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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. We have frequency table, dot plot, list of numbers. These are all showing the same data, just in different ways. | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
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. 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. | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
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. 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. | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
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. 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. | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
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? 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? | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
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. And the minimum age here, you see, is five. So there's a range of seven. | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
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. But you could have also done that over here. You could say, hey, the maximum age here is 12. | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
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. And so you find the difference between 12 and five, which is seven. Here, you still could have done it. | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
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? Let's look at five. Are there any fours here? | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
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. So five's the minimum age. And what's the largest? | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
Are there any fours here? 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? | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
And what's the largest? Is it seven? No, is it nine? Nine, not even 10? Oh, 12. 12. Are there any 13s? | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
Nine, not even 10? 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. | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
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. 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. | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
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. Or you could look over here. How many are older than nine? | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
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. 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. | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
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. And then not that person right over there. So hopefully, this is just an appreciation for yet another two ways of looking at data, frequency tables and dot plots. | Frequency tables and dot plots Data and statistics 6th grade Khan Academy.mp3 |
And I'll circle the statistical questions in yellow. And I encourage you to pause this video and try to figure this out yourself first. Look at each of these questions and think about whether you think you need statistics to answer this question or you don't need statistics, whether these are statistical questions or not. So I'm assuming you've given a pass at it, unless we can go through this together. So this first question is, how old are you? So we're talking about how old is a particular person. There is an answer here. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
So I'm assuming you've given a pass at it, unless we can go through this together. So this first question is, how old are you? So we're talking about how old is a particular person. There is an answer here. We don't need any tools of statistics to answer this. So this is not a statistical question. How old are the people who have watched this video in 2013? | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
There is an answer here. We don't need any tools of statistics to answer this. So this is not a statistical question. How old are the people who have watched this video in 2013? Now, this is interesting. We're assuming that multiple people will have watched this video in 2013. And they're not all going to be the same age. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
How old are the people who have watched this video in 2013? Now, this is interesting. We're assuming that multiple people will have watched this video in 2013. And they're not all going to be the same age. There's going to be some variability in their age. So one person might be 10 years old. Another person might be 20. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
And they're not all going to be the same age. There's going to be some variability in their age. So one person might be 10 years old. Another person might be 20. Another person might be 15. So what answer do you give here? Would you give all of the ages? | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
Another person might be 20. Another person might be 15. So what answer do you give here? Would you give all of the ages? But we want to get a sense of, in general, how old are the people? So this is where statistics might be valuable. We might want to find some type of central tendency, an average, a median age for this. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
Would you give all of the ages? But we want to get a sense of, in general, how old are the people? So this is where statistics might be valuable. We might want to find some type of central tendency, an average, a median age for this. So this is absolutely a statistical question. And you might already be seeing kind of a pattern here. The first question, we were asking about a particular person. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
We might want to find some type of central tendency, an average, a median age for this. So this is absolutely a statistical question. And you might already be seeing kind of a pattern here. The first question, we were asking about a particular person. There was only one answer here. There's no variability in the answer. The second one, we're asking about a trait of a bunch of people. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
