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Now, there's other reasons why you might introduce bias, and it might not be because of the sampling. You might introduce bias because of the wording of your survey. You could imagine a survey that says, do you consider yourself lucky to get a math education that very few other people in the world have access to? Well, that might bias you to say, well, yeah, I guess I feel lucky. Well, if the wording was, do you like the fact that a disproportionate, more students at your school tend to fail algebra than our surrounding schools? Well, that might bias you negatively. So the wording really, really, really matters in surveys, and there's a lot that would go into this. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
Well, that might bias you to say, well, yeah, I guess I feel lucky. Well, if the wording was, do you like the fact that a disproportionate, more students at your school tend to fail algebra than our surrounding schools? Well, that might bias you negatively. So the wording really, really, really matters in surveys, and there's a lot that would go into this. And the other one is just people's, it's called response bias. And once again, this isn't about response bias. And this is just people not wanting to tell the truth or maybe not wanting to respond at all. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
So the wording really, really, really matters in surveys, and there's a lot that would go into this. And the other one is just people's, it's called response bias. And once again, this isn't about response bias. And this is just people not wanting to tell the truth or maybe not wanting to respond at all. Maybe they're afraid that somehow their response is gonna show up in front of their math teacher or the administrators, or if they're too negative, it might be taken out on them in some way. And because of that, they might not be truthful. And so they might be overly positive or not fill it out at all. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And this is just people not wanting to tell the truth or maybe not wanting to respond at all. Maybe they're afraid that somehow their response is gonna show up in front of their math teacher or the administrators, or if they're too negative, it might be taken out on them in some way. And because of that, they might not be truthful. And so they might be overly positive or not fill it out at all. So anyway, this is a very high-level overview of how you could think about sampling. You wanna go random because it lowers the probability of their introducing some bias into it. And then these are some techniques. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And so if we're doing it as a Venn diagram, the universe is usually depicted as some type of a rectangle right over here. And it itself is a set and it usually is denoted with the capital U, U for universe, not to be confused with the union set notation. And you could say that the universe is all possible things that could be in a set, including farm animals and kitchen utensils and emotions and types of Italian food or even types of food. But then that just becomes somewhat crazy because you're literally thinking of all possible things. Normally when people talk about a universal set, they're talking about a universe of things that they care about. So the set of all people or the set of all real numbers or the set of all countries, whatever the discussion is being focused on. But we'll talk about an abstract terms right now. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
But then that just becomes somewhat crazy because you're literally thinking of all possible things. Normally when people talk about a universal set, they're talking about a universe of things that they care about. So the set of all people or the set of all real numbers or the set of all countries, whatever the discussion is being focused on. But we'll talk about an abstract terms right now. And let's say you have a subset of that universal set. So let's say you have a subset of that universal set, set A. And so set A literally contains everything, everything that I have just shaded in. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
But we'll talk about an abstract terms right now. And let's say you have a subset of that universal set. So let's say you have a subset of that universal set, set A. And so set A literally contains everything, everything that I have just shaded in. What we're gonna talk about now is the idea of a complement or the absolute complement of A. And the way you could think about this is this is the set of all things in the universe, in universe that aren't, that aren't, that aren't in A. And we've already looked at ways of expressing this. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
And so set A literally contains everything, everything that I have just shaded in. What we're gonna talk about now is the idea of a complement or the absolute complement of A. And the way you could think about this is this is the set of all things in the universe, in universe that aren't, that aren't, that aren't in A. And we've already looked at ways of expressing this. The set of all things in the universe that aren't in A, we could also write as a universal set minus A. Once again, this is a capital U. This is not the union symbol right over here. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
And we've already looked at ways of expressing this. The set of all things in the universe that aren't in A, we could also write as a universal set minus A. Once again, this is a capital U. This is not the union symbol right over here. Or we could literally write this as U and then we write that little slash looking thing, U slash A. So how do we represent that in the Venn diagram? Well, it would be all the stuff in U, it would be all the stuff in U that is not in A. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
This is not the union symbol right over here. Or we could literally write this as U and then we write that little slash looking thing, U slash A. So how do we represent that in the Venn diagram? Well, it would be all the stuff in U, it would be all the stuff in U that is not in A. One way to think about it, you could think about it as the relative complement of A that is in U, but when you're taking the relative complement of something that is in the universal set, you're really talking about the absolute complement. Or when people just talk about the complement, that's what they're saying. What's the set of all the things in my universe that are not, that are not in A? | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
Well, it would be all the stuff in U, it would be all the stuff in U that is not in A. One way to think about it, you could think about it as the relative complement of A that is in U, but when you're taking the relative complement of something that is in the universal set, you're really talking about the absolute complement. Or when people just talk about the complement, that's what they're saying. What's the set of all the things in my universe that are not, that are not in A? Now let's make things a little bit more concrete by talking about sets of numbers. Once again, our sets, we could have been talking about sets of TV personalities or sets of animals or whatever it might be, but numbers are a nice, simple thing to deal with. And let's say that our universe, our universe that we care about right over here, is the set of integers. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
