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This is a small chocolate. Small, small chocolate. Small chocolate. What is this one? This is going to be a small, a small strawberry. A small strawberry. And you could just keep constructing like this, where everything in this row, this is all about small, whoops. | Compound sample spaces Statistics and probability 7th grade Khan Academy.mp3 |
What is this one? This is going to be a small, a small strawberry. A small strawberry. And you could just keep constructing like this, where everything in this row, this is all about small, whoops. Let me do the, this is small. I'm having trouble changing colors. All right. | Compound sample spaces Statistics and probability 7th grade Khan Academy.mp3 |
And you could just keep constructing like this, where everything in this row, this is all about small, whoops. Let me do the, this is small. I'm having trouble changing colors. All right. There is a small, and then this is a small, this is a small vanilla. This color changing is really, it's a difficult thing. Small vanilla. | Compound sample spaces Statistics and probability 7th grade Khan Academy.mp3 |
All right. There is a small, and then this is a small, this is a small vanilla. This color changing is really, it's a difficult thing. Small vanilla. And all of these, these are all, this would be a medium chocolate, medium strawberry, medium vanilla, large chocolate, large strawberry, large vanilla. And once again, you have nine outcomes. This is another way to think about all of the possible outcomes. | Compound sample spaces Statistics and probability 7th grade Khan Academy.mp3 |
Small vanilla. And all of these, these are all, this would be a medium chocolate, medium strawberry, medium vanilla, large chocolate, large strawberry, large vanilla. And once again, you have nine outcomes. This is another way to think about all of the possible outcomes. When you're looking at these two ways in which my cupcakes could vary. Another way, a third way that you could do it is you could literally just construct a table. Well, you could say, okay, I could have a chocolate, actually I'm going to use the letters again. | Compound sample spaces Statistics and probability 7th grade Khan Academy.mp3 |
This is another way to think about all of the possible outcomes. When you're looking at these two ways in which my cupcakes could vary. Another way, a third way that you could do it is you could literally just construct a table. Well, you could say, okay, I could have a chocolate, actually I'm going to use the letters again. So let's say, let me make, this is the flavor column, and then this is the size column. Size column. And so you could say, I could have a chocolate that is, so let's see, there's three types of chocolate that I could have. | Compound sample spaces Statistics and probability 7th grade Khan Academy.mp3 |
Well, you could say, okay, I could have a chocolate, actually I'm going to use the letters again. So let's say, let me make, this is the flavor column, and then this is the size column. Size column. And so you could say, I could have a chocolate that is, so let's see, there's three types of chocolate that I could have. They're, and they could be, they could be small, medium, or large. You could say there is three types of, three types of strawberry. It could be small, medium, or large. | Compound sample spaces Statistics and probability 7th grade Khan Academy.mp3 |
And so you could say, I could have a chocolate that is, so let's see, there's three types of chocolate that I could have. They're, and they could be, they could be small, medium, or large. You could say there is three types of, three types of strawberry. It could be small, medium, or large. So let me write that in. Small, medium, large. Or you could say, well, there's three types of vanilla. | Compound sample spaces Statistics and probability 7th grade Khan Academy.mp3 |
It could be small, medium, or large. So let me write that in. Small, medium, large. Or you could say, well, there's three types of vanilla. There's three types of, color changing again, three types of vanilla. Once again, it could be small, medium, or large. So you have these nine possibilities. | Compound sample spaces Statistics and probability 7th grade Khan Academy.mp3 |
Or you could say, well, there's three types of vanilla. There's three types of, color changing again, three types of vanilla. Once again, it could be small, medium, or large. So you have these nine possibilities. Now, the sample space, the sample space isn't telling you if they're equally likely or not. It's just telling you if you're going to do an experiment, what are all the different possibilities, the possible outcomes for that experiment. Now, in the case where they are equally likely, it can be very, very useful. | Compound sample spaces Statistics and probability 7th grade Khan Academy.mp3 |
So you have these nine possibilities. Now, the sample space, the sample space isn't telling you if they're equally likely or not. It's just telling you if you're going to do an experiment, what are all the different possibilities, the possible outcomes for that experiment. Now, in the case where they are equally likely, it can be very, very useful. Because you could say, you could do something like, if you said that, okay, it's equally likely to pick any one of these nine outcomes, you could say, well, what's the probability of, what's the probability of getting a, something that is either small or chocolate? And so you could see, well, how many of those events out of the total actually meet that constraint. But we'll do more of that in future videos. | Compound sample spaces Statistics and probability 7th grade Khan Academy.mp3 |
Now, in the case where they are equally likely, it can be very, very useful. Because you could say, you could do something like, if you said that, okay, it's equally likely to pick any one of these nine outcomes, you could say, well, what's the probability of, what's the probability of getting a, something that is either small or chocolate? And so you could see, well, how many of those events out of the total actually meet that constraint. But we'll do more of that in future videos. That's just a little bit of a clue of why we even care about things like sample spaces, especially sample spaces like this, where we're looking along two ways, or multiple ways, that something can vary. And these types of sample spaces in particular are called compound sample spaces. So these right over here, this is a compound sample space, because we're looking at two different ways that it can vary, not just to heads or tails, it can vary by size or by flavor. | Compound sample spaces Statistics and probability 7th grade Khan Academy.mp3 |
