I see the range. My lowest number is 1. So my lowest number is 1. My largest number is 10. So the whiskers help me visualize the range. Now let me think about what the median of my data set is. So my median here is going to be, let's see, I have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 numbers."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "My largest number is 10. So the whiskers help me visualize the range. Now let me think about what the median of my data set is. So my median here is going to be, let's see, I have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 numbers. Since I have an even number of numbers, the middle 2 numbers are going to help define my median because there's no one middle number. I might say this number right over here, this 4, but notice there's 1, 2, 3, 4, 5, 6, 7 above it, and there's only 1, 2, 3, 4, 5, 6 below it. The same thing would have been true for this 5."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So my median here is going to be, let's see, I have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 numbers. Since I have an even number of numbers, the middle 2 numbers are going to help define my median because there's no one middle number. I might say this number right over here, this 4, but notice there's 1, 2, 3, 4, 5, 6, 7 above it, and there's only 1, 2, 3, 4, 5, 6 below it. The same thing would have been true for this 5. So this 4 and 5, the middle is actually in between these 2. So when you have an even number of numbers like this, you take the middle 2 numbers, this 4 and this 5, and you take the mean of the 2. So the mean of 4 and 5 is going to be 4 and 1 half."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "The same thing would have been true for this 5. So this 4 and 5, the middle is actually in between these 2. So when you have an even number of numbers like this, you take the middle 2 numbers, this 4 and this 5, and you take the mean of the 2. So the mean of 4 and 5 is going to be 4 and 1 half. So that's going to be the median of our entire data set, 4 and 1 half. Now I want to figure out the median of the bottom half of numbers and the top half of numbers. Here they say exclude the median."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So the mean of 4 and 5 is going to be 4 and 1 half. So that's going to be the median of our entire data set, 4 and 1 half. Now I want to figure out the median of the bottom half of numbers and the top half of numbers. Here they say exclude the median. Of course I'm going to exclude the median. It's not even included in our data points right here because our median is 4.5. So now let's take this bottom half of numbers over here and find the middle."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Here they say exclude the median. Of course I'm going to exclude the median. It's not even included in our data points right here because our median is 4.5. So now let's take this bottom half of numbers over here and find the middle. So this is the bottom 7 numbers. So the median of those is going to be the one that has 3 on either side. So it's going to be this 2 right over here."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So now let's take this bottom half of numbers over here and find the middle. So this is the bottom 7 numbers. So the median of those is going to be the one that has 3 on either side. So it's going to be this 2 right over here. So that right over there is kind of the left boundary of our box. And then for the right boundary, we need to figure out the middle of our top half of numbers. Remember, 4 and 5 were our middle 2 numbers."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "So it's going to be this 2 right over here. So that right over there is kind of the left boundary of our box. And then for the right boundary, we need to figure out the middle of our top half of numbers. Remember, 4 and 5 were our middle 2 numbers. Our median is right in between at 4 and 1 half. So our top half of numbers starts at this 5 and goes to this 10, 7 numbers. The middle one is going to have 3 on both sides."}, {"video_title": "Another example constructing box plot Data and statistics 6th grade Khan Academy.mp3", "Sentence": "Remember, 4 and 5 were our middle 2 numbers. Our median is right in between at 4 and 1 half. So our top half of numbers starts at this 5 and goes to this 10, 7 numbers. The middle one is going to have 3 on both sides. The 7 has 3 to the left, remember, of the top half, and 3 to the right. And so the 7 is, I guess you could say, the right side of our box. And we're done."}, {"video_title": "Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3", "Sentence": "She then created the following scatter plot and trend line. So this is salary in millions of dollars and the winning percentage. And so here we have a coach who made over $4 million and it looks like they won over 80% of their games. Then you have this coach over here who has a salary of a little over a million and a half dollars and they are winning over 85%. And so each of one of these data points is a coach and it's plotting their salary or their winning percentage against their salary. Assuming the line correctly shows the trend in the data and it's a bit of an assumption. There are some outliers here that are well away from the model and this isn't a, it looks like there's a linear, a positive linear correlation here, but it's not super tight and there's a bunch of coaches right over here in the lower salary area going all the way from 20 something percent to over 60%."