Sentence stringlengths 102 4.09k | video_title stringlengths 27 104 |
|---|---|
I have the middle of the second half, 13. To calculate the interquartile range, I just have to find the difference between these two things. So the interquartile range for this first example is going to be 13 minus five. The middle of the second half minus the middle of the first half, which is going to be equal to eight. Let's do some more of these. This is strangely fun. Find the interquartile range of the data in the dot plot below. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
The middle of the second half minus the middle of the first half, which is going to be equal to eight. Let's do some more of these. This is strangely fun. Find the interquartile range of the data in the dot plot below. Songs on each album in Shane's collection. And so let's see what's going on here. And like always, I encourage you to take a shot at it. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
Find the interquartile range of the data in the dot plot below. Songs on each album in Shane's collection. And so let's see what's going on here. And like always, I encourage you to take a shot at it. So this is just representing the data in a different way, but we could write this again as an ordered list. So let's do that. We have one song, or we have one album with seven songs, I guess you could say. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
And like always, I encourage you to take a shot at it. So this is just representing the data in a different way, but we could write this again as an ordered list. So let's do that. We have one song, or we have one album with seven songs, I guess you could say. So we have a seven. We have two albums with nine songs. So we have two nines. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
We have one song, or we have one album with seven songs, I guess you could say. So we have a seven. We have two albums with nine songs. So we have two nines. Let me write those. We have two nines. Then we have three tens. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
So we have two nines. Let me write those. We have two nines. Then we have three tens. Let's cross those out. So 10, 10, 10. Then we have an 11. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
Then we have three tens. Let's cross those out. So 10, 10, 10. Then we have an 11. We have an 11. We have two 12s. Two 12s. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
Then we have an 11. We have an 11. We have two 12s. Two 12s. And then finally we have, so I used those already, and then we have an album with 14 songs. 14. So all I did here is I wrote this data like this. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
Two 12s. And then finally we have, so I used those already, and then we have an album with 14 songs. 14. So all I did here is I wrote this data like this. So we could see, okay, this album has seven songs, this album has nine, this album has nine. And the way I wrote it, it's already in order. So I can immediately start calculating the median. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
So all I did here is I wrote this data like this. So we could see, okay, this album has seven songs, this album has nine, this album has nine. And the way I wrote it, it's already in order. So I can immediately start calculating the median. Let's see, I have one, two, three, four, five, six, seven, eight, nine, 10 numbers. I have an even number of numbers. So to calculate the median, I'm gonna have to look at the middle two numbers. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
So I can immediately start calculating the median. Let's see, I have one, two, three, four, five, six, seven, eight, nine, 10 numbers. I have an even number of numbers. So to calculate the median, I'm gonna have to look at the middle two numbers. So the middle two numbers look like it's these two tens here because I have four to the left of them and then four to the right of them. And so since I'm calculating the median using two numbers, it's going to be halfway between them. It's going to be the average of these two numbers. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
So to calculate the median, I'm gonna have to look at the middle two numbers. So the middle two numbers look like it's these two tens here because I have four to the left of them and then four to the right of them. And so since I'm calculating the median using two numbers, it's going to be halfway between them. It's going to be the average of these two numbers. Well, the average of 10 and 10 is just going to be 10. So the median is going to be 10. Median is going to be 10. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
It's going to be the average of these two numbers. Well, the average of 10 and 10 is just going to be 10. So the median is going to be 10. Median is going to be 10. And in a case like this where I calculated the median using the middle two numbers, I can now include this left 10 in the first half and I can include this right 10 in the second half. So let's do that. So the first half is going to be those five numbers and then the second half is going to be these five numbers. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
Median is going to be 10. And in a case like this where I calculated the median using the middle two numbers, I can now include this left 10 in the first half and I can include this right 10 in the second half. So let's do that. So the first half is going to be those five numbers and then the second half is going to be these five numbers. And it makes sense because we're literally just looking at first half, it's going to be five numbers, second half is going to be five numbers. If I had a true middle number like the previous example, then we ignore that when we look at the first and second half or at least that's the way that we're doing it in these examples. But what's the median of this first half if we look at these five numbers? | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
So the first half is going to be those five numbers and then the second half is going to be these five numbers. And it makes sense because we're literally just looking at first half, it's going to be five numbers, second half is going to be five numbers. If I had a true middle number like the previous example, then we ignore that when we look at the first and second half or at least that's the way that we're doing it in these examples. But what's the median of this first half if we look at these five numbers? Well, if you have five numbers, if you have an odd number of numbers, you're going to have one middle number and it's going to be the one that has two on either sides. This has two to the left and it has two to the right. So the median of the first half, the middle of the first half is nine right over here. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