Seven teachers said language, three teachers said history, nine teachers said geometry, one teacher said chemistry, zero teachers said physics. Create a bar chart showing everyone's favorite courses. So we've got the bar chart right over here, and let's see what we need to plot. So it said zero teachers said physics, which is surprising to me, because since physics is arguably my favorite course. But let's plot what the data has. So physics, so right now it looks like it's halfway between zero and one, so I actually have to bring the physics down to zero. See chemistry, they said, let's see, one teacher said chemistry, so we gotta bring chemistry up to one. | Creating a bar chart Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So it said zero teachers said physics, which is surprising to me, because since physics is arguably my favorite course. But let's plot what the data has. So physics, so right now it looks like it's halfway between zero and one, so I actually have to bring the physics down to zero. See chemistry, they said, let's see, one teacher said chemistry, so we gotta bring chemistry up to one. Now, nine teachers said geometry. So geometry, let's bring that up to nine. One teacher said chemistry, oh, I already read that. | Creating a bar chart Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
See chemistry, they said, let's see, one teacher said chemistry, so we gotta bring chemistry up to one. Now, nine teachers said geometry. So geometry, let's bring that up to nine. One teacher said chemistry, oh, I already read that. History, history, three teachers said history, so let's bring history up to three. And then language, seven teachers said language. So let's move this up to seven. | Creating a bar chart Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
And we record the number of cars each of the 11 salespeople sold in the past week. So this is, you know, that's one of them, two, three, four, five, six, seven, eight, nine, ten, eleven. So we record how much each of them sold. So maybe one person sold, maybe they sold five cars. Maybe the next person sold seven cars. Maybe the next person sold ten cars. And so that's how we would do it. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
So maybe one person sold, maybe they sold five cars. Maybe the next person sold seven cars. Maybe the next person sold ten cars. And so that's how we would do it. So we're going to record all of that. But then we're told that the median car sold is six. So if we ordered all of these numbers, so let's just assume that we ordered them all from least to greatest. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
And so that's how we would do it. So we're going to record all of that. But then we're told that the median car sold is six. So if we ordered all of these numbers, so let's just assume that we ordered them all from least to greatest. So maybe, and I'm just making up numbers here, maybe this is four, this is four, maybe this is five, five, five, six, seven, seven, eight, nine, ten. The median car sold means that the middle number here is going to be six. Now we don't know whether all of these other numbers are the actual numbers, I just made those up, but we know that the middle number needs to be six. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
So if we ordered all of these numbers, so let's just assume that we ordered them all from least to greatest. So maybe, and I'm just making up numbers here, maybe this is four, this is four, maybe this is five, five, five, six, seven, seven, eight, nine, ten. The median car sold means that the middle number here is going to be six. Now we don't know whether all of these other numbers are the actual numbers, I just made those up, but we know that the middle number needs to be six. So let me just fill out only the middle number. So the middle number, if you have 11 data points, the middle number is going to have five on either side. So that's that one right over there. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
Now we don't know whether all of these other numbers are the actual numbers, I just made those up, but we know that the middle number needs to be six. So let me just fill out only the middle number. So the middle number, if you have 11 data points, the middle number is going to have five on either side. So that's that one right over there. One, two, three, four, five. One, two, three, four, five. So we know that the median, we know the median, actually let me do that in purple color, the median is going to be six. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
So that's that one right over there. One, two, three, four, five. One, two, three, four, five. So we know that the median, we know the median, actually let me do that in purple color, the median is going to be six. Now the other thing we know is that the range of cars sold is four. And let's remind ourselves, the range is the maximum number of cars sold minus the minimum number of cars sold. And if we have sorted all of these out, this, if we take, if this is our max right over here, because we've sorted them all out, and this is our min right over here, our range is going to be our max minus our min. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
So we know that the median, we know the median, actually let me do that in purple color, the median is going to be six. Now the other thing we know is that the range of cars sold is four. And let's remind ourselves, the range is the maximum number of cars sold minus the minimum number of cars sold. And if we have sorted all of these out, this, if we take, if this is our max right over here, because we've sorted them all out, and this is our min right over here, our range is going to be our max minus our min. So let me write that down. So you take your min and your max, you have your range, range is equal to the maximum minus the minimum. And they're telling us that that is four, this range is four. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
And if we have sorted all of these out, this, if we take, if this is our max right over here, because we've sorted them all out, and this is our min right over here, our range is going to be our max minus our min. So let me write that down. So you take your min and your max, you have your range, range is equal to the maximum minus the minimum. And they're telling us that that is four, this range is four. So the maximum minus the minimum, the difference between the number of cars that the most productive salesperson sold and the number of cars that the least productive salesperson sold, that difference is going to be four. So given all of those assumptions, giving all of that information, I'm now going to give you a statement. And our challenge is, if we assume everything I just said is true, is this statement going to be true? | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
And they're telling us that that is four, this range is four. So the maximum minus the minimum, the difference between the number of cars that the most productive salesperson sold and the number of cars that the least productive salesperson sold, that difference is going to be four. So given all of those assumptions, giving all of that information, I'm now going to give you a statement. And our challenge is, if we assume everything I just said is true, is this statement going to be true? Or is this statement going to be false? Or do we not know? So let me write this down. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