The first question, we were asking about a particular person. There was only one answer here. There's no variability in the answer. The second one, we're asking about a trait of a bunch of people. And there's variability in that trait. They're not all the same age. And so we'll need statistics to come up with some features of the data set to be able to make some conclusions. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
The second one, we're asking about a trait of a bunch of people. And there's variability in that trait. They're not all the same age. And so we'll need statistics to come up with some features of the data set to be able to make some conclusions. We might say, on average, the people who have watched this video in 2013 are 18 years old, or 22 years old, or the median is 24 years old, whatever it might be. Do dogs run faster than cats? So once again, there are many dogs and many cats. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
And so we'll need statistics to come up with some features of the data set to be able to make some conclusions. We might say, on average, the people who have watched this video in 2013 are 18 years old, or 22 years old, or the median is 24 years old, whatever it might be. Do dogs run faster than cats? So once again, there are many dogs and many cats. And they all run at different speeds. Some dogs run faster than some cats, and some cats run faster than some dogs. So we would need some statistics to get a sense of, in general, or on average, how fast do dogs run? | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
So once again, there are many dogs and many cats. And they all run at different speeds. Some dogs run faster than some cats, and some cats run faster than some dogs. So we would need some statistics to get a sense of, in general, or on average, how fast do dogs run? And then maybe, on average, how fast do cats run? And then we could compare those averages, or we could compare the medians in some way. So this is definitely a statistical question. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
So we would need some statistics to get a sense of, in general, or on average, how fast do dogs run? And then maybe, on average, how fast do cats run? And then we could compare those averages, or we could compare the medians in some way. So this is definitely a statistical question. Once again, we're talking about, in general, a whole population of dogs, the whole species of dogs, versus cats. And there's variation in how fast dogs run and how fast cats run. If we were talking about a particular dog and a particular cat, well, then there would just be an answer. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
So this is definitely a statistical question. Once again, we're talking about, in general, a whole population of dogs, the whole species of dogs, versus cats. And there's variation in how fast dogs run and how fast cats run. If we were talking about a particular dog and a particular cat, well, then there would just be an answer. Does dog A run faster than cat B? Well, sure. That's not going to be a statistical question. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
If we were talking about a particular dog and a particular cat, well, then there would just be an answer. Does dog A run faster than cat B? Well, sure. That's not going to be a statistical question. You don't have to use the tools of statistics. And this next question actually fits that pattern. Do wolves weigh? | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
That's not going to be a statistical question. You don't have to use the tools of statistics. And this next question actually fits that pattern. Do wolves weigh? Actually, no. This fits the pattern of the previous one. Do wolves weigh more than dogs? | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
Do wolves weigh? Actually, no. This fits the pattern of the previous one. Do wolves weigh more than dogs? So once again, there are some very light dogs and some very heavy wolves. So those wolves definitely weigh more than those dogs. But there are some very, very, very heavy dogs. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
Do wolves weigh more than dogs? So once again, there are some very light dogs and some very heavy wolves. So those wolves definitely weigh more than those dogs. But there are some very, very, very heavy dogs. And so what you would want to do here, because we have variability in each of these, is you might want to come with some central tendency. On average, what's the median wolf weight? What's the average, the mean wolf weight? | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
But there are some very, very, very heavy dogs. And so what you would want to do here, because we have variability in each of these, is you might want to come with some central tendency. On average, what's the median wolf weight? What's the average, the mean wolf weight? Compare that to the mean dog's weight. So once again, since we're speaking in general about wolves, not a particular wolf, and in general about dogs, and there's variation in the data, and we're trying to glean some numbers from that to compare, this is definitely a statistical question. Definitely a statistical question. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
What's the average, the mean wolf weight? Compare that to the mean dog's weight. So once again, since we're speaking in general about wolves, not a particular wolf, and in general about dogs, and there's variation in the data, and we're trying to glean some numbers from that to compare, this is definitely a statistical question. Definitely a statistical question. Does your dog weigh more than that wolf? And we're assuming that we're pointing at a particular wolf. So now this is the particular. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
Definitely a statistical question. Does your dog weigh more than that wolf? And we're assuming that we're pointing at a particular wolf. So now this is the particular. We're comparing a dog to a particular dog to a particular wolf. We can put each of them on a weighing machine and come up with an absolute answer. There's no variability in this dog's weight, at least at the moment that we weigh it. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
So now this is the particular. We're comparing a dog to a particular dog to a particular wolf. We can put each of them on a weighing machine and come up with an absolute answer. There's no variability in this dog's weight, at least at the moment that we weigh it. No variability in this wolf's weight at the moment that we weigh it. So this is not a statistical question. So I'll put an x next to the ones that are not statistical questions. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