What's the set of all the things in my universe that are not, that are not in A? Now let's make things a little bit more concrete by talking about sets of numbers. Once again, our sets, we could have been talking about sets of TV personalities or sets of animals or whatever it might be, but numbers are a nice, simple thing to deal with. And let's say that our universe, our universe that we care about right over here, is the set of integers. So our universe is a set of integers. So I'll write U, capital U, is equal to the set of integers. And this is a little bit of an aside, but the notation for the set of integers tends to be a bold Z. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
And let's say that our universe, our universe that we care about right over here, is the set of integers. So our universe is a set of integers. So I'll write U, capital U, is equal to the set of integers. And this is a little bit of an aside, but the notation for the set of integers tends to be a bold Z. And it's Z for Zoll from German for apparently integer. And the bold is this kind of weird looking, they call it blackboard bold. And it's what mathematicians use for different types of sets of numbers. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
And this is a little bit of an aside, but the notation for the set of integers tends to be a bold Z. And it's Z for Zoll from German for apparently integer. And the bold is this kind of weird looking, they call it blackboard bold. And it's what mathematicians use for different types of sets of numbers. And in fact, I'll do a little aside here to do that. So for example, they might say, they'll write R like this for the set of real numbers, real numbers. They'll write a Q in that blackboard bold font. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
And it's what mathematicians use for different types of sets of numbers. And in fact, I'll do a little aside here to do that. So for example, they might say, they'll write R like this for the set of real numbers, real numbers. They'll write a Q in that blackboard bold font. So it looks something like this. They'll write the Q, it might look something like this. This would be the set of rational numbers. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
They'll write a Q in that blackboard bold font. So it looks something like this. They'll write the Q, it might look something like this. This would be the set of rational numbers. And you might say, why Q for rational? Well, there's a couple of reasons. One, the R is already taken up. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
This would be the set of rational numbers. And you might say, why Q for rational? Well, there's a couple of reasons. One, the R is already taken up. And Q for quotient, a rational number can be expressed as a quotient of two integers. And we just saw you have your Z for Zoll, for Zoll or integers, the set of all integers. So our universal set, the universe that we care about right now is integers. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
One, the R is already taken up. And Q for quotient, a rational number can be expressed as a quotient of two integers. And we just saw you have your Z for Zoll, for Zoll or integers, the set of all integers. So our universal set, the universe that we care about right now is integers. And let's define a subset of it. Let's call that subset, I don't know, I've been, let me put a, use a letter that I haven't been using a lot. Let's call it C. The set C, let's say it's equal to negative five, zero, and positive seven. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
So our universal set, the universe that we care about right now is integers. And let's define a subset of it. Let's call that subset, I don't know, I've been, let me put a, use a letter that I haven't been using a lot. Let's call it C. The set C, let's say it's equal to negative five, zero, and positive seven. And I'm obviously not drawing it to scale. The set of all integers is infinite, whilst the set C is a finite set. But I'll just kind of, just to draw it, that's our set C right over there. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
Let's call it C. The set C, let's say it's equal to negative five, zero, and positive seven. And I'm obviously not drawing it to scale. The set of all integers is infinite, whilst the set C is a finite set. But I'll just kind of, just to draw it, that's our set C right over there. And let's think about what is a member of C and what is not a member of C. So we know that negative five is a member of our set C. This little symbol right here, this denotes membership. Membership, it looks a lot like the Greek letter epsilon, but it is not the Greek letter epsilon. This just literally means membership of a set. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
But I'll just kind of, just to draw it, that's our set C right over there. And let's think about what is a member of C and what is not a member of C. So we know that negative five is a member of our set C. This little symbol right here, this denotes membership. Membership, it looks a lot like the Greek letter epsilon, but it is not the Greek letter epsilon. This just literally means membership of a set. We know that zero is a member of set of, sorry. We know that zero is a member of our set. We know that seven is a member, is a member of our set. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
This just literally means membership of a set. We know that zero is a member of set of, sorry. We know that zero is a member of our set. We know that seven is a member, is a member of our set. Now we also know some other things. We know that the number negative eight is not, is not a member of our set. We know that the number 53 is not a member, not a member of our set. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
We know that seven is a member, is a member of our set. Now we also know some other things. We know that the number negative eight is not, is not a member of our set. We know that the number 53 is not a member, not a member of our set. 53 is sitting someplace out here. We know the number 42 is not a member of our set. 42 might be sitting someplace out there. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
We know that the number 53 is not a member, not a member of our set. 53 is sitting someplace out here. We know the number 42 is not a member of our set. 42 might be sitting someplace out there. Now let's think about C complement or the complement of C. C complement, which is the same thing as our universe, minus C, which is the same thing as universe, or you could say the relative complement of C in our universe. These are all equivalent notation. What is this, first of all, in our Venn diagram? | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
42 might be sitting someplace out there. Now let's think about C complement or the complement of C. C complement, which is the same thing as our universe, minus C, which is the same thing as universe, or you could say the relative complement of C in our universe. These are all equivalent notation. What is this, first of all, in our Venn diagram? What's all this stuff outside of our set? Outside of our set C, right over here. And now all of a sudden we know that negative five, negative five is a member of C, so it can't be a member of C complement. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