And our challenge is, if we assume everything I just said is true, is this statement going to be true? Or is this statement going to be false? Or do we not know? So let me write this down. So is this going to be true? False, false, or do we not know? Do we not have enough information to say it's for sure going to be true, for sure going to be false, or for sure we don't know? | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
So let me write this down. So is this going to be true? False, false, or do we not know? Do we not have enough information to say it's for sure going to be true, for sure going to be false, or for sure we don't know? And so I encourage you to pause the video now and try it out. All right, so let's work through it together. This is kind of a fun puzzle here where we've been given some clues and then we have a statement and we figure out, can we make this statement? | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
Do we not have enough information to say it's for sure going to be true, for sure going to be false, or for sure we don't know? And so I encourage you to pause the video now and try it out. All right, so let's work through it together. This is kind of a fun puzzle here where we've been given some clues and then we have a statement and we figure out, can we make this statement? Can we say it's true? Or can we say for sure that it's false or do we just not know? So a couple of ways to tackle it. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
This is kind of a fun puzzle here where we've been given some clues and then we have a statement and we figure out, can we make this statement? Can we say it's true? Or can we say for sure that it's false or do we just not know? So a couple of ways to tackle it. One way to tackle it is, well look, one of the salespeople must have sold six cars. If the median sold is six, and there's 11 salespeople here, the middle number here, if we order it, one of them must have sold six cars. If we had an even number here, then that would be the average of the two middles, but if we have an odd number here, that is literally the middle number of cars that the middle number of cars sold. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
So a couple of ways to tackle it. One way to tackle it is, well look, one of the salespeople must have sold six cars. If the median sold is six, and there's 11 salespeople here, the middle number here, if we order it, one of them must have sold six cars. If we had an even number here, then that would be the average of the two middles, but if we have an odd number here, that is literally the middle number of cars that the middle number of cars sold. So someone sold six cars. So if we assume, we're trying to find a world where someone sold more than 10 cars. So if someone sold six cars, is there a way with a range of four that someone sold more than 10? | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
If we had an even number here, then that would be the average of the two middles, but if we have an odd number here, that is literally the middle number of cars that the middle number of cars sold. So someone sold six cars. So if we assume, we're trying to find a world where someone sold more than 10 cars. So if someone sold six cars, is there a way with a range of four that someone sold more than 10? Well, let's just think about it. Let's just assume that the min is six. We don't know that the min is six, but let's just try it out. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
So if someone sold six cars, is there a way with a range of four that someone sold more than 10? Well, let's just think about it. Let's just assume that the min is six. We don't know that the min is six, but let's just try it out. If the minimum is six, what's the maximum going to be? Well, remember, the range is equal to, we know that the range is four, the range is equal to the maximum minus the minimum. Maximum minus the minimum. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
We don't know that the min is six, but let's just try it out. If the minimum is six, what's the maximum going to be? Well, remember, the range is equal to, we know that the range is four, the range is equal to the maximum minus the minimum. Maximum minus the minimum. And so the maximum in this case would be 10. Would be 10. Max would be equal to 10. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
Maximum minus the minimum. And so the maximum in this case would be 10. Would be 10. Max would be equal to 10. If the min is equal to six, and instead of colons, I'm going to write an equal sign. If the min is equal to six, then the max at most can be equal to 10. And we can't take a higher min because we know that the six is going to be one of the values. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
Max would be equal to 10. If the min is equal to six, and instead of colons, I'm going to write an equal sign. If the min is equal to six, then the max at most can be equal to 10. And we can't take a higher min because we know that the six is going to be one of the values. We could try a lower minimum value. We could say the minimum value is five or four or three or two or one, but then the maximum would go down even more because remember, the maximum is no more than four larger than the minimum. So if we assume six, and we know that one of the salespeople sold six cars, then the maximum that any of the salespeople could have sold is 10. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
And we can't take a higher min because we know that the six is going to be one of the values. We could try a lower minimum value. We could say the minimum value is five or four or three or two or one, but then the maximum would go down even more because remember, the maximum is no more than four larger than the minimum. So if we assume six, and we know that one of the salespeople sold six cars, then the maximum that any of the salespeople could have sold is 10. And so the statement, at least one of the salespeople sold more than 10 cars, that's got to be false. That's got to be false. Now there's another way that you could think about it. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
So if we assume six, and we know that one of the salespeople sold six cars, then the maximum that any of the salespeople could have sold is 10. And so the statement, at least one of the salespeople sold more than 10 cars, that's got to be false. That's got to be false. Now there's another way that you could think about it. You could assume that someone sold more than 10. You could assume, in the last example or the last way of thinking about it, we assumed that the min was six, but now let's just assume that someone sold, let's just try it out. Let's see if it's possible. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