}, {"video_title": "Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3", "Sentence": "Then you have this coach over here who has a salary of a little over a million and a half dollars and they are winning over 85%. And so each of one of these data points is a coach and it's plotting their salary or their winning percentage against their salary. Assuming the line correctly shows the trend in the data and it's a bit of an assumption. There are some outliers here that are well away from the model and this isn't a, it looks like there's a linear, a positive linear correlation here, but it's not super tight and there's a bunch of coaches right over here in the lower salary area going all the way from 20 something percent to over 60%. Assuming the line correctly shows the trend in the data, what does it mean that the lie's y-intercept is 39? Well if you believe the model, then a y-intercept of being 39 would be, the model is saying that if someone makes no money, that they could, zero dollars, that they could win, that the model would expect them to win 39% of their gains, which seems a little unrealistic because you would expect most coaches to get paid something. But anyway, let's see which of these choices actually describe that."}, {"video_title": "Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3", "Sentence": "There are some outliers here that are well away from the model and this isn't a, it looks like there's a linear, a positive linear correlation here, but it's not super tight and there's a bunch of coaches right over here in the lower salary area going all the way from 20 something percent to over 60%. Assuming the line correctly shows the trend in the data, what does it mean that the lie's y-intercept is 39? Well if you believe the model, then a y-intercept of being 39 would be, the model is saying that if someone makes no money, that they could, zero dollars, that they could win, that the model would expect them to win 39% of their gains, which seems a little unrealistic because you would expect most coaches to get paid something. But anyway, let's see which of these choices actually describe that. So let me look at the choices. The average salary was $39 million. No, no one on our chart made 39 million."}, {"video_title": "Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3", "Sentence": "But anyway, let's see which of these choices actually describe that. So let me look at the choices. The average salary was $39 million. No, no one on our chart made 39 million. On average, each million dollar increase in salary was associated with a 39% increase in winning percentage. So that would be something related to the slope, and the slope was definitely not 39. The average winning percentage was 39%."}, {"video_title": "Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3", "Sentence": "No, no one on our chart made 39 million. On average, each million dollar increase in salary was associated with a 39% increase in winning percentage. So that would be something related to the slope, and the slope was definitely not 39. The average winning percentage was 39%. No, that wasn't the case either. The model indicates that teams with coaches who had a salary of zero million dollars will average a winning percentage of approximately 39%. Yeah, this is the closest statement to what we just said, that if you believe that model, and that's a big if, if you believe this model, then this model says someone making zero dollars will get 39%, and this is frankly why you have to be skeptical of models."}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "We're told that Maya and Doug are finalists in a crafting competition. For the final round, each of them will randomly select a card without replacement that will reveal what the star material must be in their craft. Here are the available cards. So I guess the star material is the primary material they need to use in this competition. Maya and Doug both want to get silk as their star material. Maya will draw first, followed by Doug. What is the probability that neither contestant draws silk?"}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "So I guess the star material is the primary material they need to use in this competition. Maya and Doug both want to get silk as their star material. Maya will draw first, followed by Doug. What is the probability that neither contestant draws silk? Pause this video and see if you can work through that before we work through this together. All right, now let's work through this together. So the probability that neither contestant draws silk."}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "What is the probability that neither contestant draws silk? Pause this video and see if you can work through that before we work through this together. All right, now let's work through this together. So the probability that neither contestant draws silk. So that would be, I'll just write it another way, the probability that, I'll write MNS for Maya no silk. So Maya no silk and Doug no silk. That's just another way of saying what is the probability that neither contestant draws silk?"}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "So the probability that neither contestant draws silk. So that would be, I'll just write it another way, the probability that, I'll write MNS for Maya no silk. So Maya no silk and Doug no silk. That's just another way of saying what is the probability that neither contestant draws silk? And so this is going to be equivalent to the probability that Maya does not get silk, Maya no silk, right over here, times the probability that Doug doesn't get silk, given that Maya did not get silk, given Maya no silk. This line right over here, this vertical line, this is shorthand for given. And so let's calculate each of these."