What is this, first of all, in our Venn diagram? What's all this stuff outside of our set? Outside of our set C, right over here. And now all of a sudden we know that negative five, negative five is a member of C, so it can't be a member of C complement. So negative five is not a member of C complement. Zero is not a member of C complement. Zero sits in C, not in C complement. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
And now all of a sudden we know that negative five, negative five is a member of C, so it can't be a member of C complement. So negative five is not a member of C complement. Zero is not a member of C complement. Zero sits in C, not in C complement. Negative, or I could say 53, 53 is a member of C complement. It's outside of C, it's in the universe, but outside of C. 42 is a member of C complement. So anyway, hopefully that helps clear things up a little bit. | Universal set and absolute complement Probability and Statistics Khan Academy.mp3 |
Or from their expected value. So let me just write that down. So if I take, I'll have x first, I'll do this in another color. So it's the expected value of random variable x minus the expected value of x. You could view this as the population mean of x, times, and then this is random variable y, so times the distance from y to its expected value. Or the population mean. The population mean of y. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
So it's the expected value of random variable x minus the expected value of x. You could view this as the population mean of x, times, and then this is random variable y, so times the distance from y to its expected value. Or the population mean. The population mean of y. And if it doesn't make a lot of intuitive sense yet, well one, you can just always kind of think about what it's doing, play around with some numbers here. But the reality is, it's saying how much they vary together. So you always take an x and a y for each of the data points. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
The population mean of y. And if it doesn't make a lot of intuitive sense yet, well one, you can just always kind of think about what it's doing, play around with some numbers here. But the reality is, it's saying how much they vary together. So you always take an x and a y for each of the data points. Let's say you have the whole population, so every x and y that go together with each other, that are a coordinate, you put into this. And what happens is, let's say that x is above its mean when y is below its mean. So let's say in the population, you had the point, so one instantiation of the random variables, you have, you sample once, I guess, from the universe. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
So you always take an x and a y for each of the data points. Let's say you have the whole population, so every x and y that go together with each other, that are a coordinate, you put into this. And what happens is, let's say that x is above its mean when y is below its mean. So let's say in the population, you had the point, so one instantiation of the random variables, you have, you sample once, I guess, from the universe. And you get x is equal to 1, and that y is equal to, let's say y is equal to 3. And let's say that you knew ahead of time that the expected value of x is 0, and let's say that the expected value of y is equal to 4. So in this situation, what just happened? | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
So let's say in the population, you had the point, so one instantiation of the random variables, you have, you sample once, I guess, from the universe. And you get x is equal to 1, and that y is equal to, let's say y is equal to 3. And let's say that you knew ahead of time that the expected value of x is 0, and let's say that the expected value of y is equal to 4. So in this situation, what just happened? Now we don't know the entire covariance, we only have one sample here of this random variable. But what just happened here? We have 1 minus, so we're not going to calculate the entire expected value, I just want to calculate what happens when we do what's inside the expected value. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
So in this situation, what just happened? Now we don't know the entire covariance, we only have one sample here of this random variable. But what just happened here? We have 1 minus, so we're not going to calculate the entire expected value, I just want to calculate what happens when we do what's inside the expected value. We'll have 1 minus 0, so you'll have a 1, times a 3 minus 4, times a negative 1. So you're going to have 1 times negative 1, which is negative 1. And what is that telling us? | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
We have 1 minus, so we're not going to calculate the entire expected value, I just want to calculate what happens when we do what's inside the expected value. We'll have 1 minus 0, so you'll have a 1, times a 3 minus 4, times a negative 1. So you're going to have 1 times negative 1, which is negative 1. And what is that telling us? Well, it's telling us at least for this sample, this one time that we sampled the random variables x and y, x was above its expected value when y was below its expected value. And if we kept doing this, let's say for the entire population this happened, then it would make sense that they have a negative covariance. When one goes up, the other one goes down. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
And what is that telling us? Well, it's telling us at least for this sample, this one time that we sampled the random variables x and y, x was above its expected value when y was below its expected value. And if we kept doing this, let's say for the entire population this happened, then it would make sense that they have a negative covariance. When one goes up, the other one goes down. When one goes down, the other one goes up. If they both go up together, they would have a positive variance. Or if they both go down together, the degree to which they do it together would tell you the magnitude of the covariance. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
When one goes up, the other one goes down. When one goes down, the other one goes up. If they both go up together, they would have a positive variance. Or if they both go down together, the degree to which they do it together would tell you the magnitude of the covariance. Hopefully that gives you a little bit of intuition about what the covariance is trying to tell us. But the more important thing that I want to do in this video is to connect this formula, is I want to connect this definition of covariance to everything we've been doing with least squared regression. And really it's just kind of a fun math thing to do to show you all of these connections and where really the definition of covariance really becomes useful. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Or if they both go down together, the degree to which they do it together would tell you the magnitude of the covariance. Hopefully that gives you a little bit of intuition about what the covariance is trying to tell us. But the more important thing that I want to do in this video is to connect this formula, is I want to connect this definition of covariance to everything we've been doing with least squared regression. And really it's just kind of a fun math thing to do to show you all of these connections and where really the definition of covariance really becomes useful. And I really do think it's motivated to a large degree by where it shows up in regressions. And this is all stuff that we've kind of seen before. You're just going to see it in a different way. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