But what's the median of this first half if we look at these five numbers? Well, if you have five numbers, if you have an odd number of numbers, you're going to have one middle number and it's going to be the one that has two on either sides. This has two to the left and it has two to the right. So the median of the first half, the middle of the first half is nine right over here. And the middle of the second half, I have one, two, three, four, five numbers and this 12 is right in the middle. You have two to the left and two to the right. So the median of the second half is 12. | How to calculate interquartile range IQR Data and statistics 6th grade Khan Academy.mp3 |
What I want to do in this video is build up some tools in our toolkit for dealing with sums and differences of random variables. So let's say that we have two random variables, X and Y, and they are completely independent. They are independent random variables. And I'm just going to go over a little bit of notation here. If we wanted to know the expected, or if we talked about the expected value of this random variable X, that is the same thing as the mean value of this random variable X. If we talk about the expected value of Y, that is the same thing as the mean of Y. If we talk about the variance of the random variable X, that is the same thing as the expected value of the squared distances between our random variable X and its mean. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
And I'm just going to go over a little bit of notation here. If we wanted to know the expected, or if we talked about the expected value of this random variable X, that is the same thing as the mean value of this random variable X. If we talk about the expected value of Y, that is the same thing as the mean of Y. If we talk about the variance of the random variable X, that is the same thing as the expected value of the squared distances between our random variable X and its mean. And that right there, squared. So the expected value of these squared differences, and that is, you can also use the notation, sigma squared for the random variable X. This is just a review of things we already know, but I just want to reintroduce it, because I'll use this to build up some of our tools. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
If we talk about the variance of the random variable X, that is the same thing as the expected value of the squared distances between our random variable X and its mean. And that right there, squared. So the expected value of these squared differences, and that is, you can also use the notation, sigma squared for the random variable X. This is just a review of things we already know, but I just want to reintroduce it, because I'll use this to build up some of our tools. So you do the same thing with this with random variable Y. The variance of random variable Y is the expected value of the squared difference between our random variable Y and the mean of Y, or the expected value of Y, squared. And that's the same thing as sigma squared of Y. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
This is just a review of things we already know, but I just want to reintroduce it, because I'll use this to build up some of our tools. So you do the same thing with this with random variable Y. The variance of random variable Y is the expected value of the squared difference between our random variable Y and the mean of Y, or the expected value of Y, squared. And that's the same thing as sigma squared of Y. There's a variance of Y. Now, you may or may not already know these properties of expected values and variances, but I will reintroduce them to you, and I won't go into some rigorous proof. Actually, I think they're fairly easy to digest. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
And that's the same thing as sigma squared of Y. There's a variance of Y. Now, you may or may not already know these properties of expected values and variances, but I will reintroduce them to you, and I won't go into some rigorous proof. Actually, I think they're fairly easy to digest. So one is that if I have some third random variable, let's say I have some third random variable that is defined as being the random variable X plus the random variable Y. Let me stay with my colors just so everything becomes clear. The random variable X plus the random variable Y. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
Actually, I think they're fairly easy to digest. So one is that if I have some third random variable, let's say I have some third random variable that is defined as being the random variable X plus the random variable Y. Let me stay with my colors just so everything becomes clear. The random variable X plus the random variable Y. What is the expected value of Z going to be? The expected value of Z is going to be equal to the expected value of X plus Y. And this is a property of expected values. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
The random variable X plus the random variable Y. What is the expected value of Z going to be? The expected value of Z is going to be equal to the expected value of X plus Y. And this is a property of expected values. I'm not going to prove it rigorously right here, but it's the expected value of X plus the expected value of Y. Or another way to think about this is that the mean of Z is going to be the mean of X plus the mean of Y. Or another way to view it is if I wanted to take, let's say I have some other random variable. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
And this is a property of expected values. I'm not going to prove it rigorously right here, but it's the expected value of X plus the expected value of Y. Or another way to think about this is that the mean of Z is going to be the mean of X plus the mean of Y. Or another way to view it is if I wanted to take, let's say I have some other random variable. I'm running out of letters here. Let's say I have the random variable A, and I define random variable A to be X minus Y. So what's its expected value going to be? | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
Or another way to view it is if I wanted to take, let's say I have some other random variable. I'm running out of letters here. Let's say I have the random variable A, and I define random variable A to be X minus Y. So what's its expected value going to be? The expected value of A is going to be equal to the expected value of X minus Y, which is equal to, you can either view it as the expected value of X plus the expected value of negative Y, or the expected value of X minus the expected value of Y, which is the same thing as the mean of X minus the mean of Y. So this is what the mean of our random variable A would be equal to. All of this is review, and I'm going to use this when we start talking about distributions that are sums and differences of other distributions. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