}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "That's just another way of saying what is the probability that neither contestant draws silk? And so this is going to be equivalent to the probability that Maya does not get silk, Maya no silk, right over here, times the probability that Doug doesn't get silk, given that Maya did not get silk, given Maya no silk. This line right over here, this vertical line, this is shorthand for given. And so let's calculate each of these. So this is going to be equal to the probability that Maya gets no silk, she picked first. There's six options out of here. Five of them are not silk."}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "And so let's calculate each of these. So this is going to be equal to the probability that Maya gets no silk, she picked first. There's six options out of here. Five of them are not silk. So it is five over six. And then the probability that Doug does not get silk, given that Maya did not get silk. So if Maya did not get silk, then that means that silk is still in the mix, but there's only five possibilities left because Maya picked one of them."}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "Five of them are not silk. So it is five over six. And then the probability that Doug does not get silk, given that Maya did not get silk. So if Maya did not get silk, then that means that silk is still in the mix, but there's only five possibilities left because Maya picked one of them. And four of them are not silk. They're still silk as an option. And it's important to recognize that the probability that Doug gets no silk is dependent on whether Maya got silk or not."}, {"video_title": "General multiplication rule example dependent events Probability & combinatorics.mp3", "Sentence": "So if Maya did not get silk, then that means that silk is still in the mix, but there's only five possibilities left because Maya picked one of them. And four of them are not silk. They're still silk as an option. And it's important to recognize that the probability that Doug gets no silk is dependent on whether Maya got silk or not. So it's very important to have this given right over here. If these were independent events, if Maya picked and then put her card back in and then Doug were to pick separately, then the probability that Doug gets no silk, given that Maya got no silk, would be the same thing as the probability that Doug gets no silk, regardless of what Maya was doing. And so this will end up becoming four over six, which is the same thing as 2 3rds."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "Suppose that each pack has probability 0.2 of containing the card Hugo is hoping for. Let the random variable X be the number of packs of cards Hugo buys. Here is the probability distribution for X. So it looks like there is a 0.2 probability that he buys one pack, and that makes sense because that first pack, there is a 0.2 probability that it contains his favorite player's card. And if it does, at that point, he'll just stop. He won't buy any more packs. Now what about the probability that he buys two packs?"}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "So it looks like there is a 0.2 probability that he buys one pack, and that makes sense because that first pack, there is a 0.2 probability that it contains his favorite player's card. And if it does, at that point, he'll just stop. He won't buy any more packs. Now what about the probability that he buys two packs? Well, over here, they give it a 0.16, and that makes sense. There is a 0.8 probability that he does not get the card he wants on the first one, and then there's another 0.2 that he gets it on the second one. So 0.8 times 0.2 does indeed equal 0.16."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "Now what about the probability that he buys two packs? Well, over here, they give it a 0.16, and that makes sense. There is a 0.8 probability that he does not get the card he wants on the first one, and then there's another 0.2 that he gets it on the second one. So 0.8 times 0.2 does indeed equal 0.16. But they're not asking us to calculate that. They give it to us. Then the probability that he gets three packs is 0.128, and then they've left blank the probability that he gets four packs."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "So 0.8 times 0.2 does indeed equal 0.16. But they're not asking us to calculate that. They give it to us. Then the probability that he gets three packs is 0.128, and then they've left blank the probability that he gets four packs. But this is the entire discrete probability distribution because Hugo has to stop at four. Even if he doesn't get the card he wants at four on the fourth pack, he's just going to stop over there. So we could actually figure out this question mark by just realizing that these four probabilities have to add up to one."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "Then the probability that he gets three packs is 0.128, and then they've left blank the probability that he gets four packs. But this is the entire discrete probability distribution because Hugo has to stop at four. Even if he doesn't get the card he wants at four on the fourth pack, he's just going to stop over there. So we could actually figure out this question mark by just realizing that these four probabilities have to add up to one. But let's just first answer the question. Find the indicated probability. What is the probability that X is greater than or equal to two?"