And really it's just kind of a fun math thing to do to show you all of these connections and where really the definition of covariance really becomes useful. And I really do think it's motivated to a large degree by where it shows up in regressions. And this is all stuff that we've kind of seen before. You're just going to see it in a different way. So this whole video I'm just going to rewrite this definition of covariance right over here. So this is going to be the same thing as the expected value of, and I'm just going to multiply these two binomials in here. So the expected value of our random variable X times our random variable Y minus, well I'll just do the X first. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
You're just going to see it in a different way. So this whole video I'm just going to rewrite this definition of covariance right over here. So this is going to be the same thing as the expected value of, and I'm just going to multiply these two binomials in here. So the expected value of our random variable X times our random variable Y minus, well I'll just do the X first. So plus X times the negative expected value of Y. So I'll just say minus X times the expected value of Y. And that negative sign comes from this negative sign right over here. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
So the expected value of our random variable X times our random variable Y minus, well I'll just do the X first. So plus X times the negative expected value of Y. So I'll just say minus X times the expected value of Y. And that negative sign comes from this negative sign right over here. And then we have minus expected value of X times Y minus the expected value of X times this Y. Just doing the distributed property twice. And then finally you have the negative expected value of X times the negative expected value of Y. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
And that negative sign comes from this negative sign right over here. And then we have minus expected value of X times Y minus the expected value of X times this Y. Just doing the distributed property twice. And then finally you have the negative expected value of X times the negative expected value of Y. And the negatives cancel out. So you're just going to have plus the expected value of X times the expected value of Y. And of course it's the expected value of this entire thing. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
And then finally you have the negative expected value of X times the negative expected value of Y. And the negatives cancel out. So you're just going to have plus the expected value of X times the expected value of Y. And of course it's the expected value of this entire thing. Now let's see if we can rewrite this. Well the expected value of the sum of a bunch of random variables is just the sum or difference of their expected values. So this is going to be the same thing. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
And of course it's the expected value of this entire thing. Now let's see if we can rewrite this. Well the expected value of the sum of a bunch of random variables is just the sum or difference of their expected values. So this is going to be the same thing. And remember expected value in a lot of context you can view it as just the arithmetic mean. Or in a continuous distribution you can view it as a probability weighted sum or probability weighted integral. Either way, we've seen it before I think. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
So this is going to be the same thing. And remember expected value in a lot of context you can view it as just the arithmetic mean. Or in a continuous distribution you can view it as a probability weighted sum or probability weighted integral. Either way, we've seen it before I think. So let's rewrite this. So this is equal to the expected value of the random variables X and Y. X times Y. Trying to keep them color coded for you. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Either way, we've seen it before I think. So let's rewrite this. So this is equal to the expected value of the random variables X and Y. X times Y. Trying to keep them color coded for you. Color coded. And then we have minus X times the expected value of Y. So then we're going to have minus the expected value of X times the expected value of Y. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Trying to keep them color coded for you. Color coded. And then we have minus X times the expected value of Y. So then we're going to have minus the expected value of X times the expected value of Y. Of X times the expected value of Y. Times the expected value of Y. Stay with the right colors. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
So then we're going to have minus the expected value of X times the expected value of Y. Of X times the expected value of Y. Times the expected value of Y. Stay with the right colors. Then you're going to have minus the expected value of this thing. Minus the expected value of, I'll close the parentheses of this thing right over here. Expected value of X. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Stay with the right colors. Then you're going to have minus the expected value of this thing. Minus the expected value of, I'll close the parentheses of this thing right over here. Expected value of X. Expected value of X times Y. Now this might look really confusing with all the embedded expected values. But one way to think about it is the things that already have the expected values, you can kind of view these as numbers. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Expected value of X. Expected value of X times Y. Now this might look really confusing with all the embedded expected values. But one way to think about it is the things that already have the expected values, you can kind of view these as numbers. You already view them as known. So we're actually going to take them out of the expected value. Because the expected value of an expected value is the same thing as the expected value. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
But one way to think about it is the things that already have the expected values, you can kind of view these as numbers. You already view them as known. So we're actually going to take them out of the expected value. Because the expected value of an expected value is the same thing as the expected value. Actually let me write this over here just to remind ourselves. The expected value of X is just going to be the expected value of X. Think of it this way. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Because the expected value of an expected value is the same thing as the expected value. Actually let me write this over here just to remind ourselves. The expected value of X is just going to be the expected value of X. Think of it this way. You could view this as the population mean for the random variable. So that's just going to be a known, it's out there, it's in the universe. So the expected value of that is just going to be itself. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Think of it this way. You could view this as the population mean for the random variable. So that's just going to be a known, it's out there, it's in the universe. So the expected value of that is just going to be itself. If the population mean or the expected value of X is 5, this is like saying the expected value of 5. Well the expected value of 5 is going to be 5. Which is the same thing as the expected value of X. Hopefully that makes sense. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