So what's its expected value going to be? The expected value of A is going to be equal to the expected value of X minus Y, which is equal to, you can either view it as the expected value of X plus the expected value of negative Y, or the expected value of X minus the expected value of Y, which is the same thing as the mean of X minus the mean of Y. So this is what the mean of our random variable A would be equal to. All of this is review, and I'm going to use this when we start talking about distributions that are sums and differences of other distributions. Now let's think about what the variance of random variable Z is and what the variance of random variable A is. So the variance of Z, and just to kind of always focus back on the intuition, it makes sense. If X is completely independent of Y, and if I have some random variable that is the sum of the two, then the expected value of that variable, of that new variable, is going to be the sum of the expected values of the other two, because they are unrelated. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
All of this is review, and I'm going to use this when we start talking about distributions that are sums and differences of other distributions. Now let's think about what the variance of random variable Z is and what the variance of random variable A is. So the variance of Z, and just to kind of always focus back on the intuition, it makes sense. If X is completely independent of Y, and if I have some random variable that is the sum of the two, then the expected value of that variable, of that new variable, is going to be the sum of the expected values of the other two, because they are unrelated. If I think, if my expected value here is 5 and my expected value here is 7, it's completely reasonable that my expected value here is 12, assuming that they are completely independent. Now, if we have a situation, so what is the variance of my random variable Z? And once again, I'm not going to do a rigorous proof here, this is really just a property of variances, but I'm going to use this to establish what the variance of our random variable A is. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
If X is completely independent of Y, and if I have some random variable that is the sum of the two, then the expected value of that variable, of that new variable, is going to be the sum of the expected values of the other two, because they are unrelated. If I think, if my expected value here is 5 and my expected value here is 7, it's completely reasonable that my expected value here is 12, assuming that they are completely independent. Now, if we have a situation, so what is the variance of my random variable Z? And once again, I'm not going to do a rigorous proof here, this is really just a property of variances, but I'm going to use this to establish what the variance of our random variable A is. So if this squared distance on average is some variance, and this one is completely independent, and its squared distance on average is some distance, then the variance of their sum is actually going to be the sum of their variances. So this is going to be equal to the variance of random variable X plus the variance of random variable Y. So another way of thinking about it is that the variance of Z, which is the same thing as the variance of X plus Y, is equal to the variance of X plus the variance of random variable Y. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
And once again, I'm not going to do a rigorous proof here, this is really just a property of variances, but I'm going to use this to establish what the variance of our random variable A is. So if this squared distance on average is some variance, and this one is completely independent, and its squared distance on average is some distance, then the variance of their sum is actually going to be the sum of their variances. So this is going to be equal to the variance of random variable X plus the variance of random variable Y. So another way of thinking about it is that the variance of Z, which is the same thing as the variance of X plus Y, is equal to the variance of X plus the variance of random variable Y. And hopefully that makes some sense, I'm not proving it too rigorously, and you'll see this in a lot of statistics books. Now, what I want to show you is that the variance of random variable A is actually this exact same thing. And that's the interesting thing, because you might say, hey, why wouldn't it be the difference? | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
So another way of thinking about it is that the variance of Z, which is the same thing as the variance of X plus Y, is equal to the variance of X plus the variance of random variable Y. And hopefully that makes some sense, I'm not proving it too rigorously, and you'll see this in a lot of statistics books. Now, what I want to show you is that the variance of random variable A is actually this exact same thing. And that's the interesting thing, because you might say, hey, why wouldn't it be the difference? We had the differences over here, so let's experiment with this a little bit. The variance of random variable A is the same thing as the variance of X minus Y, which is equal to the variance of X plus negative Y. These are equivalent statements. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
And that's the interesting thing, because you might say, hey, why wouldn't it be the difference? We had the differences over here, so let's experiment with this a little bit. The variance of random variable A is the same thing as the variance of X minus Y, which is equal to the variance of X plus negative Y. These are equivalent statements. So you could view this as being equal to, just using this over here, the sum of these two variances. So it's going to be equal to the sum of the variance of X plus the variance of negative Y. And what I need to show you is that the variance of negative Y, the negative of that random variable, is going to be the same thing as the variance of Y. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
These are equivalent statements. So you could view this as being equal to, just using this over here, the sum of these two variances. So it's going to be equal to the sum of the variance of X plus the variance of negative Y. And what I need to show you is that the variance of negative Y, the negative of that random variable, is going to be the same thing as the variance of Y. So what is the variance of negative Y? The variance of negative Y is the same thing as the variance of negative Y, which is equal to the expected value of the distance between negative Y and the expected value of negative Y squared. That's all the variance actually is. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
