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But either way, this is the model that I'm just, through eyeballing, this is my regression line, something that I'm trying to fit to these points. But clearly it can't go through, one line won't be able to go through all of these points. There is going to be, for each point, some difference, or not for all of them, but for many of them, some difference between the actual and what would have been predicted by the line. And that idea, the difference between the actual four-point and what would have been predicted given, say, the height, that is called a residual. Let me write that down. The residual for each of these data points. And so, for example, if I call this right here, if I call that 0.1, the residual for 0.1 is going to be, well, for our variable, for our height variable, 60 inches, the actual here is 100 pounds, and from that we would subtract what would be predicted. | Introduction to residuals and least squares regression.mp3 |
And that idea, the difference between the actual four-point and what would have been predicted given, say, the height, that is called a residual. Let me write that down. The residual for each of these data points. And so, for example, if I call this right here, if I call that 0.1, the residual for 0.1 is going to be, well, for our variable, for our height variable, 60 inches, the actual here is 100 pounds, and from that we would subtract what would be predicted. And so what would be predicted is right over here. I could just substitute 60 into this equation, so it would be m times 60 plus b. So I could write it as m, maybe let me write it this way, 60m plus b. | Introduction to residuals and least squares regression.mp3 |
And so, for example, if I call this right here, if I call that 0.1, the residual for 0.1 is going to be, well, for our variable, for our height variable, 60 inches, the actual here is 100 pounds, and from that we would subtract what would be predicted. And so what would be predicted is right over here. I could just substitute 60 into this equation, so it would be m times 60 plus b. So I could write it as m, maybe let me write it this way, 60m plus b. Once again, I would just take the 60 pounds and put it into my model here and say, well, what weight would that have predicted? And I could even, just for the sake of having a number here, I can look, I can, let me get my line tool out and try to get a straight line from that point. So from this point, let me get a straight line. | Introduction to residuals and least squares regression.mp3 |
So I could write it as m, maybe let me write it this way, 60m plus b. Once again, I would just take the 60 pounds and put it into my model here and say, well, what weight would that have predicted? And I could even, just for the sake of having a number here, I can look, I can, let me get my line tool out and try to get a straight line from that point. So from this point, let me get a straight line. So that doesn't look quite straight. Okay, a little bit, okay. So if I, it looks like it's about 150 pounds. | Introduction to residuals and least squares regression.mp3 |
So from this point, let me get a straight line. So that doesn't look quite straight. Okay, a little bit, okay. So if I, it looks like it's about 150 pounds. So my model would have predicted 150 pounds. So the residual here is going to be equal to negative 50. And so a negative residual is when your actual is below your predicted. | Introduction to residuals and least squares regression.mp3 |
So if I, it looks like it's about 150 pounds. So my model would have predicted 150 pounds. So the residual here is going to be equal to negative 50. And so a negative residual is when your actual is below your predicted. So this right over here, this is r one. It is a negative residual. If you had, if you tried to find, let's say this residual right over here for this point, this r two, this would be a positive residual because the actual is larger than what would have actually been predicted. | Introduction to residuals and least squares regression.mp3 |
And so a negative residual is when your actual is below your predicted. So this right over here, this is r one. It is a negative residual. If you had, if you tried to find, let's say this residual right over here for this point, this r two, this would be a positive residual because the actual is larger than what would have actually been predicted. And so a residual is good for seeing, well, how good does your line, does your regression, does your model fit a given data point or how does a given data point compare to that? But what you probably wanna do is think about some combination of all the residuals and try to minimize it. Now you might say, well, why don't I just add up all the residuals and try to minimize that? | Introduction to residuals and least squares regression.mp3 |
If you had, if you tried to find, let's say this residual right over here for this point, this r two, this would be a positive residual because the actual is larger than what would have actually been predicted. And so a residual is good for seeing, well, how good does your line, does your regression, does your model fit a given data point or how does a given data point compare to that? But what you probably wanna do is think about some combination of all the residuals and try to minimize it. Now you might say, well, why don't I just add up all the residuals and try to minimize that? But that gets tricky because some are positive and some are negative. And so a big negative residual, negative residual could counterbalance a big positive residual and it would look, they would add up to zero and then it would look like there's no residual. So you could just add up the absolute values. | Introduction to residuals and least squares regression.mp3 |
Now you might say, well, why don't I just add up all the residuals and try to minimize that? But that gets tricky because some are positive and some are negative. And so a big negative residual, negative residual could counterbalance a big positive residual and it would look, they would add up to zero and then it would look like there's no residual. So you could just add up the absolute values. So you could say, well, let me just take the sum of all of the residual, of the absolute value of all of the residuals. And then let me change m and b for my line to minimize this. And that would be a technique of trying to create a regression line. | Introduction to residuals and least squares regression.mp3 |
So you could just add up the absolute values. So you could say, well, let me just take the sum of all of the residual, of the absolute value of all of the residuals. And then let me change m and b for my line to minimize this. And that would be a technique of trying to create a regression line. But another way to do it, and this is actually the most typical way that you will see in statistics, is that people take the sum of the squares of the residuals, the sum of the squares. And when you square something, whether it's negative or positive, it's going to be a positive. So it takes care of that issue of negatives and positives canceling out with each other. | Introduction to residuals and least squares regression.mp3 |
And that would be a technique of trying to create a regression line. But another way to do it, and this is actually the most typical way that you will see in statistics, is that people take the sum of the squares of the residuals, the sum of the squares. And when you square something, whether it's negative or positive, it's going to be a positive. So it takes care of that issue of negatives and positives canceling out with each other. And when you square a number, things with large residuals are gonna become even larger, relatively speaking. If you square a large, if you think about it this way, let me put regular numbers, one, two, three, four, these are all one apart from each other. But if I were to square them, one, four, nine, 16, they get further and further apart. | Introduction to residuals and least squares regression.mp3 |