}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "So we could actually figure out this question mark by just realizing that these four probabilities have to add up to one. But let's just first answer the question. Find the indicated probability. What is the probability that X is greater than or equal to two? What is the probability? Remember, X is the number of packs of cards Hugo buys. I encourage you to pause the video and try to figure it out."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "What is the probability that X is greater than or equal to two? What is the probability? Remember, X is the number of packs of cards Hugo buys. I encourage you to pause the video and try to figure it out. So let's look at the scenarios we're talking about. Probability that our discrete random variable X is greater than or equal to two. Well, that's these three scenarios right over here."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "I encourage you to pause the video and try to figure it out. So let's look at the scenarios we're talking about. Probability that our discrete random variable X is greater than or equal to two. Well, that's these three scenarios right over here. And so what is their combined probability? Well, you might want to say, hey, we need to figure out what the probability of getting exactly four packs are. But we have to remember that these all add up to 100%."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "Well, that's these three scenarios right over here. And so what is their combined probability? Well, you might want to say, hey, we need to figure out what the probability of getting exactly four packs are. But we have to remember that these all add up to 100%. And so this right over here is 0.2. And so this is 0.2, the other three combined have to add up to 0.8. 0.8 plus 0.2 is one, or 100%."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "But we have to remember that these all add up to 100%. And so this right over here is 0.2. And so this is 0.2, the other three combined have to add up to 0.8. 0.8 plus 0.2 is one, or 100%. So just like that, we know that this is 0.8. If for kicks, we wanted to figure out this question mark right over here, we could just say that, look, have to add up to one. So we could say the probability of exactly four is going to be equal to one minus 0.2 minus 0.16 minus 0.128."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "0.8 plus 0.2 is one, or 100%. So just like that, we know that this is 0.8. If for kicks, we wanted to figure out this question mark right over here, we could just say that, look, have to add up to one. So we could say the probability of exactly four is going to be equal to one minus 0.2 minus 0.16 minus 0.128. I get one minus 0.2 minus 0.16 minus 0.128 is equal to 0.512, is equal to 0.512. 0.512. You might immediately say, wait, wait, this seems like a very high probability."}, {"video_title": "Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3", "Sentence": "So we could say the probability of exactly four is going to be equal to one minus 0.2 minus 0.16 minus 0.128. I get one minus 0.2 minus 0.16 minus 0.128 is equal to 0.512, is equal to 0.512. 0.512. You might immediately say, wait, wait, this seems like a very high probability. There's more than a 50% chance that he buys four packs. And you have to remember, he has to stop at four. Even if on the fourth, he doesn't get the card he wants, he still has to stop there."}, {"video_title": "Matched pairs experiment design Study design AP Statistics Khan Academy.mp3", "Sentence": "The last video, we constructed an experiment where we had a drug that we thought might help control people's blood sugar. We looked for something that we could measure as an indicator of whether blood sugar is being controlled, and hemoglobin A1c is actually what people measure in a blood test. We have a whole video on it on Khan Academy, but it is an average measure of your blood sugar over roughly a three-month period. So that's the explanatory variable, whether or not you're taking the pill, and the response variable is, well, what does it do to your hemoglobin A1c? We constructed a somewhat classic experiment where we had a control group and a treatment group, and we randomly assigned folks into either the control or the treatment group. And to ensure that one group or the other, or I guess both of them, don't end up with an imbalance of, in the case of the last video, an imbalance of men or women, we did what we call block design, where we took our 100 people, and we just happened to have 60 women and 40 men, and we said, okay, well, let's split the 60 women randomly between the two groups, and let's split the 40 men between these two groups so that we have at least an even distribution with respect to sex. And so we would measure folks' A1cs before they get the treatment or the placebo."}, {"video_title": "Matched pairs experiment design Study design AP Statistics Khan Academy.mp3", "Sentence": "So that's the explanatory variable, whether or not you're taking the pill, and the response variable is, well, what does it do to your hemoglobin A1c? We constructed a somewhat classic experiment where we had a control group and a treatment group, and we randomly assigned folks into either the control or the treatment group. And to ensure that one group or the other, or I guess both of them, don't end up with an imbalance of, in the case of the last video, an imbalance of men or women, we did what we call block design, where we took our 100 people, and we just happened to have 60 women and 40 men, and we said, okay, well, let's split the 60 women randomly between the two groups, and let's split the 40 men between these two groups so that we have at least an even distribution with respect to sex. And so we would measure folks' A1cs before they get the treatment or the placebo. Then we would wait three months of getting either the treatment or the placebo, and then we'll see if there's a statistically significant improvement. Now, this was a pretty good, and it's a bit of a classic experimental design. We would also do it so that the patients don't know which one they're getting, placebo or the actual treatment, so it's a blind experiment."