And what I need to show you is that the variance of negative Y, the negative of that random variable, is going to be the same thing as the variance of Y. So what is the variance of negative Y? The variance of negative Y is the same thing as the variance of negative Y, which is equal to the expected value of the distance between negative Y and the expected value of negative Y squared. That's all the variance actually is. Now, what is the expected value of negative Y right over here? Actually, even better, let me factor out a negative 1. So what's in the parentheses right here, this is the exact same thing as negative 1 squared times Y plus the expected value of negative Y. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
That's all the variance actually is. Now, what is the expected value of negative Y right over here? Actually, even better, let me factor out a negative 1. So what's in the parentheses right here, this is the exact same thing as negative 1 squared times Y plus the expected value of negative Y. So that's the exact same thing in the parentheses squared. Everything in magenta is everything in magenta here, and it is the expected value of that thing. Now, what is the expected value of negative Y? | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
So what's in the parentheses right here, this is the exact same thing as negative 1 squared times Y plus the expected value of negative Y. So that's the exact same thing in the parentheses squared. Everything in magenta is everything in magenta here, and it is the expected value of that thing. Now, what is the expected value of negative Y? The expected value of negative Y, I'll do it over here, the expected value of the negative of a random variable is just the negative of the expected value of that random variable. So if you look at this, we can rewrite this. I'll give myself a little bit more space. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
Now, what is the expected value of negative Y? The expected value of negative Y, I'll do it over here, the expected value of the negative of a random variable is just the negative of the expected value of that random variable. So if you look at this, we can rewrite this. I'll give myself a little bit more space. We can rewrite this as the expected value, the variance of negative Y is the expected value. This is just 1. Negative 1 squared is just 1. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
I'll give myself a little bit more space. We can rewrite this as the expected value, the variance of negative Y is the expected value. This is just 1. Negative 1 squared is just 1. And over here you have Y. And instead of just writing plus the expected value of negative Y, that's the same thing as minus the expected value of Y. So you have that and then all of that squared. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
Negative 1 squared is just 1. And over here you have Y. And instead of just writing plus the expected value of negative Y, that's the same thing as minus the expected value of Y. So you have that and then all of that squared. Now notice, this is the exact same thing by definition as the variance of Y. So we just showed you just now, so this is the variance of Y. So we just showed you that the variance of the difference of two independent random variables is equal to the sum of the variances. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
So you have that and then all of that squared. Now notice, this is the exact same thing by definition as the variance of Y. So we just showed you just now, so this is the variance of Y. So we just showed you that the variance of the difference of two independent random variables is equal to the sum of the variances. You could definitely believe this. It's equal to the sum of the variance of the first one plus the variance of the negative of the second one. And we just showed that that variance is the same thing as the variance of the positive version of that variable, which makes sense. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
So we just showed you that the variance of the difference of two independent random variables is equal to the sum of the variances. You could definitely believe this. It's equal to the sum of the variance of the first one plus the variance of the negative of the second one. And we just showed that that variance is the same thing as the variance of the positive version of that variable, which makes sense. Your distance from the mean is going to be, it doesn't matter whether you're taking the positive or the negative of the variable. You just care about absolute distance. So it makes complete sense that that quantity and that quantity is going to be the same thing. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
And we just showed that that variance is the same thing as the variance of the positive version of that variable, which makes sense. Your distance from the mean is going to be, it doesn't matter whether you're taking the positive or the negative of the variable. You just care about absolute distance. So it makes complete sense that that quantity and that quantity is going to be the same thing. So the whole reason why I went through this exercise, the important takeaways here, is that the mean of differences right over here, so I could rewrite it as the mean of the differences of the random variable is the same thing as the differences of their means. And then the other important takeaway, and I'm going to build on this in the next few videos, is that the variance of the difference, if I define a new random variable as the difference of two other random variables, the variance of that random variable is actually the sum of the variances of the two random variables. So these are the two important takeaways that we'll use to build on in future videos. | Variance of differences of random variables Probability and Statistics Khan Academy.mp3 |
In other videos, we've done linear regressions by hand, but we mentioned that most regressions are actually done using some type of computer or calculator. And so what we're going to do in this video is look at an example of the output that we might see from a computer and to not be intimidated by it and to see how it gives us the equation for the regression line and some of the other data it gives us. So here it tells us Cheryl Dixon is interested to see if students who consume more caffeine tend to study more as well. She randomly selects 20 students at her school and records their caffeine intake in milligrams and the number of hours spent studying. A scatterplot of the data showed a linear relationship. This is a computer output from a least squares regression analysis on the data. So we have these things called the predictors, coefficient, and then we have these other things, standard error of coefficient, T and P, and then all of these things down here. | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