So it takes care of that issue of negatives and positives canceling out with each other. And when you square a number, things with large residuals are gonna become even larger, relatively speaking. If you square a large, if you think about it this way, let me put regular numbers, one, two, three, four, these are all one apart from each other. But if I were to square them, one, four, nine, 16, they get further and further apart. And so something, the larger the residual is, when you square it, when the sum of squares is gonna represent a bigger proportion of the sum. And so what we'll see in future videos is that there's a technique called least squares regression, least squares regression, where you can find an M and a B for a given set of data so it minimizes the sum of the squares of the residual. And that's valuable. | Introduction to residuals and least squares regression.mp3 |
But if I were to square them, one, four, nine, 16, they get further and further apart. And so something, the larger the residual is, when you square it, when the sum of squares is gonna represent a bigger proportion of the sum. And so what we'll see in future videos is that there's a technique called least squares regression, least squares regression, where you can find an M and a B for a given set of data so it minimizes the sum of the squares of the residual. And that's valuable. And the reason why this is used most is it really tries to take into account things that are significant outliers, things that sit from pretty far away from the model. Something like this is going to really, with a least squares regression, is going to try to be minimized, or it's going to be weighted a little bit heavier because when you square it, it becomes even a bigger factor in this. But this is just a conceptual introduction. | Introduction to residuals and least squares regression.mp3 |
Using the store's total selection, she documented the price of each movie title and how many years it has been since it was featured in movie theaters. She plotted the points below. So let's see what's going on below here. So let's see, it looks like there's two curves that she tries to fit, and I'm assuming we're gonna read about it in a second, but these blue points are the data points. So, for example, this data point right over here shows a movie that the title costs $6 and it has been released for almost two years, a little under two years. This data point right over here, this is a movie that has been released for, looks like, almost four years, looks like maybe three and three quarters years, and they're selling that, looks like, for a dollar or even a little bit less than a dollar. So those are her data points. | Comparing models to fit data Regression Probability and Statistics Khan Academy.mp3 |
So let's see, it looks like there's two curves that she tries to fit, and I'm assuming we're gonna read about it in a second, but these blue points are the data points. So, for example, this data point right over here shows a movie that the title costs $6 and it has been released for almost two years, a little under two years. This data point right over here, this is a movie that has been released for, looks like, almost four years, looks like maybe three and three quarters years, and they're selling that, looks like, for a dollar or even a little bit less than a dollar. So those are her data points. So once again, she documented the price of each movie title as a function of how long it's been, how many years it's been since it was featured in movie theaters. She is looking for a function that models her data. Since the trend of the data is decreasing and convex, and you see it here, it's decreasing, it's definitely decreasing, and convex, it's opening upwards. | Comparing models to fit data Regression Probability and Statistics Khan Academy.mp3 |
So those are her data points. So once again, she documented the price of each movie title as a function of how long it's been, how many years it's been since it was featured in movie theaters. She is looking for a function that models her data. Since the trend of the data is decreasing and convex, and you see it here, it's decreasing, it's definitely decreasing, and convex, it's opening upwards. If you imagine a curve, it looks like it's opening upwards a little bit like that. So, decreasing and convex. She found a decreasing convex exponential model and a decreasing convex quadratic model. | Comparing models to fit data Regression Probability and Statistics Khan Academy.mp3 |
Since the trend of the data is decreasing and convex, and you see it here, it's decreasing, it's definitely decreasing, and convex, it's opening upwards. If you imagine a curve, it looks like it's opening upwards a little bit like that. So, decreasing and convex. She found a decreasing convex exponential model and a decreasing convex quadratic model. So which of the following functions better fits the data? So, function A, this is an exponential, this is the one in green right over here, and function B, this one right over here is a quadratic, and you can see this one in purple. And so, which one of those better fits the data? | Comparing models to fit data Regression Probability and Statistics Khan Academy.mp3 |
She found a decreasing convex exponential model and a decreasing convex quadratic model. So which of the following functions better fits the data? So, function A, this is an exponential, this is the one in green right over here, and function B, this one right over here is a quadratic, and you can see this one in purple. And so, which one of those better fits the data? And so, if we look at what's going on here, the green function, the exponential one, most of the data points for any given duration, for how long the title's been out, it looks like it's consistently underestimating. That it's always, you know, it's the model's guess or what the model would say the price is, is always, at least for, except for only, essentially except for only one data point right over here, for all of these other data points, it's underestimating what the price would be. The purple model, or the purple function right over here, it is, it has more of a balance between overestimating right over here, and so it's overestimating by a little bit, and underestimating, and its underestimates are closer, and its overestimates are closer than this green model. | Comparing models to fit data Regression Probability and Statistics Khan Academy.mp3 |
And so, which one of those better fits the data? And so, if we look at what's going on here, the green function, the exponential one, most of the data points for any given duration, for how long the title's been out, it looks like it's consistently underestimating. That it's always, you know, it's the model's guess or what the model would say the price is, is always, at least for, except for only, essentially except for only one data point right over here, for all of these other data points, it's underestimating what the price would be. The purple model, or the purple function right over here, it is, it has more of a balance between overestimating right over here, and so it's overestimating by a little bit, and underestimating, and its underestimates are closer, and its overestimates are closer than this green model. So I would say that function B is definitely a better model. Use the function of best fits, so we're gonna say function B, to predict the price of a movie that was featured in theaters 5.5 years ago. Round your answer to the nearest cent. | Comparing models to fit data Regression Probability and Statistics Khan Academy.mp3 |