Now there's another way that you could think about it. You could assume that someone sold more than 10. You could assume, in the last example or the last way of thinking about it, we assumed that the min was six, but now let's just assume that someone sold, let's just try it out. Let's see if it's possible. Let's just assume that the max is 11, that someone sold more than 10 cars. If the max is equal to 11, what's the min going to need to be? Well, we just have to remind ourselves. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
Let's see if it's possible. Let's just assume that the max is 11, that someone sold more than 10 cars. If the max is equal to 11, what's the min going to need to be? Well, we just have to remind ourselves. Range is equal to max minus min. So four is equal to the maximum, 11 minus the minimum. So 11 minus what is equal to four? | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
Well, we just have to remind ourselves. Range is equal to max minus min. So four is equal to the maximum, 11 minus the minimum. So 11 minus what is equal to four? Well, 11 minus seven is equal to four. So if we assume that the maximum is 11, then the minimum is going to have to be, the minimum is going to have to be equal to seven. Now can the minimum be seven when your median is six? | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
So 11 minus what is equal to four? Well, 11 minus seven is equal to four. So if we assume that the maximum is 11, then the minimum is going to have to be, the minimum is going to have to be equal to seven. Now can the minimum be seven when your median is six? No. If your median is six, that means you have, that means if you have an odd number of data points, that means one of the data points is six, and if you had an even number of data points, that means that the middle two are going to average out to be six, which means you even have data points that are less than six. So your minimum can't be seven. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
Now can the minimum be seven when your median is six? No. If your median is six, that means you have, that means if you have an odd number of data points, that means one of the data points is six, and if you had an even number of data points, that means that the middle two are going to average out to be six, which means you even have data points that are less than six. So your minimum can't be seven. You're going to have a value that is at least as low, at least as low as six. So your assumption can't be true, can't be true. So once again, the assumption based on the statement can't be true. | Median and range puzzle Data and statistics 6th grade Khan Academy.mp3 |
So they say the probability, I'll just say P for probability, the probability of picking a yellow marble. And so this is sometimes the event in question right over here, is picking the yellow marble. I'll even write down the word picking. And when you say probability, it's really just a way of measuring the likelihood that something is going to happen. And the way we're going to think about it is how many of the outcomes from this trial, from this picking a marble out of a bag, how many meet our constraints, satisfy this event, and how many possible outcomes are there? So let me write the possible outcomes right over here. So possible outcomes. | Finding probability example Probability and Statistics Khan Academy.mp3 |
And when you say probability, it's really just a way of measuring the likelihood that something is going to happen. And the way we're going to think about it is how many of the outcomes from this trial, from this picking a marble out of a bag, how many meet our constraints, satisfy this event, and how many possible outcomes are there? So let me write the possible outcomes right over here. So possible outcomes. And you'll see it's actually a very straightforward idea, but I'll just make sure that we understand all the words that people might say. So the set of all the possible outcomes, well, there's three yellow marbles, so I could pick that yellow marble, that yellow marble, or that yellow marble. These are clearly all yellow. | Finding probability example Probability and Statistics Khan Academy.mp3 |
So possible outcomes. And you'll see it's actually a very straightforward idea, but I'll just make sure that we understand all the words that people might say. So the set of all the possible outcomes, well, there's three yellow marbles, so I could pick that yellow marble, that yellow marble, or that yellow marble. These are clearly all yellow. There's two red marbles in the bag, so I could pick that red marble or that red marble. There's two green marbles in the bag, so I could pick that green marble or that green marble. And then there's one blue marble in the bag. | Finding probability example Probability and Statistics Khan Academy.mp3 |
These are clearly all yellow. There's two red marbles in the bag, so I could pick that red marble or that red marble. There's two green marbles in the bag, so I could pick that green marble or that green marble. And then there's one blue marble in the bag. So this is all the possible outcomes, and sometimes this is referred to as the sample space. Sample, the set of all the possible outcomes. Fancy word for just a simple idea, that the sample space, when I pick something out of the bag and that picking out of the bag is called a trial, there's eight possible things I can do. | Finding probability example Probability and Statistics Khan Academy.mp3 |
And then there's one blue marble in the bag. So this is all the possible outcomes, and sometimes this is referred to as the sample space. Sample, the set of all the possible outcomes. Fancy word for just a simple idea, that the sample space, when I pick something out of the bag and that picking out of the bag is called a trial, there's eight possible things I can do. So when I think about the probability of picking a yellow marble, I want to think about, well, what are all of the possibilities? Well, there's eight possibilities. Eight possibilities for my trial. | Finding probability example Probability and Statistics Khan Academy.mp3 |
Fancy word for just a simple idea, that the sample space, when I pick something out of the bag and that picking out of the bag is called a trial, there's eight possible things I can do. So when I think about the probability of picking a yellow marble, I want to think about, well, what are all of the possibilities? Well, there's eight possibilities. Eight possibilities for my trial. So the number of outcomes, number of possible outcomes, you could view it as the size of the sample space, number of possible outcomes. And it's as simple as saying, well, look, I have eight marbles. And then you say, well, how many of those marbles meet my constraint that satisfy this event here? | Finding probability example Probability and Statistics Khan Academy.mp3 |
Eight possibilities for my trial. So the number of outcomes, number of possible outcomes, you could view it as the size of the sample space, number of possible outcomes. And it's as simple as saying, well, look, I have eight marbles. And then you say, well, how many of those marbles meet my constraint that satisfy this event here? Well, there's three marbles that satisfy my event. There's three outcomes that will allow this event to occur, I guess is one way to say it. So there's three right over here. | Finding probability example Probability and Statistics Khan Academy.mp3 |