She randomly selects 20 students at her school and records their caffeine intake in milligrams and the number of hours spent studying. A scatterplot of the data showed a linear relationship. This is a computer output from a least squares regression analysis on the data. So we have these things called the predictors, coefficient, and then we have these other things, standard error of coefficient, T and P, and then all of these things down here. How do we make sense of this in order to come up with an equation for our linear regression? So let's just get straight on our variables. Let's just say that we say that Y is the thing that we're trying to predict. | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
So we have these things called the predictors, coefficient, and then we have these other things, standard error of coefficient, T and P, and then all of these things down here. How do we make sense of this in order to come up with an equation for our linear regression? So let's just get straight on our variables. Let's just say that we say that Y is the thing that we're trying to predict. So this is the hours spent studying, hours studying, and then let's say X is what we think explains the hours studying or is one of the things that explains the hours studying, and this is the amount of caffeine ingested. So this is caffeine consumed in milligrams. And so our regression line would have the form Y hat. | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
Let's just say that we say that Y is the thing that we're trying to predict. So this is the hours spent studying, hours studying, and then let's say X is what we think explains the hours studying or is one of the things that explains the hours studying, and this is the amount of caffeine ingested. So this is caffeine consumed in milligrams. And so our regression line would have the form Y hat. This tells us that this is a linear regression. It's trying to estimate the actual Y values for given Xs. It's going to be equal to MX plus B. | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
And so our regression line would have the form Y hat. This tells us that this is a linear regression. It's trying to estimate the actual Y values for given Xs. It's going to be equal to MX plus B. Now how do we figure out what M and B are based on this computer output? So when you look at this table here, this first column says predictor, and it says constant, and it has caffeine. And so all this is saying is when you're trying to predict the number of hours studying, when you're trying to predict Y, there's essentially two inputs there. | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
It's going to be equal to MX plus B. Now how do we figure out what M and B are based on this computer output? So when you look at this table here, this first column says predictor, and it says constant, and it has caffeine. And so all this is saying is when you're trying to predict the number of hours studying, when you're trying to predict Y, there's essentially two inputs there. There is the constant value, and there is your variable, in this case caffeine, that you're using to predict the amount that you study. And so this tells you the coefficients on each. The coefficient on a constant is the constant. | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
And so all this is saying is when you're trying to predict the number of hours studying, when you're trying to predict Y, there's essentially two inputs there. There is the constant value, and there is your variable, in this case caffeine, that you're using to predict the amount that you study. And so this tells you the coefficients on each. The coefficient on a constant is the constant. You could view this as the coefficient on the X to the zeroth term. And so the coefficient on the constant, that is the constant, 2.544. And then the coefficient on the caffeine, well, we just said that X is the caffeine consumed, so this is that coefficient, 0.164. | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
The coefficient on a constant is the constant. You could view this as the coefficient on the X to the zeroth term. And so the coefficient on the constant, that is the constant, 2.544. And then the coefficient on the caffeine, well, we just said that X is the caffeine consumed, so this is that coefficient, 0.164. So just like that, we actually have the equation for the regression line. That is why these computer things are useful. So we can just write it out. | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
And then the coefficient on the caffeine, well, we just said that X is the caffeine consumed, so this is that coefficient, 0.164. So just like that, we actually have the equation for the regression line. That is why these computer things are useful. So we can just write it out. Y hat is equal to 0.164X plus 2.544, 2.544. So that's the regression line. What is this other information they give us? | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
So we can just write it out. Y hat is equal to 0.164X plus 2.544, 2.544. So that's the regression line. What is this other information they give us? Well, I won't give you a very satisfying answer because all of this is actually useful for inferential statistics. To think about things like, well, what is the probability that this is chance that we got something to fit this well? So this right over here is the R squared. | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
What is this other information they give us? Well, I won't give you a very satisfying answer because all of this is actually useful for inferential statistics. To think about things like, well, what is the probability that this is chance that we got something to fit this well? So this right over here is the R squared. And if you wanted to figure out the R from this, you would just take the square root here. We could say that R is going to be equal to the square root of 0.60032, depending on how much precision you have. But you might say, well, how do we know if R is a positive square root or the negative square root of that? | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
So this right over here is the R squared. And if you wanted to figure out the R from this, you would just take the square root here. We could say that R is going to be equal to the square root of 0.60032, depending on how much precision you have. But you might say, well, how do we know if R is a positive square root or the negative square root of that? R can take on values between negative one and positive one. And the answer is, you would look at the slope here. We have a positive slope, which tells us that R is going to be positive. | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