The purple model, or the purple function right over here, it is, it has more of a balance between overestimating right over here, and so it's overestimating by a little bit, and underestimating, and its underestimates are closer, and its overestimates are closer than this green model. So I would say that function B is definitely a better model. Use the function of best fits, so we're gonna say function B, to predict the price of a movie that was featured in theaters 5.5 years ago. Round your answer to the nearest cent. So 5.5 years ago, that's gonna be right over here, we're gonna go to function B, which is this purple one, so it's gonna be, you know, it's gonna be under a dollar, but we wanna get something to the nearest cent, so let's actually use the actual definition of the function. So this is price as a function of how long the movie has been released, where x is the, how long it's been released, and y is its price. So let's just, if x is 5.5, let's figure out what y is going to be. | Comparing models to fit data Regression Probability and Statistics Khan Academy.mp3 |
Round your answer to the nearest cent. So 5.5 years ago, that's gonna be right over here, we're gonna go to function B, which is this purple one, so it's gonna be, you know, it's gonna be under a dollar, but we wanna get something to the nearest cent, so let's actually use the actual definition of the function. So this is price as a function of how long the movie has been released, where x is the, how long it's been released, and y is its price. So let's just, if x is 5.5, let's figure out what y is going to be. So it's going to be, so y is going to be equal to 0.5 times x squared, so x is 5.5, 5.5 squared, alright? So then we have minus five times x again, so minus five times 5.5, and then we have plus 13, and what does that get us? That gets us 62.5 cents. | Comparing models to fit data Regression Probability and Statistics Khan Academy.mp3 |
So let's say you put 50 of these magenta marbles. So one, two, three, four, five, six, seven. I'm not going to draw all of them, but you get the general idea. There are going to be 50 magenta marbles. 15 magenta marbles. And there's also going to be 50 blue marbles. So 50 blue marbles. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
There are going to be 50 magenta marbles. 15 magenta marbles. And there's also going to be 50 blue marbles. So 50 blue marbles. And what you do is you have these 100 marbles in there, half of them magenta, half of them blue, and before picking a marble out, and you're going to be blindfolded when you pick a marble out, you shake the bag really good so you can mix them up a little bit. And so if you were to say, well, theoretically, what is the probability, if you stuck your hand in and you're not looking, what is the probability of picking a magenta? I have to feel the need to write the word magenta in magenta. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
So 50 blue marbles. And what you do is you have these 100 marbles in there, half of them magenta, half of them blue, and before picking a marble out, and you're going to be blindfolded when you pick a marble out, you shake the bag really good so you can mix them up a little bit. And so if you were to say, well, theoretically, what is the probability, if you stuck your hand in and you're not looking, what is the probability of picking a magenta? I have to feel the need to write the word magenta in magenta. What is the probability of picking a magenta marble? Well, theoretically, there's 100 equally likely possibilities. There's 100 marbles in the bag. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
I have to feel the need to write the word magenta in magenta. What is the probability of picking a magenta marble? Well, theoretically, there's 100 equally likely possibilities. There's 100 marbles in the bag. And 50 of them involve picking a magenta. So 50 out of 100, and this is the same thing as a 1 half probability. So you could say, well, theoretically, there is a 1 half probability. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
There's 100 marbles in the bag. And 50 of them involve picking a magenta. So 50 out of 100, and this is the same thing as a 1 half probability. So you could say, well, theoretically, there is a 1 half probability. I just did the math. If you say these are 100 equally likely possibilities, 50 of them are picking magenta. Now let's say you actually start doing the experiment. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
So you could say, well, theoretically, there is a 1 half probability. I just did the math. If you say these are 100 equally likely possibilities, 50 of them are picking magenta. Now let's say you actually start doing the experiment. So you literally take a bag with 50 magenta marbles, 50 blue marbles, and then you start picking the marbles, and then you see what marble color you picked, and you put it back in, and then you do it again. And so let's say that after, every time you put your hand in the bag and you take something out of the bag and you observe what it is, we're going to call that an experiment. So after 10 experiments, let's say that you have picked out 7 magenta and 3 blue. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
Now let's say you actually start doing the experiment. So you literally take a bag with 50 magenta marbles, 50 blue marbles, and then you start picking the marbles, and then you see what marble color you picked, and you put it back in, and then you do it again. And so let's say that after, every time you put your hand in the bag and you take something out of the bag and you observe what it is, we're going to call that an experiment. So after 10 experiments, let's say that you have picked out 7 magenta and 3 blue. So does this cause, is this strange that after 10, out of the first 10 experiments, you haven't picked out exactly half of them being magenta. You've picked out 7 magenta, and then the other 3 were blue. Well, no, this is definitely a reasonable thing. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
So after 10 experiments, let's say that you have picked out 7 magenta and 3 blue. So does this cause, is this strange that after 10, out of the first 10 experiments, you haven't picked out exactly half of them being magenta. You've picked out 7 magenta, and then the other 3 were blue. Well, no, this is definitely a reasonable thing. If the true probability of picking out a magenta is 1 half, it's definitely possible that you could still pick out 7 magenta. That just happened to be what your fingers touched. And this isn't a lot of experiments. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
Well, no, this is definitely a reasonable thing. If the true probability of picking out a magenta is 1 half, it's definitely possible that you could still pick out 7 magenta. That just happened to be what your fingers touched. And this isn't a lot of experiments. It's completely reasonable that out of 10, yeah, you could have a, and later on in statistics, we'll define these things in more detail, but there's enough variation in where you might pick that you're not going to always get, especially with only 10 experiments, you're not definitely going to get exactly 1 half. Instead of having 5 magenta, it's completely reasonable to have 7 magenta. So this really wouldn't cause me a lot of pause. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