And then you say, well, how many of those marbles meet my constraint that satisfy this event here? Well, there's three marbles that satisfy my event. There's three outcomes that will allow this event to occur, I guess is one way to say it. So there's three right over here. So number that satisfy the event or the constraint right over here. So it's very simple ideas. Many times, the words make them more complicated than they need to. | Finding probability example Probability and Statistics Khan Academy.mp3 |
So there's three right over here. So number that satisfy the event or the constraint right over here. So it's very simple ideas. Many times, the words make them more complicated than they need to. If I say, what's the probability of picking a yellow marble? Well, how many different types of marbles can I pick? Well, there's eight different marbles I could pick. | Finding probability example Probability and Statistics Khan Academy.mp3 |
Our goal is to simplify this expression for the squared error between those end points. Just to remind ourselves what we're doing. We have these end points and we're taking the sum of the squared error between each of those end points and our actual line y equals mx plus b. And we get this expression over here which we've been simplifying over the last couple of videos. We're going to try to simplify the expression as much as possible and then we're going to try to optimize, we're going to try to minimize this expression or find the m and b values that minimize it or that's, I guess you can call it, the best fitting line. Now to do that, it looks like we were just getting, making the algebra even hairier and hairier but this next step is going to simplify things a good bit. So just to show you that, what is, if I want to take the mean of all of the squared values of the y's. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
And we get this expression over here which we've been simplifying over the last couple of videos. We're going to try to simplify the expression as much as possible and then we're going to try to optimize, we're going to try to minimize this expression or find the m and b values that minimize it or that's, I guess you can call it, the best fitting line. Now to do that, it looks like we were just getting, making the algebra even hairier and hairier but this next step is going to simplify things a good bit. So just to show you that, what is, if I want to take the mean of all of the squared values of the y's. So that would be this, that would be y1 squared plus y2 squared plus all the way to yn squared. So I've summed n values, n squared values and then I want to divide it by n, since there are n values here, and this is the mean of the y's squared. That's how we can denote it just like that. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
So just to show you that, what is, if I want to take the mean of all of the squared values of the y's. So that would be this, that would be y1 squared plus y2 squared plus all the way to yn squared. So I've summed n values, n squared values and then I want to divide it by n, since there are n values here, and this is the mean of the y's squared. That's how we can denote it just like that. That is the mean of the y squared. Or if you multiply both sides of this equation by n, you get y1 squared plus y2 squared plus all the way to, all the way to yn squared, yn squared is equal to n, is equal to n, let me do this in different colors, is equal to, so this is, this is equal to n, this n times the mean of the squared values of y. And notice, this is exactly, this is exactly what we have over here. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
That's how we can denote it just like that. That is the mean of the y squared. Or if you multiply both sides of this equation by n, you get y1 squared plus y2 squared plus all the way to, all the way to yn squared, yn squared is equal to n, is equal to n, let me do this in different colors, is equal to, so this is, this is equal to n, this n times the mean of the squared values of y. And notice, this is exactly, this is exactly what we have over here. That is n times the mean of the squared values of y or the mean of the y squareds. That's exactly what that is and we can do that with each of these terms. What is, what is x1, x1, y1 plus x2, y2 plus all the way to, all the way to xn, yn. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
And notice, this is exactly, this is exactly what we have over here. That is n times the mean of the squared values of y or the mean of the y squareds. That's exactly what that is and we can do that with each of these terms. What is, what is x1, x1, y1 plus x2, y2 plus all the way to, all the way to xn, yn. Well if we take this whole sum and we divide it by n terms, this is going to be the mean value for x times y. For each of those points, you multiply x times y and you find the mean of all of those products. That's exactly what this is. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
What is, what is x1, x1, y1 plus x2, y2 plus all the way to, all the way to xn, yn. Well if we take this whole sum and we divide it by n terms, this is going to be the mean value for x times y. For each of those points, you multiply x times y and you find the mean of all of those products. That's exactly what this is. Well once again, you multiply both sides of this equation by n and you get, you get x1, y1 plus x2, y2 plus all the way, all the way to xn, yn is equal to, is equal to n times the mean of xy's, the mean of the xy's. So n times the mean of the xy's and I think you see where this is going. This term right here, so this term right here is going to be equal to n times the mean of the products of xy. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
That's exactly what this is. Well once again, you multiply both sides of this equation by n and you get, you get x1, y1 plus x2, y2 plus all the way, all the way to xn, yn is equal to, is equal to n times the mean of xy's, the mean of the xy's. So n times the mean of the xy's and I think you see where this is going. This term right here, so this term right here is going to be equal to n times the mean of the products of xy. This term right here is n times the mean of the y values. That's what this term right here is. And then this term right here is n times the mean of the x, the mean of the x's squared. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
This term right here, so this term right here is going to be equal to n times the mean of the products of xy. This term right here is n times the mean of the y values. That's what this term right here is. And then this term right here is n times the mean of the x, the mean of the x's squared. The mean of the x squared values, I should say. This term right here is the mean of the x's times n. Right, if you divided this by n, you'd get the mean. Since we're not dividing it by n, this is the mean times n. And then this is obviously, we don't have to simplify anything. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