But you might say, well, how do we know if R is a positive square root or the negative square root of that? R can take on values between negative one and positive one. And the answer is, you would look at the slope here. We have a positive slope, which tells us that R is going to be positive. If we had a negative slope, then we would take the negative square root. Now, this right here is the adjusted R squared. And we really don't have to worry about it too much when we're thinking about just bivariate data. | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
We have a positive slope, which tells us that R is going to be positive. If we had a negative slope, then we would take the negative square root. Now, this right here is the adjusted R squared. And we really don't have to worry about it too much when we're thinking about just bivariate data. We're talking about caffeine and hour studying in this case. If we started to have more variables that tried to explain the hour studying, then we would care about adjusted R squared, but we're not gonna do that just yet. Last but not least, this S variable, this is the standard deviation of the residuals, which we study in other videos. | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
And we really don't have to worry about it too much when we're thinking about just bivariate data. We're talking about caffeine and hour studying in this case. If we started to have more variables that tried to explain the hour studying, then we would care about adjusted R squared, but we're not gonna do that just yet. Last but not least, this S variable, this is the standard deviation of the residuals, which we study in other videos. And why is that useful? Well, that's a measure of how well the regression line fits the data. It's a measure of, we could say, the typical error. | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
Last but not least, this S variable, this is the standard deviation of the residuals, which we study in other videos. And why is that useful? Well, that's a measure of how well the regression line fits the data. It's a measure of, we could say, the typical error. So big takeaway, computers are useful. They'll give you a lot of data. And the key thing is how do you pick out the things that you actually need? | Interpreting computer regression data AP Statistics Khan Academy.mp3 |
So you go around the restaurant, and you write down everyone's age. And so these are the ages of everyone in the restaurant at that moment. And so you're interested in somehow presenting this, somehow visualizing the distribution of the ages, because you want to just say, well, are there more young people, are there more teenagers, are there more middle-aged people, are there more seniors here? And so when you just look at these numbers, it really doesn't give you a good sense of it. It's just a bunch of numbers. And so how could you do that? Well, one way to think about it is is to put these ages into different buckets, and then to think about how many people are there in each of those buckets, or sometimes someone might say, how many in each of those bins? | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
And so when you just look at these numbers, it really doesn't give you a good sense of it. It's just a bunch of numbers. And so how could you do that? Well, one way to think about it is is to put these ages into different buckets, and then to think about how many people are there in each of those buckets, or sometimes someone might say, how many in each of those bins? So let's do that. So let's do buckets or categories. So I like, sometimes it's called a bin. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
Well, one way to think about it is is to put these ages into different buckets, and then to think about how many people are there in each of those buckets, or sometimes someone might say, how many in each of those bins? So let's do that. So let's do buckets or categories. So I like, sometimes it's called a bin. So the bucket, I like to think of it more as a bucket. The bucket, and then the number in the bucket. The number in the bucket. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
So I like, sometimes it's called a bin. So the bucket, I like to think of it more as a bucket. The bucket, and then the number in the bucket. The number in the bucket. Number, I'll just write the number, whoops. It's the, whoops. It's the number. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
The number in the bucket. Number, I'll just write the number, whoops. It's the, whoops. It's the number. It's the number in the bucket. All right, so let's just make buckets, let's make them 10-year ranges. So let's say the first one is ages zero to nine. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
It's the number. It's the number in the bucket. All right, so let's just make buckets, let's make them 10-year ranges. So let's say the first one is ages zero to nine. So how many people, and we just define all of the buckets here. So the next one is ages 10 to 19, then 20 to 29, then 30 to 39, and 40 to 49, 50 to 59. Make sure you can read that properly. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
So let's say the first one is ages zero to nine. So how many people, and we just define all of the buckets here. So the next one is ages 10 to 19, then 20 to 29, then 30 to 39, and 40 to 49, 50 to 59. Make sure you can read that properly. Then you have 60 to 69. I think that covers everyone. I don't see anyone 70 years old or older here. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
Make sure you can read that properly. Then you have 60 to 69. I think that covers everyone. I don't see anyone 70 years old or older here. So then how many people fall into the zero to nine-year-old bucket? Well, it's gonna be one, two, three, four, five, six people fall into that bucket. How many people fall into the, how many people fall into the 10 to 19-year-old bucket? | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
I don't see anyone 70 years old or older here. So then how many people fall into the zero to nine-year-old bucket? Well, it's gonna be one, two, three, four, five, six people fall into that bucket. How many people fall into the, how many people fall into the 10 to 19-year-old bucket? Well, let's see. One, two, three. Three people. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
How many people fall into the, how many people fall into the 10 to 19-year-old bucket? Well, let's see. One, two, three. Three people. And I think you see where this is going. What about 20 to 29? So it's one, two, three, four, five people. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
Three people. And I think you see where this is going. What about 20 to 29? So it's one, two, three, four, five people. Five people fall into that bucket. All right, what about 30 to 39? We have one, and that's it. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