And this isn't a lot of experiments. It's completely reasonable that out of 10, yeah, you could have a, and later on in statistics, we'll define these things in more detail, but there's enough variation in where you might pick that you're not going to always get, especially with only 10 experiments, you're not definitely going to get exactly 1 half. Instead of having 5 magenta, it's completely reasonable to have 7 magenta. So this really wouldn't cause me a lot of pause. I still wouldn't say, hey, I still wouldn't question what I did here when I calculated this theoretical probability. But let's say, and let's say you have a lot of time on your hands, and let's say after 10,000 trials here, after 10,000 experiments, and remember, an experiment, you're sticking your hand in the bag without looking, with your fingers kind of feeling around, picks out a marble, and you observe the marble, and you record what you found. And so let's say after 10,000 experiments, you get 7,000 magenta. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
So this really wouldn't cause me a lot of pause. I still wouldn't say, hey, I still wouldn't question what I did here when I calculated this theoretical probability. But let's say, and let's say you have a lot of time on your hands, and let's say after 10,000 trials here, after 10,000 experiments, and remember, an experiment, you're sticking your hand in the bag without looking, with your fingers kind of feeling around, picks out a marble, and you observe the marble, and you record what you found. And so let's say after 10,000 experiments, you get 7,000 magenta. Actually, let me do slightly different numbers. Actually, let me make it even more extreme. Let's say you get 8,000 magenta, and you have 2,000 blue. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
And so let's say after 10,000 experiments, you get 7,000 magenta. Actually, let me do slightly different numbers. Actually, let me make it even more extreme. Let's say you get 8,000 magenta, and you have 2,000 blue. Now this is interesting, because here what you're seeing experimentally seems to be very different, and now you have a large number of trials right over here, not just 10. 10 is completely reasonable, that hey, I got 7 magenta and 3 blue instead of 5 and 5, but now you've done 10,000. 10,000 is definitely, you would have expected if this was the true probability, you would have expected that half of these would have been magenta, only 5,000 magenta and 5,000 blue, but you had 8,000 magenta. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
Let's say you get 8,000 magenta, and you have 2,000 blue. Now this is interesting, because here what you're seeing experimentally seems to be very different, and now you have a large number of trials right over here, not just 10. 10 is completely reasonable, that hey, I got 7 magenta and 3 blue instead of 5 and 5, but now you've done 10,000. 10,000 is definitely, you would have expected if this was the true probability, you would have expected that half of these would have been magenta, only 5,000 magenta and 5,000 blue, but you had 8,000 magenta. Now this is within the realm of possibility if the true probability of picking a magenta is 1 half, but it's very unlikely that you would have gotten this result with this many experiments, this many trials, if the true probability is 1 half. Here your experimental probability is showing look, out of 10,000 trials, let me write that here, experimental probability, experimental probability here is you had 10,000 trials or 10,000 experiments, I guess you could say, and in 8,000 of them, you got a magenta marble, and so this is going to be 80% or 8 tenths, so 80%, so there seems to be a difference here. The reason why I would take this more seriously is that you had a lot of trials here. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
10,000 is definitely, you would have expected if this was the true probability, you would have expected that half of these would have been magenta, only 5,000 magenta and 5,000 blue, but you had 8,000 magenta. Now this is within the realm of possibility if the true probability of picking a magenta is 1 half, but it's very unlikely that you would have gotten this result with this many experiments, this many trials, if the true probability is 1 half. Here your experimental probability is showing look, out of 10,000 trials, let me write that here, experimental probability, experimental probability here is you had 10,000 trials or 10,000 experiments, I guess you could say, and in 8,000 of them, you got a magenta marble, and so this is going to be 80% or 8 tenths, so 80%, so there seems to be a difference here. The reason why I would take this more seriously is that you had a lot of trials here. You did this 10,000 times. If the true probability was 1 half, it's very low likelihood that you would have gotten this many magenta. So when you think about it, you're like, what's going on here? | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
The reason why I would take this more seriously is that you had a lot of trials here. You did this 10,000 times. If the true probability was 1 half, it's very low likelihood that you would have gotten this many magenta. So when you think about it, you're like, what's going on here? What are possible explanations for this? This I wouldn't have fretted about. After 10 experiments, not a big deal, but after 10,000, this would have caused me pause. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
So when you think about it, you're like, what's going on here? What are possible explanations for this? This I wouldn't have fretted about. After 10 experiments, not a big deal, but after 10,000, this would have caused me pause. Say, well, why would this happen? I mixed up the bag every time, and there are some different possibilities. Maybe the blue marbles are slightly heavier, and so when you shake the bag up enough, the blue marbles settle to the bottom, and you're more likely to pick a magenta marble. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
After 10 experiments, not a big deal, but after 10,000, this would have caused me pause. Say, well, why would this happen? I mixed up the bag every time, and there are some different possibilities. Maybe the blue marbles are slightly heavier, and so when you shake the bag up enough, the blue marbles settle to the bottom, and you're more likely to pick a magenta marble. Maybe the blue marbles have a slightly different texture to them, in which case maybe they slip out of your hands or they're less likely to be gripped on, and so you're more likely to pick a magenta. I don't know the explanation. I don't know what's going on in that bag, but if I thought theoretically that the probability should be 1 half, because half of the marbles are magenta, but I'm seeing through my experiments that 80% of what I'm picking out, especially if I did 10,000 of it, if I did this 10,000 times, well, this is going to cause me some pause. | Comparing theoretical to experimental probabilites 7th grade Khan Academy.mp3 |
What we're going to do in this video is start to compare distributions. So for example here we have two distributions that show the various temperatures different cities get during the month of January. 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. | Example Comparing distributions AP Statistics Khan Academy.mp3 |
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. In either of these cases, there are multiple measures in our statistical toolkit. | Example Comparing distributions AP Statistics Khan Academy.mp3 |
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. But sometimes you can just kind of gauge it by looking. | Example Comparing distributions AP Statistics Khan Academy.mp3 |
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. While for Minneapolis, it looks like our center is much closer to maybe negative two or negative three degrees Celsius. | Example Comparing distributions AP Statistics Khan Academy.mp3 |