There's no variability in this dog's weight, at least at the moment that we weigh it. No variability in this wolf's weight at the moment that we weigh it. So this is not a statistical question. So I'll put an x next to the ones that are not statistical questions. Does it rain more in Seattle than Singapore? So once again, there's variation here. And we would also probably want to notice it rained more in Seattle than Singapore in a given year, over a decade, or whatever. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
So I'll put an x next to the ones that are not statistical questions. Does it rain more in Seattle than Singapore? So once again, there's variation here. And we would also probably want to notice it rained more in Seattle than Singapore in a given year, over a decade, or whatever. But regardless of those questions, however we ask it, in some years it might rain more in Seattle. In other years it might rain more in Singapore. Or if we just pick Seattle, it rains a different amount from year to year. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
And we would also probably want to notice it rained more in Seattle than Singapore in a given year, over a decade, or whatever. But regardless of those questions, however we ask it, in some years it might rain more in Seattle. In other years it might rain more in Singapore. Or if we just pick Seattle, it rains a different amount from year to year. In Singapore, it rains a different amount from year to year. So how do we compare? Well, that's where the statistics could be valuable. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
Or if we just pick Seattle, it rains a different amount from year to year. In Singapore, it rains a different amount from year to year. So how do we compare? Well, that's where the statistics could be valuable. There's variability in the data. So we can look at the data set for Seattle and come up with some type of an average, some type of a central tendency, and compare that to the average, the mean, the mode, whatever you want to. The mode probably wouldn't be that useful here. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
Well, that's where the statistics could be valuable. There's variability in the data. So we can look at the data set for Seattle and come up with some type of an average, some type of a central tendency, and compare that to the average, the mean, the mode, whatever you want to. The mode probably wouldn't be that useful here. To Singapore. So this is definitely a statistical question. Definitely statistical. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
The mode probably wouldn't be that useful here. To Singapore. So this is definitely a statistical question. Definitely statistical. What was the difference in rainfall between Singapore and Seattle in 2013? Well, these two numbers aren't known. They can be measured. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
Definitely statistical. What was the difference in rainfall between Singapore and Seattle in 2013? Well, these two numbers aren't known. They can be measured. Both the rainfall in Singapore can be measured. The rainfall in Seattle can be measured. And assuming that this has already happened and we can measure it, then we can just find the difference. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
They can be measured. Both the rainfall in Singapore can be measured. The rainfall in Seattle can be measured. And assuming that this has already happened and we can measure it, then we can just find the difference. So you don't need statistics here. You just have to have both of these measurements and subtract the difference. So not a statistical question. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
And assuming that this has already happened and we can measure it, then we can just find the difference. So you don't need statistics here. You just have to have both of these measurements and subtract the difference. So not a statistical question. In general, will I use less gas driving at 55 miles an hour than 70 miles per hour? So this feels statistical because it probably depends on the circumstance. It might depend on the car. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
So not a statistical question. In general, will I use less gas driving at 55 miles an hour than 70 miles per hour? So this feels statistical because it probably depends on the circumstance. It might depend on the car. Or even for a given car, when you drive at 55 miles per hour, there's some variation in your gas mileage. It might be how recent an oil change happened, what the wind conditions are like, what the road conditions are like, I mean, exactly how you're driving the car. Are you turning? | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
It might depend on the car. Or even for a given car, when you drive at 55 miles per hour, there's some variation in your gas mileage. It might be how recent an oil change happened, what the wind conditions are like, what the road conditions are like, I mean, exactly how you're driving the car. Are you turning? Are you going in a straight line? And same thing for 70 miles an hour. So when we're seeing in general, there's variation in what the gas mileage is at 55 miles an hour and at 70 miles an hour. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
Are you turning? Are you going in a straight line? And same thing for 70 miles an hour. So when we're seeing in general, there's variation in what the gas mileage is at 55 miles an hour and at 70 miles an hour. So what you'd probably want to do is say, well, what's my average mileage when I drive at 55 miles an hour? And compare that to the average mileage when I drive at 70. So because we have this variability in each of those cases, this is definitely a statistical question. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
So when we're seeing in general, there's variation in what the gas mileage is at 55 miles an hour and at 70 miles an hour. So what you'd probably want to do is say, well, what's my average mileage when I drive at 55 miles an hour? And compare that to the average mileage when I drive at 70. So because we have this variability in each of those cases, this is definitely a statistical question. Do English professors get paid less than math professors? So once again, all English professors don't get paid the same amount, and all math professors don't get paid the same amount. Some English professors might do quite well. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