So the expected value of that is just going to be itself. If the population mean or the expected value of X is 5, this is like saying the expected value of 5. Well the expected value of 5 is going to be 5. Which is the same thing as the expected value of X. Hopefully that makes sense. We're going to use that in a second. So we're almost done. We did the expected value of this and we have one term left. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Which is the same thing as the expected value of X. Hopefully that makes sense. We're going to use that in a second. So we're almost done. We did the expected value of this and we have one term left. And then the final term, the expected value of this guy. And here we can actually use the property right from the get-go. I'll write it down. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
We did the expected value of this and we have one term left. And then the final term, the expected value of this guy. And here we can actually use the property right from the get-go. I'll write it down. So the expected value of, put some big brackets up, of this thing right over here. Expected value of X times the expected value of Y. Let's see if we can simplify it right here. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
I'll write it down. So the expected value of, put some big brackets up, of this thing right over here. Expected value of X times the expected value of Y. Let's see if we can simplify it right here. So this is just going to be the expected value of the product of these two random variables. I'll just leave that the way it is. So let me just, the stuff that I'm going to leave the way it is, I'm just going to kind of freeze them. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Let's see if we can simplify it right here. So this is just going to be the expected value of the product of these two random variables. I'll just leave that the way it is. So let me just, the stuff that I'm going to leave the way it is, I'm just going to kind of freeze them. So the expected value of XY. Now what do we have over here? We have the expected value of X times, once again, you can kind of view it if you go back to what we just said, is this is just going to be a number, expected value of Y. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
So let me just, the stuff that I'm going to leave the way it is, I'm just going to kind of freeze them. So the expected value of XY. Now what do we have over here? We have the expected value of X times, once again, you can kind of view it if you go back to what we just said, is this is just going to be a number, expected value of Y. So we can just bring this out. If this was the expected value of 3X, it would be the same thing as 3 times the expected value of X. So we can rewrite this as negative expected value of Y times the expected value of X. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
We have the expected value of X times, once again, you can kind of view it if you go back to what we just said, is this is just going to be a number, expected value of Y. So we can just bring this out. If this was the expected value of 3X, it would be the same thing as 3 times the expected value of X. So we can rewrite this as negative expected value of Y times the expected value of X. So you can kind of view this as we took it out of the expected value. We factored it out. So just like that. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
So we can rewrite this as negative expected value of Y times the expected value of X. So you can kind of view this as we took it out of the expected value. We factored it out. So just like that. And then you have minus, same thing over here. You can factor out this expected value of X minus the expected value of X times the expected value of Y times the expected value of Y. Let me write it. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
So just like that. And then you have minus, same thing over here. You can factor out this expected value of X minus the expected value of X times the expected value of Y times the expected value of Y. Let me write it. Times the expected value of Y. This is getting confusing with all the E's laying around. And then finally, you have the expected value of this thing, of two expected values. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Let me write it. Times the expected value of Y. This is getting confusing with all the E's laying around. And then finally, you have the expected value of this thing, of two expected values. Well, that's just going to be the product of those two expected values. So that's just going to be plus, I'll freeze this, expected value of X times the expected value of Y. Now what do we have here? | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
And then finally, you have the expected value of this thing, of two expected values. Well, that's just going to be the product of those two expected values. So that's just going to be plus, I'll freeze this, expected value of X times the expected value of Y. Now what do we have here? We have expected value of Y times the expected value of X, and then we are subtracting the expected value of X times the expected value of Y. These two things are the exact same thing. So this is going to be, and actually look at this, we're subtracting it twice and then we have one more. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Now what do we have here? We have expected value of Y times the expected value of X, and then we are subtracting the expected value of X times the expected value of Y. These two things are the exact same thing. So this is going to be, and actually look at this, we're subtracting it twice and then we have one more. These are all the same thing. This is the expected value of Y times the expected value of X. This is the expected value of Y times the expected value of X, just written in a different order. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
So this is going to be, and actually look at this, we're subtracting it twice and then we have one more. These are all the same thing. This is the expected value of Y times the expected value of X. This is the expected value of Y times the expected value of X, just written in a different order. And this is the expected value of Y times the expected value of X. We're subtracting it twice and then we're adding it. Or, one way to think about it is that this guy and that guy will cancel out. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