So it's one, two, three, four, five people. Five people fall into that bucket. All right, what about 30 to 39? We have one, and that's it. Only one person in that 30 to 39 bin or bucket or category. All right, what about 40 to 49? We have one, two people. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
We have one, and that's it. Only one person in that 30 to 39 bin or bucket or category. All right, what about 40 to 49? We have one, two people. Two people are in that bucket. And then 50 to 59. So you have one, two people. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
We have one, two people. Two people are in that bucket. And then 50 to 59. So you have one, two people. Two people. And then finally, finally, ages 60 to 69, let me do it in a different color, 60 to 69, there is one person right over there. So this is one way of thinking about how the ages are distributed, but let's actually make a visualization of this. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
So you have one, two people. Two people. And then finally, finally, ages 60 to 69, let me do it in a different color, 60 to 69, there is one person right over there. So this is one way of thinking about how the ages are distributed, but let's actually make a visualization of this. And the visualization that we're gonna create, this is called a histogram. Histogram. Histogram. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
So this is one way of thinking about how the ages are distributed, but let's actually make a visualization of this. And the visualization that we're gonna create, this is called a histogram. Histogram. Histogram. We're taking data that can take on a whole bunch of different values. We're putting them into categories, and we're gonna plot how many folks are in each category. How big are each of those categories? | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
Histogram. We're taking data that can take on a whole bunch of different values. We're putting them into categories, and we're gonna plot how many folks are in each category. How big are each of those categories? And actually, I wrote histogram. I wrote histograph, I should have written histogram. So a histogram. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
How big are each of those categories? And actually, I wrote histogram. I wrote histograph, I should have written histogram. So a histogram. So let's do this. All right, so on this axis, let's see, the largest category has six. So this is the number. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
So a histogram. So let's do this. All right, so on this axis, let's see, the largest category has six. So this is the number. Number of folks, and it's gonna go one, two, three, four, five, six. One, two, three, four, five, six. This is the number, and on this axis, I'm gonna make the buckets. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
So this is the number. Number of folks, and it's gonna go one, two, three, four, five, six. One, two, three, four, five, six. This is the number, and on this axis, I'm gonna make the buckets. The buckets, and let me scroll up a little bit now that I have my data here, I don't have to look at my data set again. So I have one bucket. This is going to be the zero to nine bucket, right over here. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
This is the number, and on this axis, I'm gonna make the buckets. The buckets, and let me scroll up a little bit now that I have my data here, I don't have to look at my data set again. So I have one bucket. This is going to be the zero to nine bucket, right over here. Zero to nine. Then I'm going to have the three, actually, let me just plot them, since I have my pen that color. So zero to nine, there are six people. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
This is going to be the zero to nine bucket, right over here. Zero to nine. Then I'm going to have the three, actually, let me just plot them, since I have my pen that color. So zero to nine, there are six people. Zero to nine, there are six people. So I'll just plot it like that. And then we have the 10 to 19. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
So zero to nine, there are six people. Zero to nine, there are six people. So I'll just plot it like that. And then we have the 10 to 19. There are three people. So 10 to 19, there are three people. So I'll do a bar like this. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
And then we have the 10 to 19. There are three people. So 10 to 19, there are three people. So I'll do a bar like this. Then 20 to 29, I have five people. 20 to 29, which is gonna be this one, which is getting, I'm writing too big. So 20 to 29, this is gonna be this bar. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
So I'll do a bar like this. Then 20 to 29, I have five people. 20 to 29, which is gonna be this one, which is getting, I'm writing too big. So 20 to 29, this is gonna be this bar. There's five people. Five people there. So it'll look like this. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
So 20 to 29, this is gonna be this bar. There's five people. Five people there. So it'll look like this. I should have made the bars wide enough so I could write below them, but I've already, that train has already left. All right, then 30 to 39. I'll try to write smaller. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
So it'll look like this. I should have made the bars wide enough so I could write below them, but I've already, that train has already left. All right, then 30 to 39. I'll try to write smaller. 30 to 39, that's gonna be this bar right over here. We have one person. One person. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
I'll try to write smaller. 30 to 39, that's gonna be this bar right over here. We have one person. One person. And then we have 40 to 49. We have two people. 40 to 49, two people. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
One person. And then we have 40 to 49. We have two people. 40 to 49, two people. So it looks like this. 40 to 49, two people. Almost there, 50 to 59. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
40 to 49, two people. So it looks like this. 40 to 49, two people. Almost there, 50 to 59. We have two people. 50 to 59, we also have two people. So that's that right over there. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
Almost there, 50 to 59. We have two people. 50 to 59, we also have two people. So that's that right over there. That's this category. And then finally, 60 to 69, we have one person. 60 to 69, we have one person. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