So because we have this variability in each of those cases, this is definitely a statistical question. Do English professors get paid less than math professors? So once again, all English professors don't get paid the same amount, and all math professors don't get paid the same amount. Some English professors might do quite well. Some might make very little. Same thing for math professors. So we'd probably want to find some type of an average to represent the central tendency for each of these. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
Some English professors might do quite well. Some might make very little. Same thing for math professors. So we'd probably want to find some type of an average to represent the central tendency for each of these. So once again, this is a statistical question. This is a statistical question. Does the most highly paid English professor at Harvard get paid more than the most highly paid math professor at MIT? | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
So we'd probably want to find some type of an average to represent the central tendency for each of these. So once again, this is a statistical question. This is a statistical question. Does the most highly paid English professor at Harvard get paid more than the most highly paid math professor at MIT? Well, now we're talking about two particular individuals. You could go look at their tax forms, see how much each of them get paid. And especially if we assume that this is in a particular year, let's say, and let's just make it that way, say in 2013, just so that we can remove some variability that they might make from year to year, make it a little bit more concrete. | Statistical and non statistical questions Probability and Statistics Khan Academy.mp3 |
Jamie's dad gave her a die for her birthday. She wanted to make sure it was fair, so she took her die to school and rolled it 500 times and kept track of how many times the die rolled each number. Afterwards, she calculated the expected value of the sum of 20 rolls to be 67.4. The expected value of the sum of 20 rolls to be 67.4. On her way home from school, it was raining, and two values were washed away from her data table. Find the two missing absolute frequencies from Jamie's data table. So you see here, she rolled her die 500 times, and she wrote down how many times she got a two. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
The expected value of the sum of 20 rolls to be 67.4. On her way home from school, it was raining, and two values were washed away from her data table. Find the two missing absolute frequencies from Jamie's data table. So you see here, she rolled her die 500 times, and she wrote down how many times she got a two. She got a two 110 times, a 395 times, a 470 times, a 575 times, and then she had written down what she got, how many times she got a one and a six, but then it got washed away, so we need to figure out how many times she got a one and a six. Given the information on this table right over here, and given the information that the expected value of the sum of 20 rolls is 67.4. So I encourage you to pause this video and think about it on your own before I give a go at it. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
So you see here, she rolled her die 500 times, and she wrote down how many times she got a two. She got a two 110 times, a 395 times, a 470 times, a 575 times, and then she had written down what she got, how many times she got a one and a six, but then it got washed away, so we need to figure out how many times she got a one and a six. Given the information on this table right over here, and given the information that the expected value of the sum of 20 rolls is 67.4. So I encourage you to pause this video and think about it on your own before I give a go at it. So first, let's think about what this expected value, the sum of 20 rolls, being 67.4 tells us. That means that the expected value of one roll, the expected value of the sum of 20 rolls is just 20 times the expected value of one roll. So the expected value of a roll, let me do it here, expected value of a roll is going to be equal to 67.4 divided by 20. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
So I encourage you to pause this video and think about it on your own before I give a go at it. So first, let's think about what this expected value, the sum of 20 rolls, being 67.4 tells us. That means that the expected value of one roll, the expected value of the sum of 20 rolls is just 20 times the expected value of one roll. So the expected value of a roll, let me do it here, expected value of a roll is going to be equal to 67.4 divided by 20. We can get our calculator out. Let's see, so we have 67.4 divided by 20 is 3.37. So this is equal to 3.37. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
So the expected value of a roll, let me do it here, expected value of a roll is going to be equal to 67.4 divided by 20. We can get our calculator out. Let's see, so we have 67.4 divided by 20 is 3.37. So this is equal to 3.37. So how does that help us? Well, we know how to calculate an expected value given this frequency table right over here. If we say that this number right over here, let's say that's capital A, and let's say that this number here is capital B. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
So this is equal to 3.37. So how does that help us? Well, we know how to calculate an expected value given this frequency table right over here. If we say that this number right over here, let's say that's capital A, and let's say that this number here is capital B. If we were to try to calculate the expected value of a roll, what we really want to do is take the weighted frequency of each of these values, the weighted sum. So for example, if we got a one A out of 500 times, it would be A out of 500 times one, plus, I'll do these in different colors, plus 110 out of 500 times two, plus 110 out of 500 times two. Notice, this is the frequency, which was they got two, times two, we're taking a weighted sum of these values. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