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? 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. | Example Comparing distributions AP Statistics Khan Academy.mp3 |
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. Let's do another example, and we'll use a different representation for the data here. | Example Comparing distributions AP Statistics Khan Academy.mp3 |
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. 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. | Example Comparing distributions AP Statistics Khan Academy.mp3 |
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? Well, once again, I mean, and here, you can actually, it's a little bit easier to eyeball even what the median might be. | Example Comparing distributions AP Statistics Khan Academy.mp3 |
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? 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. | Example Comparing distributions AP Statistics Khan Academy.mp3 |
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. A distribution like this might have a higher range, but lower standard deviation than a distribution like this. | Example Comparing distributions AP Statistics Khan Academy.mp3 |
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. In fact, I can make that even better. | Example Comparing distributions AP Statistics Khan Academy.mp3 |
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. This is very high-level comparison. | Example Comparing distributions AP Statistics Khan Academy.mp3 |
Researchers used these results to test the null hypothesis is that the proportion is 0.5, the alternative hypothesis is that it's greater than 0.5, where P is the true proportion of adults that support the tax increase. They calculated a test statistic of z is approximately equal to 1.84 and a corresponding P value of approximately 0.033. Assuming the conditions for inference were met, which of these is an appropriate conclusion? And we have our four conclusions here. At any point, I encourage you to pause this video and see if you can answer it for yourself. But now we will do it together and just make sure we understand what's going on. Before we even cut to the chase and get to the answer. | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
And we have our four conclusions here. At any point, I encourage you to pause this video and see if you can answer it for yourself. But now we will do it together and just make sure we understand what's going on. Before we even cut to the chase and get to the answer. So what we do is we have this population and we are going to sample it. So n is equal to 200. From that sample, we calculate a sample proportion of adults that support the tax increase. | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
Before we even cut to the chase and get to the answer. So what we do is we have this population and we are going to sample it. So n is equal to 200. From that sample, we calculate a sample proportion of adults that support the tax increase. We see 113 out of 200 support it, which is going to be equal to, let's see, that is the same thing as 56.5%. So 56.5%. And so the key is is to figure out the P value. | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
From that sample, we calculate a sample proportion of adults that support the tax increase. We see 113 out of 200 support it, which is going to be equal to, let's see, that is the same thing as 56.5%. So 56.5%. And so the key is is to figure out the P value. What is the probability of getting a result this much above the assumed proportion or greater, at least this much above the assumed proportion if we assume that the null hypothesis is true? And if that probability, if that P value is below a preset threshold, if it's below our significance level, they haven't told it to us yet, it looks like they're gonna give some in the choices, well, then we would reject the null hypothesis, which would suggest the alternative. If the P value is not lower than this, then we will fail to reject the null hypothesis. | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
And so the key is is to figure out the P value. What is the probability of getting a result this much above the assumed proportion or greater, at least this much above the assumed proportion if we assume that the null hypothesis is true? And if that probability, if that P value is below a preset threshold, if it's below our significance level, they haven't told it to us yet, it looks like they're gonna give some in the choices, well, then we would reject the null hypothesis, which would suggest the alternative. If the P value is not lower than this, then we will fail to reject the null hypothesis. Now, to calculate that P value, to calculate that probability, what we figure out is, well, how many, in our sampling distribution, how many standard deviations above the mean of the sampling distribution, and the mean of the sampling distribution would be our assumed population proportion, how many standard deviations above that mean is this right over here? And that is what this test statistic is. And then we can use this to look at a z-table and say, all right, well, in a normal distribution, what percentage or what is the area under the normal curve that is further than 1.84 standard deviations or at least 1.84 or more standard deviations above the mean? | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
If the P value is not lower than this, then we will fail to reject the null hypothesis. Now, to calculate that P value, to calculate that probability, what we figure out is, well, how many, in our sampling distribution, how many standard deviations above the mean of the sampling distribution, and the mean of the sampling distribution would be our assumed population proportion, how many standard deviations above that mean is this right over here? And that is what this test statistic is. And then we can use this to look at a z-table and say, all right, well, in a normal distribution, what percentage or what is the area under the normal curve that is further than 1.84 standard deviations or at least 1.84 or more standard deviations above the mean? And they give that for us as well. So really, what we just need to do is compare this P value right over here to the significance level. If the P value is less than our significance level, then we reject, reject our null hypothesis, and that would suggest the alternative. | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
And then we can use this to look at a z-table and say, all right, well, in a normal distribution, what percentage or what is the area under the normal curve that is further than 1.84 standard deviations or at least 1.84 or more standard deviations above the mean? And they give that for us as well. So really, what we just need to do is compare this P value right over here to the significance level. If the P value is less than our significance level, then we reject, reject our null hypothesis, and that would suggest the alternative. If this is not true, then we would fail to reject the null hypothesis. So let's look at these choices. And if you didn't answer it the first time, I encourage you to pause the video again. | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
If the P value is less than our significance level, then we reject, reject our null hypothesis, and that would suggest the alternative. If this is not true, then we would fail to reject the null hypothesis. So let's look at these choices. And if you didn't answer it the first time, I encourage you to pause the video again. So at the alpha is equal to 0.01 significance level, they should conclude that more than 50% of adults support the tax increase. So if the alpha is 1 hundredth, the P value right over here is over 3 hundredths. It's roughly 3.3%. | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