}, {"video_title": "Matched pairs experiment design Study design AP Statistics Khan Academy.mp3", "Sentence": "And so we would measure folks' A1cs before they get the treatment or the placebo. Then we would wait three months of getting either the treatment or the placebo, and then we'll see if there's a statistically significant improvement. Now, this was a pretty good, and it's a bit of a classic experimental design. We would also do it so that the patients don't know which one they're getting, placebo or the actual treatment, so it's a blind experiment. And it would probably be good if even the nurses or the doctors who are administering the pills, who are giving the pills, also don't know which one they're giving, so it would be a double blind experiment. But this doesn't mean that it's a perfect experiment, and there seldom is a perfect experiment, and that's why it should be able to be replicated. Other people should try to prove the same thing, maybe in different ways."}, {"video_title": "Matched pairs experiment design Study design AP Statistics Khan Academy.mp3", "Sentence": "We would also do it so that the patients don't know which one they're getting, placebo or the actual treatment, so it's a blind experiment. And it would probably be good if even the nurses or the doctors who are administering the pills, who are giving the pills, also don't know which one they're giving, so it would be a double blind experiment. But this doesn't mean that it's a perfect experiment, and there seldom is a perfect experiment, and that's why it should be able to be replicated. Other people should try to prove the same thing, maybe in different ways. But even the way that we designed it, there's still a possibility that there are some lurking variables in here. Maybe, you know, we took care to make sure that our distribution of men and women was roughly even across both of these groups, but maybe by, through that random sampling, we got a disproportionate number of young people in the treatment group, and maybe young people responded better to taking a pill. Maybe it changes their behaviors in other ways, or maybe older people, when they take a pill, they decide to eat worse because they say, oh, this pill's gonna solve all my problems."}, {"video_title": "Matched pairs experiment design Study design AP Statistics Khan Academy.mp3", "Sentence": "Other people should try to prove the same thing, maybe in different ways. But even the way that we designed it, there's still a possibility that there are some lurking variables in here. Maybe, you know, we took care to make sure that our distribution of men and women was roughly even across both of these groups, but maybe by, through that random sampling, we got a disproportionate number of young people in the treatment group, and maybe young people responded better to taking a pill. Maybe it changes their behaviors in other ways, or maybe older people, when they take a pill, they decide to eat worse because they say, oh, this pill's gonna solve all my problems. And so you could have these other lurking variables, like age, or where in the country they live, or other types of things, that just by the random process, you might have things get uneven in one way or another. Now, one technique to help control for this a little bit, and I shouldn't use the word control too much, another technique to help mitigate this is something called matched pairs design. Matched, matched pairs, pairs design of an experiment, and it's essentially, instead of going through all of this trouble saying, oh boy, maybe we do block design, all this random sampling, instead, you randomly put people first into either the control or the treatment group, and then we do another round, you measure, and then you do another round where you switch, where the people who are in the treatment go into the control, and the people who are in the control go into the treatment."}, {"video_title": "Matched pairs experiment design Study design AP Statistics Khan Academy.mp3", "Sentence": "Maybe it changes their behaviors in other ways, or maybe older people, when they take a pill, they decide to eat worse because they say, oh, this pill's gonna solve all my problems. And so you could have these other lurking variables, like age, or where in the country they live, or other types of things, that just by the random process, you might have things get uneven in one way or another. Now, one technique to help control for this a little bit, and I shouldn't use the word control too much, another technique to help mitigate this is something called matched pairs design. Matched, matched pairs, pairs design of an experiment, and it's essentially, instead of going through all of this trouble saying, oh boy, maybe we do block design, all this random sampling, instead, you randomly put people first into either the control or the treatment group, and then we do another round, you measure, and then you do another round where you switch, where