If we say that this number right over here, let's say that's capital A, and let's say that this number here is capital B. If we were to try to calculate the expected value of a roll, what we really want to do is take the weighted frequency of each of these values, the weighted sum. So for example, if we got a one A out of 500 times, it would be A out of 500 times one, plus, I'll do these in different colors, plus 110 out of 500 times two, plus 110 out of 500 times two. Notice, this is the frequency, which was they got two, times two, we're taking a weighted sum of these values. And then plus 95 out of 500 times three, plus 95 out of 500 times three, plus, I think you see where this is going, 70 over 500 times four, plus 70 over 500 times four, almost there, plus, let's see, I haven't used this brown color, plus 75 over 500 times, I'll do it here, plus 75 over 500 times five, finally, plus B over 500 times six, this is going to give us our expected value of a roll, which is going to be equal to 3.37. So all of this is equal to 3.37. So one thing that we can do, since we have all these 500s in this denominator right over here, let's multiply both sides of this equation times 500. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
Notice, this is the frequency, which was they got two, times two, we're taking a weighted sum of these values. And then plus 95 out of 500 times three, plus 95 out of 500 times three, plus, I think you see where this is going, 70 over 500 times four, plus 70 over 500 times four, almost there, plus, let's see, I haven't used this brown color, plus 75 over 500 times, I'll do it here, plus 75 over 500 times five, finally, plus B over 500 times six, this is going to give us our expected value of a roll, which is going to be equal to 3.37. So all of this is equal to 3.37. So one thing that we can do, since we have all these 500s in this denominator right over here, let's multiply both sides of this equation times 500. If we do that, the left-hand side becomes, well, 500 times A over 500 is just going to be A, plus 110, plus, oh, 110 times two, so it's going to be 220, plus 95 times three, that's going to be 15 less than 300, so it's going to be plus 285, plus 285, and then 70 times four is 280, plus 280. 75 times five is going to be 350 plus 25, 375, so plus 375, plus 6B, make sure I'm not skipping any steps here, plus 6B is going to be equal to, is going to be equal to this times 500, and that is going to be equal to 3.37 times 500 is equal to 1685, 1,685. So all I did to go from this step right over here, which I set up saying, hey, this is the expected value of one roll, which we already know to be 3.37, is I just multiplied both sides of this equation by 500. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
So one thing that we can do, since we have all these 500s in this denominator right over here, let's multiply both sides of this equation times 500. If we do that, the left-hand side becomes, well, 500 times A over 500 is just going to be A, plus 110, plus, oh, 110 times two, so it's going to be 220, plus 95 times three, that's going to be 15 less than 300, so it's going to be plus 285, plus 285, and then 70 times four is 280, plus 280. 75 times five is going to be 350 plus 25, 375, so plus 375, plus 6B, make sure I'm not skipping any steps here, plus 6B is going to be equal to, is going to be equal to this times 500, and that is going to be equal to 3.37 times 500 is equal to 1685, 1,685. So all I did to go from this step right over here, which I set up saying, hey, this is the expected value of one roll, which we already know to be 3.37, is I just multiplied both sides of this equation by 500. I just did this times 500, and I did this times, I did this times 500, and this 500 obviously cancels with all of these, and then 500 times 3.37 is 1685, and so I got this right over here. Now, I got one, two, three, four, five, six, yep, I did enough, I have the right number of terms. I just want to make sure I'm not making a careless mistake, and so if we want to simplify this, we can subtract 220, 285, 280, and 375 from both sides, and so we would be, if we did that, we would get A, if we subtract that from the left-hand side, we're just going to get A plus 6B, and on the right-hand side, we are going to get, let's get our calculator out, 1,685 minus 220, 220 minus 285, minus 285, minus 280, minus 280, minus 375, minus 375 gets us to 525. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
So all I did to go from this step right over here, which I set up saying, hey, this is the expected value of one roll, which we already know to be 3.37, is I just multiplied both sides of this equation by 500. I just did this times 500, and I did this times, I did this times 500, and this 500 obviously cancels with all of these, and then 500 times 3.37 is 1685, and so I got this right over here. Now, I got one, two, three, four, five, six, yep, I did enough, I have the right number of terms. I just want to make sure I'm not making a careless mistake, and so if we want to simplify this, we can subtract 220, 285, 280, and 375 from both sides, and so we would be, if we did that, we would get A, if we subtract that from the left-hand side, we're just going to get A plus 6B, and on the right-hand side, we are going to get, let's get our calculator out, 1,685 minus 220, 220 minus 285, minus 285, minus 280, minus 280, minus 375, minus 375 gets us to 525. So we get A plus 6B is equal to 525, and you say, okay, you did all that work, but we still have one equation with two unknowns. How do we figure out what A and B, how do we figure out what A and B actually are? Well, we know something else. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
I just want to make sure I'm not making a careless mistake, and so if we want to simplify this, we can subtract 220, 285, 280, and 375 from both sides, and so we would be, if we did that, we would get A, if we subtract that from the left-hand side, we're just going to get A plus 6B, and on the right-hand side, we are going to get, let's get our calculator out, 1,685 minus 220, 220 minus 285, minus 285, minus 280, minus 280, minus 375, minus 375 gets us to 525. So we get A plus 6B is equal to 525, and you say, okay, you did all that work, but we still have one equation with two unknowns. How do we figure out what A and B, how do we figure out what A and B actually are? Well, we know something else. We know, and this was actually much easier to figure out, we know that the sum of this whole table right over here, A plus 110 plus 95 plus 70 plus 75 plus B is equal to 500, or if we, let me write that down. So we know that A plus 110 plus 95 plus 70 plus 75 plus B needs to be equal to 500, needs to be equal to 500, or we could subtract 110 plus 95 plus 70 plus 75 from both sides and get, if you subtract it from the left-hand side, you're just left with A plus B, A plus B, and on the right-hand side, if we start with 500, so 500 minus 110, minus 95, minus 70, minus 75 gets us to 150. So A plus B must be equal to 150, is equal to 150. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