And if you didn't answer it the first time, I encourage you to pause the video again. So at the alpha is equal to 0.01 significance level, they should conclude that more than 50% of adults support the tax increase. So if the alpha is 1 hundredth, the P value right over here is over 3 hundredths. It's roughly 3.3%. So this is a situation where our P value, our P value is greater than or equal to alpha. In fact, it's definitely greater than alpha here. And so here we would fail to reject, we would fail to reject our null hypothesis. | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
It's roughly 3.3%. So this is a situation where our P value, our P value is greater than or equal to alpha. In fact, it's definitely greater than alpha here. And so here we would fail to reject, we would fail to reject our null hypothesis. And so we wouldn't conclude that more than 50% of adults support the tax increase. Because remember, our null hypothesis is that 50% do, and we're failing to reject this. So that's not gonna be true. | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
And so here we would fail to reject, we would fail to reject our null hypothesis. And so we wouldn't conclude that more than 50% of adults support the tax increase. Because remember, our null hypothesis is that 50% do, and we're failing to reject this. So that's not gonna be true. At that same significance level, they should conclude that less than 50% of adults support the tax increase. No, we can't say that either. We just failed to reject this null hypothesis, that the true proportion is 50%. | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
So that's not gonna be true. At that same significance level, they should conclude that less than 50% of adults support the tax increase. No, we can't say that either. We just failed to reject this null hypothesis, that the true proportion is 50%. At the alpha is equal to, so at the alpha equals to 5 hundredth significance level, they should conclude that more than 50% of adults support the tax increase. Well yeah, in this situation, we have our P value, which is 0.033. It is indeed less than our significance level, in which case we reject, reject the null hypothesis. | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
We just failed to reject this null hypothesis, that the true proportion is 50%. At the alpha is equal to, so at the alpha equals to 5 hundredth significance level, they should conclude that more than 50% of adults support the tax increase. Well yeah, in this situation, we have our P value, which is 0.033. It is indeed less than our significance level, in which case we reject, reject the null hypothesis. And if we reject the null hypothesis, that would suggest the alternative, that the true proportion is greater than 50%. And so I would pick this choice right over here. And then choice D, at that same significance level, they should conclude that less than 50% of adults support the tax increase. | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
Thomas's favorite colors are blue and green. He has one blue shirt, one green shirt, one blue hat, one green scarf, one blue pair of pants, and one green pair of pants. Thomas selects one of these garments at random. Let A be the event that he selects a blue garment, and let B be the event that he chooses a shirt. Which of the following statements are true? And they all, let's see, before I even read them, they all deal with probability of event A, probability of event B, probability of B given A, probability of A given B, probability of A and B. So actually, let's just calculate these things ahead of time before we even look at these right over here. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
Let A be the event that he selects a blue garment, and let B be the event that he chooses a shirt. Which of the following statements are true? And they all, let's see, before I even read them, they all deal with probability of event A, probability of event B, probability of B given A, probability of A given B, probability of A and B. So actually, let's just calculate these things ahead of time before we even look at these right over here. So let's just think about probability of A. The probability of A, that's the probability that he picks a blue, that he selects a blue garment. So how many equally likely outcomes are there? | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
So actually, let's just calculate these things ahead of time before we even look at these right over here. So let's just think about probability of A. The probability of A, that's the probability that he picks a blue, that he selects a blue garment. So how many equally likely outcomes are there? Well, there's one, two, three, four, five, six equally likely outcomes. And how many involve selecting a blue garment? Well, there's one, two, three of the equally likely outcomes involve selecting a blue garment. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
So how many equally likely outcomes are there? Well, there's one, two, three, four, five, six equally likely outcomes. And how many involve selecting a blue garment? Well, there's one, two, three of the equally likely outcomes involve selecting a blue garment. So he has a three-sixths or one-half probability of selecting a blue garment. Now let's put the probability of B. What's probability of B? | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
Well, there's one, two, three of the equally likely outcomes involve selecting a blue garment. So he has a three-sixths or one-half probability of selecting a blue garment. Now let's put the probability of B. What's probability of B? And I'll do this in a neutral color since we're just saying that B is just the event that he chooses a shirt. So once again, there's six possible items, equally likely outcomes here. And which involve a shirt? | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
What's probability of B? And I'll do this in a neutral color since we're just saying that B is just the event that he chooses a shirt. So once again, there's six possible items, equally likely outcomes here. And which involve a shirt? Well, there's one, there is two. So it looks like two of the six involve picking a shirt. Or we could say the probability of B is equal to one-third. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
And which involve a shirt? Well, there's one, there is two. So it looks like two of the six involve picking a shirt. Or we could say the probability of B is equal to one-third. Now what's the probability of A given B? Let's write that down. What's the probability of, let me just do it in a new color. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
Or we could say the probability of B is equal to one-third. Now what's the probability of A given B? Let's write that down. What's the probability of, let me just do it in a new color. What's the probability, probability of A given B? I'll do those in the colors. A, given that B has happened. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
What's the probability of, let me just do it in a new color. What's the probability, probability of A given B? I'll do those in the colors. A, given that B has happened. So this is saying, what's the probability, what's the probability, probability of A given B is the probability that he picks a blue garment given that he has picked a shirt. So this, the given B, that restricts our outcomes to these two. And so the probability that he's picked a blue item, well that's one out of the two equally likely ones. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