the people who are in the treatment go into the control, and the people who are in the control go into the treatment. So we could even extend from what we have here, we could imagine a world where the first three months, we have the 50 people in this treatment group, we have another 50 people in this control group that are taking the placebo, we see what happens to the A1Cs, and then we switch, where this group over here, then, and they don't know, they don't know, first of all, ideally, it's a blind experiment, so they don't even know they were in the treatment groups, and hopefully the pills look identical, so now, that same group, for the next three months, is now going to be the control group, and so they got the medicine for the first three months, and we saw what happens to their A1C, and now they're gonna get the placebo, they're going to get the placebo for the second three months, and then we are going to see what happens to their A1C, and likewise, the other group is going to be switched around. The thing that, the folks that used to be getting the placebo could now get, could now get the treatment. They are now going to get the treatment, and the value here is, is that because everyone is going through, is that for one period is in the control group, and for one period is in the treatment group, and they don't know when, which one is happening, you are less likely to have a lurking variable like age or geographic region or behavior cause an imbalance or somehow skew the results or give you biased results."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "She recorded the height in centimeters of each customer and the frame size in centimeters of the bicycle that customer rented. After plotting her results, Vera noticed that the relationship between the two variables was fairly linear, so she used the data to calculate the following least squares regression equation for predicting bicycle frame size from the height of the customer, and this is the equation. So before I even look at this question, let's just think about what she did. So she had a bunch of customers, and she recorded, given the height of the customer, what size frame that person rented, and so she might have had something like this, where in the horizontal axis, you have height measured in centimeters, and in the vertical axis, you have frame size that's also measured in centimeters, and so there might have been someone who measures 100 centimeters in height who gets a 25-centimeter frame. I don't know if that's reasonable or not for you bicycle experts, but let's just go with it, and so she would have plotted it there. Maybe there was another person of 100 centimeters in height who got a frame that was slightly larger, and she plotted it there, and then she did a least squares regression, and a least squares regression is trying to fit a line to this data. Oftentimes, you would use a spreadsheet or you use a computer, and that line is trying to minimize the square of the distance between these points, and so the least squares regression, maybe it would look something like this, and this is just a rough estimate of it."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So she had a bunch of customers, and she recorded, given the height of the customer, what size frame that person rented, and so she might have had something like this, where in the horizontal axis, you have height measured in centimeters, and in the vertical axis, you have frame size that's also measured in centimeters, and so there might have been someone who measures 100 centimeters in height who gets a 25-centimeter frame. I don't know if that's reasonable or not for you bicycle experts, but let's just go with it, and so she would have plotted it there. Maybe there was another person of 100 centimeters in height who got a frame that was slightly larger, and she plotted it there, and then she did a least squares regression, and a least squares regression is trying to fit a line to this data. Oftentimes, you would use a spreadsheet or you use a computer, and that line is trying to minimize the square of the distance between these points, and so the least squares regression, maybe it would look something like this, and this is just a rough estimate of it. It might look something, actually, let me get my ruler tool. It might look something like, it might look something like this. So let me plot it."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "Oftentimes, you would use a spreadsheet or you use a computer, and that line is trying to minimize the square of the distance between these points, and so the least squares regression, maybe it would look something like this, and this is just a rough estimate of it. It might look something, actually, let me get my ruler tool. It might look something like, it might look something like this. So let me plot it. So this, that would be the line, so our regression line, y-hat, is equal to 1 3rd plus 1 3rd x, and so you could view this as a way of predicting or either modeling the relationship or predicting that, hey, if I get a new person, I could take their height and put it as an x and figure out what frame size they're likely to rent, but they ask us, what is the residual of a customer with a frame, with a height of 155 centimeters who rents a bike with a 51-centimeter frame? So how do we think about this? Well, the residual is going to be the difference between what they actually produce and what the line, what our regression line would have predicted, so we could say residual, let me write it this way, residual is going to be actual, actual minus predicted."