This is the expected value of Y times the expected value of X, just written in a different order. And this is the expected value of Y times the expected value of X. We're subtracting it twice and then we're adding it. Or, one way to think about it is that this guy and that guy will cancel out. You could have also picked that guy and that guy. But what do we have left? We have the covariance of these two random variables, X and Y, are equal to the expected value of, I'll switch back to my colors just because this is the final result, the expected value of X times the expected value of the product of XY, the expected value of the product, the expected value of the product, minus, what is this? | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Or, one way to think about it is that this guy and that guy will cancel out. You could have also picked that guy and that guy. But what do we have left? We have the covariance of these two random variables, X and Y, are equal to the expected value of, I'll switch back to my colors just because this is the final result, the expected value of X times the expected value of the product of XY, the expected value of the product, the expected value of the product, minus, what is this? The expected value of Y, the expected value of Y times the expected value of X, times the expected value of X. Now, you can calculate these expected values if you know everything about the probability distribution or density functions for each of these random variables, or if you had the entire population that you're sampling from whenever you take an instantiation of these random variables. But let's say you just had a sample of these random variables. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
We have the covariance of these two random variables, X and Y, are equal to the expected value of, I'll switch back to my colors just because this is the final result, the expected value of X times the expected value of the product of XY, the expected value of the product, the expected value of the product, minus, what is this? The expected value of Y, the expected value of Y times the expected value of X, times the expected value of X. Now, you can calculate these expected values if you know everything about the probability distribution or density functions for each of these random variables, or if you had the entire population that you're sampling from whenever you take an instantiation of these random variables. But let's say you just had a sample of these random variables. How could you estimate them? Well, if you were estimating it, the expected value, let's say you just have a bunch of data points, a bunch of coordinates, and I think you'll start to see how this relates to what we did with regression. The expected value of X times Y, it can be approximated by the sample mean of the products of X and Y. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
But let's say you just had a sample of these random variables. How could you estimate them? Well, if you were estimating it, the expected value, let's say you just have a bunch of data points, a bunch of coordinates, and I think you'll start to see how this relates to what we did with regression. The expected value of X times Y, it can be approximated by the sample mean of the products of X and Y. This is going to be the sample mean of X and Y. You take each of your XY associations, take their product, and then take the mean of all of them. So that's going to be the product of X and Y, and then this thing right over here, the expected value of Y, that can be approximated by the sample mean of Y, and the expected value of X can be approximated by the sample mean of X. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
The expected value of X times Y, it can be approximated by the sample mean of the products of X and Y. This is going to be the sample mean of X and Y. You take each of your XY associations, take their product, and then take the mean of all of them. So that's going to be the product of X and Y, and then this thing right over here, the expected value of Y, that can be approximated by the sample mean of Y, and the expected value of X can be approximated by the sample mean of X. So what can the covariance of two random variables be approximated by? What can they be approximated by? Well, this right here is the mean of their product from your sample, minus the mean of your sample Ys times the mean of your sample Xs. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
So that's going to be the product of X and Y, and then this thing right over here, the expected value of Y, that can be approximated by the sample mean of Y, and the expected value of X can be approximated by the sample mean of X. So what can the covariance of two random variables be approximated by? What can they be approximated by? Well, this right here is the mean of their product from your sample, minus the mean of your sample Ys times the mean of your sample Xs. And this should start looking familiar. This should look a little bit familiar, because what is this? This was the numerator. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Well, this right here is the mean of their product from your sample, minus the mean of your sample Ys times the mean of your sample Xs. And this should start looking familiar. This should look a little bit familiar, because what is this? This was the numerator. This right here is the numerator when we were trying to figure out the slope of the regression line. So we tried to figure out the slope of the regression line. Let me just rewrite the formula here just to remind you. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
This was the numerator. This right here is the numerator when we were trying to figure out the slope of the regression line. So we tried to figure out the slope of the regression line. Let me just rewrite the formula here just to remind you. It was literally the mean of the products of each of our data points, minus, or the XYs, minus the mean of Ys times the mean of the Xs. All of that over the mean of the X squareds, and you could even view it as this, over the mean of the X times the Xs, but I could just write the X squareds over here, minus the mean of X squared. This is how we figured out the slope of our regression line. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Let me just rewrite the formula here just to remind you. It was literally the mean of the products of each of our data points, minus, or the XYs, minus the mean of Ys times the mean of the Xs. All of that over the mean of the X squareds, and you could even view it as this, over the mean of the X times the Xs, but I could just write the X squareds over here, minus the mean of X squared. This is how we figured out the slope of our regression line. Or maybe a better way to think about it, if we assume in our regression line that the points that we have were a sample from an entire universe of possible points, then you could say that we are approximating the slope of our regression line. You might see this little hat notation in a lot of books. I don't want you to be confused. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
This is how we figured out the slope of our regression line. Or maybe a better way to think about it, if we assume in our regression line that the points that we have were a sample from an entire universe of possible points, then you could say that we are approximating the slope of our regression line. You might see this little hat notation in a lot of books. I don't want you to be confused. You're saying that you're approximating the population's regression line from a sample of it. Now, this right here, so everything we've learned right now, this right here is the covariance, or this is an estimate of the covariance of X and Y. Now, what is this over here? | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