And then this term right here is n times the mean of the x, the mean of the x's squared. The mean of the x squared values, I should say. This term right here is the mean of the x's times n. Right, if you divided this by n, you'd get the mean. Since we're not dividing it by n, this is the mean times n. And then this is obviously, we don't have to simplify anything. So let's rewrite everything using our new notation, knowing that these are the means of these, you know, of y squared, of xy and all of that. So our squared error of the line, our squared error to the line, from the sum of the squared error to the line from the end points is going to equal to, this term right here is n, let me color code it a little bit. This term right here is n times the mean of the y squared values. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
Since we're not dividing it by n, this is the mean times n. And then this is obviously, we don't have to simplify anything. So let's rewrite everything using our new notation, knowing that these are the means of these, you know, of y squared, of xy and all of that. So our squared error of the line, our squared error to the line, from the sum of the squared error to the line from the end points is going to equal to, this term right here is n, let me color code it a little bit. This term right here is n times the mean of the y squared values. This term right here is, I'll do it all in this green color, is equal to negative 2m, that's just that right there, times n times the mean of the xy values, the arithmetic mean. And then we have this term over here. I think you can appreciate this is simplifying the algebraic expression a good bit. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
This term right here is n times the mean of the y squared values. This term right here is, I'll do it all in this green color, is equal to negative 2m, that's just that right there, times n times the mean of the xy values, the arithmetic mean. And then we have this term over here. I think you can appreciate this is simplifying the algebraic expression a good bit. This term right over here is going to be minus 2b, minus 2bn, times the mean of the y values. And then we have plus, we have this term right here, plus m squared times n times the mean of the x squared values. And then we have, almost there, home stretch, we have this over here which is plus 2mb times n times the mean of the x values. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
I think you can appreciate this is simplifying the algebraic expression a good bit. This term right over here is going to be minus 2b, minus 2bn, times the mean of the y values. And then we have plus, we have this term right here, plus m squared times n times the mean of the x squared values. And then we have, almost there, home stretch, we have this over here which is plus 2mb times n times the mean of the x values. And then finally we have plus nb squared. So really in the last two, three videos, all we've done is we've simplified the expression for the sum of the squared differences from those end points to this line y equals mx plus b, right over here. So that is kind of, we're finished the hardcore algebra stage. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
And then we have, almost there, home stretch, we have this over here which is plus 2mb times n times the mean of the x values. And then finally we have plus nb squared. So really in the last two, three videos, all we've done is we've simplified the expression for the sum of the squared differences from those end points to this line y equals mx plus b, right over here. So that is kind of, we're finished the hardcore algebra stage. The next stage we actually want to optimize. We actually want to optimize this. We actually want to optimize, or actually maybe a better way to talk about it, we want to minimize this expression right over here. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
So that is kind of, we're finished the hardcore algebra stage. The next stage we actually want to optimize. We actually want to optimize this. We actually want to optimize, or actually maybe a better way to talk about it, we want to minimize this expression right over here. We want to find the m and the b values that minimize it. And to help visualize it, we're going to start breaking into a little bit of three-dimensional calculus here. But hopefully it won't be too daunting. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
We actually want to optimize, or actually maybe a better way to talk about it, we want to minimize this expression right over here. We want to find the m and the b values that minimize it. And to help visualize it, we're going to start breaking into a little bit of three-dimensional calculus here. But hopefully it won't be too daunting. If you've done any partial derivatives it won't be difficult. This is a surface. If you view that you have the x and y data points, everything here is a constant except for the m's and the b's. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
But hopefully it won't be too daunting. If you've done any partial derivatives it won't be difficult. This is a surface. If you view that you have the x and y data points, everything here is a constant except for the m's and the b's. We're assuming that we have the x's and y's. So we can figure out the mean of the squared values of y, the mean of the xy product, the mean of the y's, the mean of the x squareds. We assume that those are all actual numbers. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
If you view that you have the x and y data points, everything here is a constant except for the m's and the b's. We're assuming that we have the x's and y's. So we can figure out the mean of the squared values of y, the mean of the xy product, the mean of the y's, the mean of the x squareds. We assume that those are all actual numbers. So this expression right here is going to be, it's actually going to be a surface in three dimensions. So you can imagine this right here, that is the m axis. This right here is the b axis. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
We assume that those are all actual numbers. So this expression right here is going to be, it's actually going to be a surface in three dimensions. So you can imagine this right here, that is the m axis. This right here is the b axis. That is the b axis. Let me continue the m axis and let me continue the b axis. And then you can imagine the vertical axis here to be the squared error. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
This right here is the b axis. That is the b axis. Let me continue the m axis and let me continue the b axis. And then you can imagine the vertical axis here to be the squared error. This is the squared error of the line axis. So for any combination of m and b's, if you're in the mb plane, you pick some combination of m and b, you put it into this expression for the squared error of the line, it will give you a point. If you do that for all of the combinations of m's and b's, you're going to get a surface. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