A, given that B has happened. So this is saying, what's the probability, what's the probability, probability of A given B is the probability that he picks a blue garment given that he has picked a shirt. So this, the given B, that restricts our outcomes to these two. And so the probability that he's picked a blue item, well that's one out of the two equally likely ones. So there is a one-half probability that he picks a blue garment given that he's picked a shirt. And that's because there's one blue shirt and one green shirt. Now let's look at the probability of B given A. Probability of B given A. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
And so the probability that he's picked a blue item, well that's one out of the two equally likely ones. So there is a one-half probability that he picks a blue garment given that he's picked a shirt. And that's because there's one blue shirt and one green shirt. Now let's look at the probability of B given A. Probability of B given A. So assuming that we've picked a blue garment, so assuming we've picked a blue garment, so it's either that one, that one, or that one, what's the probability that we have also chosen a shirt? Well, there's one, two, three possibilities, equally likely possibilities where we have a blue garment, and only one of those involves a shirt. So probability of B given A is one-third. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
Now let's look at the probability of B given A. Probability of B given A. So assuming that we've picked a blue garment, so assuming we've picked a blue garment, so it's either that one, that one, or that one, what's the probability that we have also chosen a shirt? Well, there's one, two, three possibilities, equally likely possibilities where we have a blue garment, and only one of those involves a shirt. So probability of B given A is one-third. And then finally, we could think about probability of A and B. So the probability of A and B. So this is the probability of picking a blue shirt. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
So probability of B given A is one-third. And then finally, we could think about probability of A and B. So the probability of A and B. So this is the probability of picking a blue shirt. So only one out of the six equally likely outcomes is a blue shirt. So this one right over here is going to be one over six. So now that we've figured out all of that, let's see if we can answer these questions. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
So this is the probability of picking a blue shirt. So only one out of the six equally likely outcomes is a blue shirt. So this one right over here is going to be one over six. So now that we've figured out all of that, let's see if we can answer these questions. The probability of A given B equals the probability of A, and that does work out. Probability of A given B is one-half, and that's the same thing as the probability of A. The probability that Thomas selects a blue garment given that he has chosen a shirt is equal to the probability that Thomas selects a blue garment. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
So now that we've figured out all of that, let's see if we can answer these questions. The probability of A given B equals the probability of A, and that does work out. Probability of A given B is one-half, and that's the same thing as the probability of A. The probability that Thomas selects a blue garment given that he has chosen a shirt is equal to the probability that Thomas selects a blue garment. Yep, that's exactly. So I guess the words are just rephrasing what they wrote here in, I guess, more mathy notation. So this is absolutely true. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
The probability that Thomas selects a blue garment given that he has chosen a shirt is equal to the probability that Thomas selects a blue garment. Yep, that's exactly. So I guess the words are just rephrasing what they wrote here in, I guess, more mathy notation. So this is absolutely true. The probability of B given A is equal to the probability of B. Yep, probability of B given A is one-third, and the probability of B is one-third. The probability that Thomas selects a shirt given that he has chosen a blue garment is equal to the probability that Thomas selects a shirt. Yep, that's right. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
So this is absolutely true. The probability of B given A is equal to the probability of B. Yep, probability of B given A is one-third, and the probability of B is one-third. The probability that Thomas selects a shirt given that he has chosen a blue garment is equal to the probability that Thomas selects a shirt. Yep, that's right. Events A and B are independent events. So two events are independent if the, well, let me write it more in math notation. These are independent if the probability of A given B is equal to the probability of A. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
Yep, that's right. Events A and B are independent events. So two events are independent if the, well, let me write it more in math notation. These are independent if the probability of A given B is equal to the probability of A. Then we could say A and B are independent because the probability of A, if this is true, then this means the probability of A given B actually isn't dependent on whether B happened or not. It's the same thing as the probability of A. This would lead to these events being independent. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
These are independent if the probability of A given B is equal to the probability of A. Then we could say A and B are independent because the probability of A, if this is true, then this means the probability of A given B actually isn't dependent on whether B happened or not. It's the same thing as the probability of A. This would lead to these events being independent. Also, if you had probability of B given A is equal to the probability of B, same argument. That would mean they're independent. Or if we said that the probability of A and B is equal to the probability of A times the probability of B, then these would also mean they're independent. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
This would lead to these events being independent. Also, if you had probability of B given A is equal to the probability of B, same argument. That would mean they're independent. Or if we said that the probability of A and B is equal to the probability of A times the probability of B, then these would also mean they're independent. We know that this one's true. The probability of A and B is one-sixth, and the probability of A times the probability of B is one-half times one-third, which is one-sixth. So all of these are clearly true, so we can say that A and B are independent. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
Or if we said that the probability of A and B is equal to the probability of A times the probability of B, then these would also mean they're independent. We know that this one's true. The probability of A and B is one-sixth, and the probability of A times the probability of B is one-half times one-third, which is one-sixth. So all of these are clearly true, so we can say that A and B are independent. The probability of A is independent of whether B has happened or not, and the probability of B happening is independent of whether A has happened or not. The outcome of events A and B are dependent on each other. No, that's the opposite of saying that they're independent, so we're not going to, we can cross that out. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