I don't want you to be confused. You're saying that you're approximating the population's regression line from a sample of it. Now, this right here, so everything we've learned right now, this right here is the covariance, or this is an estimate of the covariance of X and Y. Now, what is this over here? I just said you could rewrite this very easily. This bottom part right here, you could write as the mean of X times X, that's the same thing as X squared, minus the mean of X times the mean of X. That's what the mean of X squared is. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
Now, what is this over here? I just said you could rewrite this very easily. This bottom part right here, you could write as the mean of X times X, that's the same thing as X squared, minus the mean of X times the mean of X. That's what the mean of X squared is. What's this? You could view this as the covariance of X with X. We've actually already seen this, and I've actually shown you many, many videos ago when we first learned about it what this is. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
That's what the mean of X squared is. What's this? You could view this as the covariance of X with X. We've actually already seen this, and I've actually shown you many, many videos ago when we first learned about it what this is. The covariance of a random variable with itself is really just the variance of that random variable. You could verify it for yourself. If you change this Y to an X, this becomes X minus the expected value of X times X minus the expected value of X, or that's the expected value of X minus the expected value of X squared. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
We've actually already seen this, and I've actually shown you many, many videos ago when we first learned about it what this is. The covariance of a random variable with itself is really just the variance of that random variable. You could verify it for yourself. If you change this Y to an X, this becomes X minus the expected value of X times X minus the expected value of X, or that's the expected value of X minus the expected value of X squared. That's your definition of variance. Another way of thinking about the slope of our regression line, it can be literally viewed as the covariance of our two random variables over the variance of X. You can view it as the independent random variable. | Covariance and the regression line Regression Probability and Statistics Khan Academy.mp3 |
City Councilwoman Kelly wants to know how the residents of her district feel about a proposed school redistricting plan. Which of the following survey methods will allow Councilwoman Kelly to make a valid conclusion about how residents of her district feel about the proposed plan? So before we even look at these, we have to realize that if you're trying to make a valid conclusion about how the residents of her entire district feel about the proposed plan, she has to find a representative sample, and not kind of a skewed sample that would just sample parts of her district. So let's look at her choices. Should she just ask her neighbors? Well, her neighbors, she might live in a part of the neighborhood that might unusually benefit from the redistricting plan or might get hurt from the redistricting plan. And so just her neighbors wouldn't be representative of the district as a whole. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
So let's look at her choices. Should she just ask her neighbors? Well, her neighbors, she might live in a part of the neighborhood that might unusually benefit from the redistricting plan or might get hurt from the redistricting plan. And so just her neighbors wouldn't be representative of the district as a whole. So just asking her neighbors probably does not make sense. Ask the residents of Whispering Pines Retirement Community. So once again, this is not, the first one's skews by geography. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
And so just her neighbors wouldn't be representative of the district as a whole. So just asking her neighbors probably does not make sense. Ask the residents of Whispering Pines Retirement Community. So once again, this is not, the first one's skews by geography. She's oversampling her neighbors and not the entire district. Here, she's oversampling a specific age demographic. So here she's oversampling older residents who might have very different opinions than middle-aged or younger residents. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
So once again, this is not, the first one's skews by geography. She's oversampling her neighbors and not the entire district. Here, she's oversampling a specific age demographic. So here she's oversampling older residents who might have very different opinions than middle-aged or younger residents. So that doesn't make sense either. Ask 200 residents of her district whose names are chosen at random. Well, that seems reasonable. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
So here she's oversampling older residents who might have very different opinions than middle-aged or younger residents. So that doesn't make sense either. Ask 200 residents of her district whose names are chosen at random. Well, that seems reasonable. It doesn't seem like there's some chance that you somehow oversample one direction or another, but it's most likely to give a reasonably representative sample. And this is a pretty large sample size. So it's important to say what is the random process, where is she getting these names from, but this actually does seem reasonable. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
Well, that seems reasonable. It doesn't seem like there's some chance that you somehow oversample one direction or another, but it's most likely to give a reasonably representative sample. And this is a pretty large sample size. So it's important to say what is the random process, where is she getting these names from, but this actually does seem reasonable. Ask a group of parents at the local playground. Well, once again, this is just like asking your neighbors, and it's also sampling a specific demographic. Now, this might be the demographic that cares most about the schools, but she wants to know how her whole district feels about the redistricting plan. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
So it's important to say what is the random process, where is she getting these names from, but this actually does seem reasonable. Ask a group of parents at the local playground. Well, once again, this is just like asking your neighbors, and it's also sampling a specific demographic. Now, this might be the demographic that cares most about the schools, but she wants to know how her whole district feels about the redistricting plan. And once again, this is at a local playground. This isn't at all the playgrounds in the district somehow, so I wouldn't do this one either. Let's do one more of these. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