And then you can imagine the vertical axis here to be the squared error. This is the squared error of the line axis. So for any combination of m and b's, if you're in the mb plane, you pick some combination of m and b, you put it into this expression for the squared error of the line, it will give you a point. If you do that for all of the combinations of m's and b's, you're going to get a surface. The surface is going to look something like this. I'm going to try my best to draw it. It's going to look something like this. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
If you do that for all of the combinations of m's and b's, you're going to get a surface. The surface is going to look something like this. I'm going to try my best to draw it. It's going to look something like this. You can almost imagine it as somewhat of a kind of a bowl. Or you can even think of it as a three-dimensional parabola, if you want to think of it that way. Instead of a parabola that just goes like this, if you were to kind of rotate it around and distort it a little bit, you would get this thing that looks kind of like a cup or a thimble or whatever. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
It's going to look something like this. You can almost imagine it as somewhat of a kind of a bowl. Or you can even think of it as a three-dimensional parabola, if you want to think of it that way. Instead of a parabola that just goes like this, if you were to kind of rotate it around and distort it a little bit, you would get this thing that looks kind of like a cup or a thimble or whatever. What we want to do is we want to find the m and b values that minimize. Notice this is a three-dimensional surface. I don't know if I'm doing justice to it. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
Instead of a parabola that just goes like this, if you were to kind of rotate it around and distort it a little bit, you would get this thing that looks kind of like a cup or a thimble or whatever. What we want to do is we want to find the m and b values that minimize. Notice this is a three-dimensional surface. I don't know if I'm doing justice to it. Let me see if I can imagine a three-dimensional surface that looks something like this. This is the back part that you're not seeing. That's the inside of our three-dimensional surface. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
I don't know if I'm doing justice to it. Let me see if I can imagine a three-dimensional surface that looks something like this. This is the back part that you're not seeing. That's the inside of our three-dimensional surface. We want to find the m and b values that minimize the value on the surface. There's some m and b value right over here that minimizes it. To do that, and I'll actually do the calculation in the next video, we're going to find the partial derivative of this with respect to m, and we're going to find the partial derivative of this with respect to b and set both of them equal to zero. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
That's the inside of our three-dimensional surface. We want to find the m and b values that minimize the value on the surface. There's some m and b value right over here that minimizes it. To do that, and I'll actually do the calculation in the next video, we're going to find the partial derivative of this with respect to m, and we're going to find the partial derivative of this with respect to b and set both of them equal to zero. Because at this minimum point, I guess you could say in three dimensions, this minimum point on the surface is going to occur is when the slope with respect to m and the slope with respect to b is zero. At that point, the partial derivative of our squared error with respect to m is going to be equal to zero, and the partial derivative of our squared error with respect to b is going to be equal to zero. All we're going to do in the next video is take the partial derivative of this expression with respect to m, set that equal to zero, and the partial derivative of this with respect to b, set that equal to zero, and then we're ready to solve for the m and the b, or the particular m and b. | Proof (part 2) minimizing squared error to regression line Khan Academy.mp3 |
And we want to get a sense of how these students feel about the quality of math instruction at this school. So we construct a survey, and we just need to decide who are we going to get to actually answer this survey. One option is to just go to every member of the population, but let's just say it's a really large school. Let's say we're a college, and there's 10,000 people in the college. We say, well, we can't just talk to everyone. So instead we say, let's sample this population to get an indication of how the entire school feels. So we are going to sample it. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
Let's say we're a college, and there's 10,000 people in the college. We say, well, we can't just talk to everyone. So instead we say, let's sample this population to get an indication of how the entire school feels. So we are going to sample it. We're going to sample that population. Now, in order to avoid having bias in our response, in order for it to have the best chance of it being indicative of the entire population, we want our sample to be random. So our sample could either be random, random, or not random, not random. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
So we are going to sample it. We're going to sample that population. Now, in order to avoid having bias in our response, in order for it to have the best chance of it being indicative of the entire population, we want our sample to be random. So our sample could either be random, random, or not random, not random. And it might seem at first pretty straightforward to do a random sample, but when you actually get down to it, it's not always as straightforward as you would think. So one type of random sample is just a simple random sample. So simple, simple, random, random sample. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
So our sample could either be random, random, or not random, not random. And it might seem at first pretty straightforward to do a random sample, but when you actually get down to it, it's not always as straightforward as you would think. So one type of random sample is just a simple random sample. So simple, simple, random, random sample. And this is saying, all right, let me maybe assign a number to every person in the school. Maybe they already have a student ID number. And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