Well, we know something else. We know, and this was actually much easier to figure out, we know that the sum of this whole table right over here, A plus 110 plus 95 plus 70 plus 75 plus B is equal to 500, or if we, let me write that down. So we know that A plus 110 plus 95 plus 70 plus 75 plus B needs to be equal to 500, needs to be equal to 500, or we could subtract 110 plus 95 plus 70 plus 75 from both sides and get, if you subtract it from the left-hand side, you're just left with A plus B, A plus B, and on the right-hand side, if we start with 500, so 500 minus 110, minus 95, minus 70, minus 75 gets us to 150. So A plus B must be equal to 150, is equal to 150. And now we have a system of two equations and two unknowns, and so we know how to solve those. We could do it by substitution, or we could subtract the second equation from the first. So let's do that. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
So A plus B must be equal to 150, is equal to 150. And now we have a system of two equations and two unknowns, and so we know how to solve those. We could do it by substitution, or we could subtract the second equation from the first. So let's do that. Let's subtract the left-hand side of this equation from that. So, or essentially, we could add these two, so we can multiply this one times a negative one, and then add these two equations. The A's are going to cancel out, and we are going to be left with six B minus B is five B, is equal to 375, is equal to 375. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
So let's do that. Let's subtract the left-hand side of this equation from that. So, or essentially, we could add these two, so we can multiply this one times a negative one, and then add these two equations. The A's are going to cancel out, and we are going to be left with six B minus B is five B, is equal to 375, is equal to 375. Did I do that right? If I add 125 to this, I get to 500, and then another 25, I get to 525. So five B is equal to 375, or if we divide both sides by five, we get B is equal to, B is equal to 75. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
The A's are going to cancel out, and we are going to be left with six B minus B is five B, is equal to 375, is equal to 375. Did I do that right? If I add 125 to this, I get to 500, and then another 25, I get to 525. So five B is equal to 375, or if we divide both sides by five, we get B is equal to, B is equal to 75. B is equal to 75. So this right over here is equal to 75. If B is equal to 75, what is A? | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
So five B is equal to 375, or if we divide both sides by five, we get B is equal to, B is equal to 75. B is equal to 75. So this right over here is equal to 75. If B is equal to 75, what is A? Well, we know that A plus B is equal to 500. We figured that out a little while ago before we multiplied both sides of this times a negative one. We knew that A plus B, when B is now 75, so we could say A plus 75, is equal to 150. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
If B is equal to 75, what is A? Well, we know that A plus B is equal to 500. We figured that out a little while ago before we multiplied both sides of this times a negative one. We knew that A plus B, when B is now 75, so we could say A plus 75, is equal to 150. And that's just, from this, we figured out that A plus B is equal to 150 before we multiplied both sides times a negative. Subtract 75 from both sides, you get A is also equal to 75. And we are done. | Getting data from expected value Probability and Statistics Khan Academy.mp3 |
Which of the following plots suits the above description? So let's see, this looks like a negative linear correlation or association. As the years go by, you have a smaller percent of smokers. This one does too. As years go by, you have the number of smokers go down and down. This one down here also looks like that. Although it's not as smooth. | People smoking less over time scatter plot Regression Probability and Statistics Khan Academy.mp3 |
This one does too. As years go by, you have the number of smokers go down and down. This one down here also looks like that. Although it's not as smooth. If you were to fit a line here, it looks like you have a few outliers. Well, this is a positive correlation. So we can definitely rule out graph four. | People smoking less over time scatter plot Regression Probability and Statistics Khan Academy.mp3 |
Although it's not as smooth. If you were to fit a line here, it looks like you have a few outliers. Well, this is a positive correlation. So we can definitely rule out graph four. Now the other thing that they told us is that there are no outliers. Suggests a negative linear association with no outliers. If you were to try to fit a line to graph three, you could fit a line pretty reasonably that would go someplace like that. | People smoking less over time scatter plot Regression Probability and Statistics Khan Academy.mp3 |
So we can definitely rule out graph four. Now the other thing that they told us is that there are no outliers. Suggests a negative linear association with no outliers. If you were to try to fit a line to graph three, you could fit a line pretty reasonably that would go someplace like that. But it would have this outlier right over here. It looks like it's 12 or 13 years. It looks about 13 years after 1967. | People smoking less over time scatter plot Regression Probability and Statistics Khan Academy.mp3 |
If you were to try to fit a line to graph three, you could fit a line pretty reasonably that would go someplace like that. But it would have this outlier right over here. It looks like it's 12 or 13 years. It looks about 13 years after 1967. So that would be 1980. Looks like an outlier there. But they said it doesn't have any outliers. | People smoking less over time scatter plot Regression Probability and Statistics Khan Academy.mp3 |