So all of these are clearly true, so we can say that A and B are independent. The probability of A is independent of whether B has happened or not, and the probability of B happening is independent of whether A has happened or not. The outcome of events A and B are dependent on each other. No, that's the opposite of saying that they're independent, so we're not going to, we can cross that out. Probability of A and B is equal to probability of A times probability of B. We already said that to be true. One-sixth is one-half times one-third. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
No, that's the opposite of saying that they're independent, so we're not going to, we can cross that out. Probability of A and B is equal to probability of A times probability of B. We already said that to be true. One-sixth is one-half times one-third. The probability that Tom selects a blue garment that is a shirt is equal to the probability that Tom selects a blue garment multiplied by the probability that he selects a shirt. Yep, that's absolutely right. So actually, a lot of these statements are true. | Analyzing event probability for independence Probability and Statistics Khan Academy.mp3 |
The scores of the first four rounds and the lowest round are shown in the following dot plot. And we see it right over here, the lowest round she scores an 80. She also scores a 90 once, a 92 once, a 94 once, and a 96 once. It was discovered that Anna broke some rules when she scored 80. So that score, so I guess cheating didn't help her. So that score will be removed from the data set. So they removed that 80 right over there, or just left with the scores from the other four rounds. | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
It was discovered that Anna broke some rules when she scored 80. So that score, so I guess cheating didn't help her. So that score will be removed from the data set. So they removed that 80 right over there, or just left with the scores from the other four rounds. How will the removal of the lowest round affect the mean and the median? So let's actually think about the median first. So the median is the middle number. | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
So they removed that 80 right over there, or just left with the scores from the other four rounds. How will the removal of the lowest round affect the mean and the median? So let's actually think about the median first. So the median is the middle number. So over here, when you had five data points, the middle data point is gonna be the one that has two to the left and two to the right. So the median up here is going to be 92. The median up there is 92. | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
So the median is the middle number. So over here, when you had five data points, the middle data point is gonna be the one that has two to the left and two to the right. So the median up here is going to be 92. The median up there is 92. And what's the median once you remove this? Now you only have four data points. When you're trying to find the median of an even number of numbers, you look at the middle two numbers. | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
The median up there is 92. And what's the median once you remove this? Now you only have four data points. When you're trying to find the median of an even number of numbers, you look at the middle two numbers. So that's a 92 and a 94. And then you take the average of them. You go halfway between them to figure out the median. | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
When you're trying to find the median of an even number of numbers, you look at the middle two numbers. So that's a 92 and a 94. And then you take the average of them. You go halfway between them to figure out the median. So the median here is going to be, let me do that a little bit clearer. The median over here is gonna be halfway between 92 and 94, which is 93. So the median, the median is 93. | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
You go halfway between them to figure out the median. So the median here is going to be, let me do that a little bit clearer. The median over here is gonna be halfway between 92 and 94, which is 93. So the median, the median is 93. Median is 93. So removing the lowest data point, in this case, increased the median. So the median, let me write it down here. | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
So the median, the median is 93. Median is 93. So removing the lowest data point, in this case, increased the median. So the median, let me write it down here. So the median increased by a little bit. The median increases. Now what's going to happen to the mean? | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
So the median, let me write it down here. So the median increased by a little bit. The median increases. Now what's going to happen to the mean? What's going to happen to the mean? Well, one way to think about it, without even doing any calculations, is if you remove a number that is lower than the mean, lower than the existing mean, and I haven't calculated what the existing mean is, but if you remove that, the mean is going to go up. The mean is going to go up. | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
Now what's going to happen to the mean? What's going to happen to the mean? Well, one way to think about it, without even doing any calculations, is if you remove a number that is lower than the mean, lower than the existing mean, and I haven't calculated what the existing mean is, but if you remove that, the mean is going to go up. The mean is going to go up. So hopefully that gives you some intuition. If you removed a number that's larger than the mean, your mean is going to go down, because you don't have that large number anymore. If you remove a number that's lower than the mean, well, you take that out, you don't have that small number bringing the average down, and so the mean will go up. | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
The mean is going to go up. So hopefully that gives you some intuition. If you removed a number that's larger than the mean, your mean is going to go down, because you don't have that large number anymore. If you remove a number that's lower than the mean, well, you take that out, you don't have that small number bringing the average down, and so the mean will go up. But let's verify it mathematically. So let's calculate the mean over here. So we're going to add 80 plus 90 plus 92 plus 94 plus 96. | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
If you remove a number that's lower than the mean, well, you take that out, you don't have that small number bringing the average down, and so the mean will go up. But let's verify it mathematically. So let's calculate the mean over here. So we're going to add 80 plus 90 plus 92 plus 94 plus 96. Those are our data points. And that gets us, two plus four is six, plus six is 12. And then we have one plus eight is nine. | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
So we're going to add 80 plus 90 plus 92 plus 94 plus 96. Those are our data points. And that gets us, two plus four is six, plus six is 12. And then we have one plus eight is nine. And we essentially, this is, so these are nine, and you have another nine, another nine, another nine, another nine. You essentially have, this is five nines right over here. So this is going to be 450, 452. | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
And then we have one plus eight is nine. And we essentially, this is, so these are nine, and you have another nine, another nine, another nine, another nine. You essentially have, this is five nines right over here. So this is going to be 450, 452. So that's the sum of the scores of these five rounds. And then you divide it by the number of rounds you have. So it'd be 452 divided by five. | Impact on median and mean when removing lowest value example 6th grade Khan Academy.mp3 |
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