So that's that right over there. That's this category. And then finally, 60 to 69, we have one person. 60 to 69, we have one person. We have one person. And what I have just constructed, I took our data, I put it into buckets that are kind of representative of the categories I care about, zero to nine is kind of young kids, 10 to 19, I guess you could call them adolescents or roughly teenagers, although obviously if you're 10 you're not quite a teenager yet. And then all the different age groups. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
60 to 69, we have one person. We have one person. And what I have just constructed, I took our data, I put it into buckets that are kind of representative of the categories I care about, zero to nine is kind of young kids, 10 to 19, I guess you could call them adolescents or roughly teenagers, although obviously if you're 10 you're not quite a teenager yet. And then all the different age groups. And then when I counted the number in each bucket and I plotted it, now I can visually get a sense of how are the ages distributed in this restaurant. This must be some type of a restaurant that gives away toys or something because there's a lot of younger people. Maybe it's very family friendly. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
And then all the different age groups. And then when I counted the number in each bucket and I plotted it, now I can visually get a sense of how are the ages distributed in this restaurant. This must be some type of a restaurant that gives away toys or something because there's a lot of younger people. Maybe it's very family friendly. So every adult that comes in, maybe there's a lot of young adults with kids or maybe grandparents up here, and they just bring a lot of kids to this restaurant. So it gives you a view of what's going on here. Just a lot of kids here, a lot fewer senior citizens. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
Maybe it's very family friendly. So every adult that comes in, maybe there's a lot of young adults with kids or maybe grandparents up here, and they just bring a lot of kids to this restaurant. So it gives you a view of what's going on here. Just a lot of kids here, a lot fewer senior citizens. So once again, this is just a way of visualizing things. We took a lot of data that can take multiple data points. Instead of plotting each data point like we might do in a dot plot, instead of saying how many one year olds are there, well there's only one one year old, how many three year olds are there? | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
Just a lot of kids here, a lot fewer senior citizens. So once again, this is just a way of visualizing things. We took a lot of data that can take multiple data points. Instead of plotting each data point like we might do in a dot plot, instead of saying how many one year olds are there, well there's only one one year old, how many three year olds are there? There's only one three year old. That wouldn't give us much information. We would just have like these single dots if we were doing a dot plot. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
Instead of plotting each data point like we might do in a dot plot, instead of saying how many one year olds are there, well there's only one one year old, how many three year olds are there? There's only one three year old. That wouldn't give us much information. We would just have like these single dots if we were doing a dot plot. But as a histogram, we're able to put them into buckets. And we're just like, hey, generally between the ages zero and nine, we have six people. And so you see that plotted out just like that. | How to create a histogram Data and statistics 6th grade Khan Academy.mp3 |
Hey everybody, LeBron here. Got another quick brain teaser for you. Do I have a better odds of making three free throws in a row or one three pointer? Here's my friend Sal with the answer. Excellent question LeBron. But before I answer it, I want to point out an interesting trend related to your question that I just dug up. This is from the New York Times, October 2009, so it's a couple of years old. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
Here's my friend Sal with the answer. Excellent question LeBron. But before I answer it, I want to point out an interesting trend related to your question that I just dug up. This is from the New York Times, October 2009, so it's a couple of years old. But it's really interesting. It shows that since three pointers were first introduced, they were first introduced in the 1979-1980 season, that three pointers have become more and more frequent. So what they are showing here is the average attempts per team season by season. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
This is from the New York Times, October 2009, so it's a couple of years old. But it's really interesting. It shows that since three pointers were first introduced, they were first introduced in the 1979-1980 season, that three pointers have become more and more frequent. So what they are showing here is the average attempts per team season by season. It looks like there's just a steady upward trend here related to our question. There's a couple of anomalies here. The ones that really jump out are these three seasons in the late 90s. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
So what they are showing here is the average attempts per team season by season. It looks like there's just a steady upward trend here related to our question. There's a couple of anomalies here. The ones that really jump out are these three seasons in the late 90s. That's because the actual three pointer line was pulled in to get higher scoring games, so people attempted more, but then it was put back to where it was originally. This was a shortened season. I'm not really sure what happened in the 2000-2001-2002 season. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
The ones that really jump out are these three seasons in the late 90s. That's because the actual three pointer line was pulled in to get higher scoring games, so people attempted more, but then it was put back to where it was originally. This was a shortened season. I'm not really sure what happened in the 2000-2001-2002 season. But it's something to think about. There is just this trend that more and more people are taking three pointers. With that out of the way, let's think about your actual question. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
I'm not really sure what happened in the 2000-2001-2002 season. But it's something to think about. There is just this trend that more and more people are taking three pointers. With that out of the way, let's think about your actual question. To answer it, I dug up your stats right over here from NBA.com. We'll use your career stats. We want to compare three free throws to a three pointer. | Three pointer vs free throwing probability Probability and Statistics Khan Academy.mp3 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.