It looks about 13 years after 1967. So that would be 1980. Looks like an outlier there. But they said it doesn't have any outliers. So we would rule out graph number three. And so we have to pick between graph one and graph two. And so the other hint they give us or piece of data they give us is that the percent drops by 0.5% each year. | People smoking less over time scatter plot Regression Probability and Statistics Khan Academy.mp3 |
But they said it doesn't have any outliers. So we would rule out graph number three. And so we have to pick between graph one and graph two. And so the other hint they give us or piece of data they give us is that the percent drops by 0.5% each year. So here, what's happening? As 10 years go by, let's see. As we go from, let's start with, let's see, in 1967, where it looks like we're at about 55%. | People smoking less over time scatter plot Regression Probability and Statistics Khan Academy.mp3 |
And so the other hint they give us or piece of data they give us is that the percent drops by 0.5% each year. So here, what's happening? As 10 years go by, let's see. As we go from, let's start with, let's see, in 1967, where it looks like we're at about 55%. And then 10 years go by, we are roughly at around 45%, a little under 45%. So we dropped 10% in 10 years. That seems to be how much this is dropping, about roughly 10% in 10 years. | People smoking less over time scatter plot Regression Probability and Statistics Khan Academy.mp3 |
As we go from, let's start with, let's see, in 1967, where it looks like we're at about 55%. And then 10 years go by, we are roughly at around 45%, a little under 45%. So we dropped 10% in 10 years. That seems to be how much this is dropping, about roughly 10% in 10 years. Another 10 years go by. We go from 45% or a little more than 10% in 10 years. And so that would mean that we're dropping, on average, more than 1 percentage point per year. | People smoking less over time scatter plot Regression Probability and Statistics Khan Academy.mp3 |
That seems to be how much this is dropping, about roughly 10% in 10 years. Another 10 years go by. We go from 45% or a little more than 10% in 10 years. And so that would mean that we're dropping, on average, more than 1 percentage point per year. That seems more than what's going on here. Now let's look over here. Over here, we're starting, it looks like, at around 42%. | People smoking less over time scatter plot Regression Probability and Statistics Khan Academy.mp3 |
And so that would mean that we're dropping, on average, more than 1 percentage point per year. That seems more than what's going on here. Now let's look over here. Over here, we're starting, it looks like, at around 42%. And then after 10 years, it looks like we're at 37%. So it looks like we've dropped about 5% in 10 years, which is consistent with this. If you drop 5% in 10 years, that means you drop half a percent per year. | People smoking less over time scatter plot Regression Probability and Statistics Khan Academy.mp3 |
But because it's so applicable to so many things, it's often a misused law or sometimes a slightly misunderstood. So just to be a little bit formal in our mathematics, let me just define it for you first. And then we'll talk a little bit about the intuition. So let's say I have a random variable X. And we know its expected value or its population mean. The law of large numbers just says that if we take a sample of n observations of our random variable, and if we were to average all of those observations, and let me define another variable, let's call that X sub n with a line on top of it, this is the mean of n observations of our random variable. So it's literally, this is my first observation. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
So let's say I have a random variable X. And we know its expected value or its population mean. The law of large numbers just says that if we take a sample of n observations of our random variable, and if we were to average all of those observations, and let me define another variable, let's call that X sub n with a line on top of it, this is the mean of n observations of our random variable. So it's literally, this is my first observation. So you could kind of say I run the experiment once and I get this observation and I run it again, I get that observation. And I keep running it n times. And then I divide by my number of observations. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
So it's literally, this is my first observation. So you could kind of say I run the experiment once and I get this observation and I run it again, I get that observation. And I keep running it n times. And then I divide by my number of observations. So this is my sample mean. This is the mean of all of the observations I've made. The law of large numbers just tells us that my sample mean will approach my expected value of the random variable. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
And then I divide by my number of observations. So this is my sample mean. This is the mean of all of the observations I've made. The law of large numbers just tells us that my sample mean will approach my expected value of the random variable. Or I could also write it as my sample mean will approach my population mean for n approaching infinity. And I'll be a little informal with what is approach or what is convergence mean, but I think you have the general intuitive sense that if I take a large enough sample here, that I'm going to end up getting the expected value of the population as a whole. And I think to a lot of us that's kind of intuitive, that if I do enough trials that over large samples, that the trials would kind of give me the numbers that I would expect given the expected value and the probability of all that. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
The law of large numbers just tells us that my sample mean will approach my expected value of the random variable. Or I could also write it as my sample mean will approach my population mean for n approaching infinity. And I'll be a little informal with what is approach or what is convergence mean, but I think you have the general intuitive sense that if I take a large enough sample here, that I'm going to end up getting the expected value of the population as a whole. And I think to a lot of us that's kind of intuitive, that if I do enough trials that over large samples, that the trials would kind of give me the numbers that I would expect given the expected value and the probability of all that. But I think it's often a little bit misunderstood in terms of why that happens. And before I go into that, let me give you a particular example. So the law of large numbers will just tell us that, let's say I have a random variable x is equal to the number of heads after 100 tosses of a fair coin. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