Now, this might be the demographic that cares most about the schools, but she wants to know how her whole district feels about the redistricting plan. And once again, this is at a local playground. This isn't at all the playgrounds in the district somehow, so I wouldn't do this one either. Let's do one more of these. Mimi wants to conduct a survey of her 300 classmates to determine which candidate for class president Napoleon Dynamite or Blair Waldorf is in the lead in the upcoming election. Mimi will ask the question, if the election were today, which candidate would get your vote? Which of the following methods of surveying her classmates will allow Mimi to make valid conclusions about which candidate is in the lead? | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
Let's do one more of these. Mimi wants to conduct a survey of her 300 classmates to determine which candidate for class president Napoleon Dynamite or Blair Waldorf is in the lead in the upcoming election. Mimi will ask the question, if the election were today, which candidate would get your vote? Which of the following methods of surveying her classmates will allow Mimi to make valid conclusions about which candidate is in the lead? So let's see. Ask all of the students at Blair's lunch table. No, that would skew it in Blair's favor. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
Which of the following methods of surveying her classmates will allow Mimi to make valid conclusions about which candidate is in the lead? So let's see. Ask all of the students at Blair's lunch table. No, that would skew it in Blair's favor. Probably. That's not a representative sample. Ask all the members of Napoleon's soccer team. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
No, that would skew it in Blair's favor. Probably. That's not a representative sample. Ask all the members of Napoleon's soccer team. Same thing. They're likely to go Napoleon's way, or maybe they don't like Napoleon. Maybe they'll go against Napoleon, but either way, this seems like a skewed sample. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
Ask all the members of Napoleon's soccer team. Same thing. They're likely to go Napoleon's way, or maybe they don't like Napoleon. Maybe they'll go against Napoleon, but either way, this seems like a skewed sample. Put the names of all the students in a hat and draw 50 names. Ask those students whose names are drawn. Well, this seems like a nice random sample that could be nicely representative of the entire population. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
Maybe they'll go against Napoleon, but either way, this seems like a skewed sample. Put the names of all the students in a hat and draw 50 names. Ask those students whose names are drawn. Well, this seems like a nice random sample that could be nicely representative of the entire population. Ask all students whose names begin with N or B. Well, this could be perceived as kind of random, but notice N is the same starting letter as Napoleon. B is the same starting letter as Blair. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
Well, this seems like a nice random sample that could be nicely representative of the entire population. Ask all students whose names begin with N or B. Well, this could be perceived as kind of random, but notice N is the same starting letter as Napoleon. B is the same starting letter as Blair. You might say, well, that's fair. You're doing it for each of their letters, but maybe there's like 10 people whose names start with an N and only two people whose names start with a B. Once again, you're not even getting a large sample. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
B is the same starting letter as Blair. You might say, well, that's fair. You're doing it for each of their letters, but maybe there's like 10 people whose names start with an N and only two people whose names start with a B. Once again, you're not even getting a large sample. And then on top of that, maybe there's some type of people with the same starting letters somehow like each other more. So I would steer clear of this one. Ask every student in the class. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
Once again, you're not even getting a large sample. And then on top of that, maybe there's some type of people with the same starting letters somehow like each other more. So I would steer clear of this one. Ask every student in the class. Well, that would work. You know, there's 300 classmates. That might not be that time consuming, and that would be a pretty good. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
Ask every student in the class. Well, that would work. You know, there's 300 classmates. That might not be that time consuming, and that would be a pretty good. You can't get a better sample than asking everyone in the population. Which of the following methods of surveying your classmates will allow Mimi to make a valid conclusion about which candidate is in the lead? Well, that's a pretty good conclusion. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
That might not be that time consuming, and that would be a pretty good. You can't get a better sample than asking everyone in the population. Which of the following methods of surveying your classmates will allow Mimi to make a valid conclusion about which candidate is in the lead? Well, that's a pretty good conclusion. People might change their minds, so it's not a done deal, but you can't get a better sample size than the entire population. Assign numbers to each student in the class and use a computer program to generate 50 random numbers between 1 and 300. Ask those students whose numbers are selected. | Reasonable samples Statistical studies Probability and Statistics Khan Academy.mp3 |
So I have a box and whiskers plot showing us the ages of students at a party. And what I'm hoping to do in this video is get a little bit of practice interpreting this. And what I have here are five different statements. And I want you to look at these statements. Pause the video, look at these statements, and think about which of these, based on the information in the box and whiskers plot, which of these are for sure true, which of these are for sure false, and which of these we don't have enough information. It could go either way. All right, so let's work through these. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
And I want you to look at these statements. Pause the video, look at these statements, and think about which of these, based on the information in the box and whiskers plot, which of these are for sure true, which of these are for sure false, and which of these we don't have enough information. It could go either way. All right, so let's work through these. So the first statement is that all of the students are less than 17 years old. Well, we see right over here that the maximum age, that's the right end of this right whisker, is 16. So it is the case that all of the students are less than 17 years old. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
All right, so let's work through these. So the first statement is that all of the students are less than 17 years old. Well, we see right over here that the maximum age, that's the right end of this right whisker, is 16. So it is the case that all of the students are less than 17 years old. So this is definitely going to be true. The next statement, at least 75% of the students are 10 years old or older. So when you look at this, this feels right because 10 is the value, that is at the beginning of the second quartile. | Interpreting box plots Data and statistics 6th grade Khan Academy.mp3 |
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