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So let me plot it. So this, that would be the line, so our regression line, y-hat, is equal to 1 3rd plus 1 3rd x, and so you could view this as a way of predicting or either modeling the relationship or predicting that, hey, if I get a new person, I could take their height and put it as an x and figure out what frame size they're likely to rent, but they ask us, what is the residual of a customer with a frame, with a height of 155 centimeters who rents a bike with a 51-centimeter frame? So how do we think about this? Well, the residual is going to be the difference between what they actually produce and what the line, what our regression line would have predicted, so we could say residual, let me write it this way, residual is going to be actual, actual minus predicted. So if predicted is larger than actual, this is actually going to be a negative number. If predicted is smaller than actual, this is going to be a positive number. Well, we know the actual, they tell us that, they tell us that they rent, it's a, the 155-centimeter person rents a bike with a 51-centimeter frame, so this is 51 centimeters, but what is the predicted?"}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "Well, the residual is going to be the difference between what they actually produce and what the line, what our regression line would have predicted, so we could say residual, let me write it this way, residual is going to be actual, actual minus predicted. So if predicted is larger than actual, this is actually going to be a negative number. If predicted is smaller than actual, this is going to be a positive number. Well, we know the actual, they tell us that, they tell us that they rent, it's a, the 155-centimeter person rents a bike with a 51-centimeter frame, so this is 51 centimeters, but what is the predicted? Well, that's where we can use our regression equation that Vera came up with. The predicted, I'll do that in orange, the predicted is going to be equal to 1 3rd plus 1 3rd times a person's height. Their height is 155."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "Well, we know the actual, they tell us that, they tell us that they rent, it's a, the 155-centimeter person rents a bike with a 51-centimeter frame, so this is 51 centimeters, but what is the predicted? Well, that's where we can use our regression equation that Vera came up with. The predicted, I'll do that in orange, the predicted is going to be equal to 1 3rd plus 1 3rd times a person's height. Their height is 155. That's the predicted. Y hat is what our linear regression predicts, our line predicts, so what is this going to be? This is going to be equal to 1 3rd plus 155 over three, which is equal to 156 over three, which comes out nicely to 52."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "Their height is 155. That's the predicted. Y hat is what our linear regression predicts, our line predicts, so what is this going to be? This is going to be equal to 1 3rd plus 155 over three, which is equal to 156 over three, which comes out nicely to 52. So the predicted on our line is 52, and so here, so this person is 155, we can plot them right over here, 155. They're coming in slightly below the line. So they're coming in slightly below the line right there, and that distance, which is, and we can see that they are below the line, so that distance is going to be, or in this case, the residual is going to be negative, so this is going to be negative one."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "This is going to be equal to 1 3rd plus 155 over three, which is equal to 156 over three, which comes out nicely to 52. So the predicted on our line is 52, and so here, so this person is 155, we can plot them right over here, 155. They're coming in slightly below the line. So they're coming in slightly below the line right there, and that distance, which is, and we can see that they are below the line, so that distance is going to be, or in this case, the residual is going to be negative, so this is going to be negative one. And so if we were to zoom in right over here, you can't see it that well, but let me draw, so if we zoom in, let's say we were to zoom in the line, and it looks like this, and our data point is right, our data point is right over here. We know we're below the line, and this is going to be a negative residual, and the magnitude of that residual is how far we are below the line. And in this case, it is negative one."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "So they're coming in slightly below the line right there, and that distance, which is, and we can see that they are below the line, so that distance is going to be, or in this case, the residual is going to be negative, so this is going to be negative one. And so if we were to zoom in right over here, you can't see it that well, but let me draw, so if we zoom in, let's say we were to zoom in the line, and it looks like this, and our data point is right, our data point is right over here. We know we're below the line, and this is going to be a negative residual, and the magnitude of that residual is how far we are below the line. And in this case, it is negative one. And so that is our residual. This is what the actual data minus what was predicted by our regression line."}, {"video_title": "Calculating residual example Exploring bivariate numerical data AP Statistics Khan Academy.mp3", "Sentence": "And in this case, it is negative one. And so that is our residual. This is what the actual data minus what was predicted by our regression line."}]