And I think to a lot of us that's kind of intuitive, that if I do enough trials that over large samples, that the trials would kind of give me the numbers that I would expect given the expected value and the probability of all that. But I think it's often a little bit misunderstood in terms of why that happens. And before I go into that, let me give you a particular example. So the law of large numbers will just tell us that, let's say I have a random variable x is equal to the number of heads after 100 tosses of a fair coin. Tosses or flips of a fair coin. The law of large numbers just tells us, well, first of all, we know what the expected value of this random variable is. It's the number of tosses, the number of trials, times the probability of success of any trial. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
So the law of large numbers will just tell us that, let's say I have a random variable x is equal to the number of heads after 100 tosses of a fair coin. Tosses or flips of a fair coin. The law of large numbers just tells us, well, first of all, we know what the expected value of this random variable is. It's the number of tosses, the number of trials, times the probability of success of any trial. So that's equal to 50. So the law of large numbers just says if I were to take a sample, or if I were to average the sample of a bunch of these trials, so I get, I don't know, my first time I run this trial I flip 100 coins or I have 100 coins in a shoebox and I shake the shoebox and I count the number of heads and I get 55. So that would be x1. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
It's the number of tosses, the number of trials, times the probability of success of any trial. So that's equal to 50. So the law of large numbers just says if I were to take a sample, or if I were to average the sample of a bunch of these trials, so I get, I don't know, my first time I run this trial I flip 100 coins or I have 100 coins in a shoebox and I shake the shoebox and I count the number of heads and I get 55. So that would be x1. Then I shake the box again and I get 65. Then I shake the box again and I get 45. And I do this n times and then I divide it by the number of times I did it. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
So that would be x1. Then I shake the box again and I get 65. Then I shake the box again and I get 45. And I do this n times and then I divide it by the number of times I did it. The law of large numbers just tells us that this average, the average of all of my observations, is going to converge to 50 as n approaches infinity. Or for n approaching infinity. And I want to talk a little bit about why this happens or intuitively why this is. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
And I do this n times and then I divide it by the number of times I did it. The law of large numbers just tells us that this average, the average of all of my observations, is going to converge to 50 as n approaches infinity. Or for n approaching infinity. And I want to talk a little bit about why this happens or intuitively why this is. A lot of people kind of feel that like, oh, this means that if after 100 trials, that if I'm above the average, that somehow the laws of probability are going to give me more heads or fewer heads to kind of make up the difference. And that's not quite what's going to happen. And that's often called the gambler's fallacy. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
And I want to talk a little bit about why this happens or intuitively why this is. A lot of people kind of feel that like, oh, this means that if after 100 trials, that if I'm above the average, that somehow the laws of probability are going to give me more heads or fewer heads to kind of make up the difference. And that's not quite what's going to happen. And that's often called the gambler's fallacy. And let me differentiate. And I'll use this example. So let's say, let me make a graph. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
And that's often called the gambler's fallacy. And let me differentiate. And I'll use this example. So let's say, let me make a graph. And I'll switch colors. So let's say that this is, so let me make, so on the, this is n, my x-axis is n. This is the number of trials I take. And my y-axis, let me make that the sample mean. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
So let's say, let me make a graph. And I'll switch colors. So let's say that this is, so let me make, so on the, this is n, my x-axis is n. This is the number of trials I take. And my y-axis, let me make that the sample mean. And we know what the expected value is, right? We know the expected value of this random variable is. It's 50. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
And my y-axis, let me make that the sample mean. And we know what the expected value is, right? We know the expected value of this random variable is. It's 50. Let me draw that here. This is 50. So just going to the example I did. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
It's 50. Let me draw that here. This is 50. So just going to the example I did. So when n is equal to, let me just plot it here. So my first trial, I got 55. And so that was my average, right? | Law of large numbers Probability and Statistics Khan Academy.mp3 |
So just going to the example I did. So when n is equal to, let me just plot it here. So my first trial, I got 55. And so that was my average, right? I only had one data point. Then after two trials, let's see, then I have 65. And so my average is going to be 65 plus 55 divided by 2. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
And so that was my average, right? I only had one data point. Then after two trials, let's see, then I have 65. And so my average is going to be 65 plus 55 divided by 2. Which is 60. So then my average went up a little bit. Then I had a 45, which will bring my average down a little bit. | Law of large numbers Probability and Statistics Khan Academy.mp3 |
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