So simple, simple, random, random sample. And this is saying, all right, let me maybe assign a number to every person in the school. Maybe they already have a student ID number. And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students. So let's say there's a sample of 100 students that I'm going to apply the survey to. So that would be a simple random sample. We are just going into this whole population and randomly, let me just draw this. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And I'm just going to get a computer, a random number generator, to generate the 100 people, the 100 students. So let's say there's a sample of 100 students that I'm going to apply the survey to. So that would be a simple random sample. We are just going into this whole population and randomly, let me just draw this. So this is the population. We are just randomly picking people out. And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
We are just going into this whole population and randomly, let me just draw this. So this is the population. We are just randomly picking people out. And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students. Now that's pretty good. It's unlikely that you're going to have bias from this sample. But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And we know it's random because a random number generator, or we have a string of numbers or something like that, that is allowing us to pick these students. Now that's pretty good. It's unlikely that you're going to have bias from this sample. But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors. And that's a possibility. So even though you're taking a simple random sample and it's truly random, once again, it's some probability that's not indicative of the entire population. And so to mitigate that, there are other techniques at our disposal. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
But there is some probability that just by chance, your random number generator just happened to select maybe a disproportionate number of boys over girls, or a disproportionate number of freshmen, or a disproportionate number of engineering majors versus English majors. And that's a possibility. So even though you're taking a simple random sample and it's truly random, once again, it's some probability that's not indicative of the entire population. And so to mitigate that, there are other techniques at our disposal. One technique is a stratified sample. Stratified. And so this is the idea of taking our entire population and essentially stratifying it. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And so to mitigate that, there are other techniques at our disposal. One technique is a stratified sample. Stratified. And so this is the idea of taking our entire population and essentially stratifying it. So let's say we take that same population, we take that same population, I'll draw it as a square here just for convenience, and we're gonna stratify it by, let's say we're concerned that we get an appropriate sample of freshmen, sophomores, juniors, and seniors. So we'll stratify it by freshmen, sophomores, juniors, and seniors. And then we sample 25 from each of these groups. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And so this is the idea of taking our entire population and essentially stratifying it. So let's say we take that same population, we take that same population, I'll draw it as a square here just for convenience, and we're gonna stratify it by, let's say we're concerned that we get an appropriate sample of freshmen, sophomores, juniors, and seniors. So we'll stratify it by freshmen, sophomores, juniors, and seniors. And then we sample 25 from each of these groups. So these are the stratifications. This is freshmen, sophomore, juniors, and seniors. And instead of just sampling 100 out of the entire pool, we sample 25 from each of these. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And then we sample 25 from each of these groups. So these are the stratifications. This is freshmen, sophomore, juniors, and seniors. And instead of just sampling 100 out of the entire pool, we sample 25 from each of these. So just like that. And so that makes sure that you are getting indicative responses from at least all of the different age groups or levels within your university. Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And instead of just sampling 100 out of the entire pool, we sample 25 from each of these. So just like that. And so that makes sure that you are getting indicative responses from at least all of the different age groups or levels within your university. Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school. And there is some probability, if I do 100 random people, it's very likely that it's close to 50-50, but there's some chance just due to randomness that it's disproportionately male or disproportionately female. And that's even possible in the stratified case. And so what you might say is, well, you know what I'm gonna do? | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
Now there might be another issue where you say, well, I'm actually more concerned that we have accurate representation of males and females in the school. And there is some probability, if I do 100 random people, it's very likely that it's close to 50-50, but there's some chance just due to randomness that it's disproportionately male or disproportionately female. And that's even possible in the stratified case. And so what you might say is, well, you know what I'm gonna do? I'm going to, there's a technique called a clustered sample. Let me write this right over here. Clustered, a clustered sample. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And so what you might say is, well, you know what I'm gonna do? I'm going to, there's a technique called a clustered sample. Let me write this right over here. Clustered, a clustered sample. And what we do is we sample groups. Each of those groups, we feel confident, has a good balance of male, females. So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
Clustered, a clustered sample. And what we do is we sample groups. Each of those groups, we feel confident, has a good balance of male, females. So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do. We'll, let's just say that we can split our, let's say we can split our population into groups. Maybe these are classrooms. And each of these classrooms have an even distribution of males and females, or pretty close to even distributions. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
So for example, we might, instead of sampling individuals from the entire population, we might say, look, you know, on Tuesdays and Thursdays, and this, well, even there, as you can tell, this is not a trivial thing to do. We'll, let's just say that we can split our, let's say we can split our population into groups. Maybe these are classrooms. And each of these classrooms have an even distribution of males and females, or pretty close to even distributions. And so what we do is we sample the actual classrooms. So that's why it's called cluster, or cluster technique, or clustered random sample, because we're going to randomly sample our classrooms, each of which have a close, or maybe an exact balance of males and females. So we know that we're gonna get good representation, but we are still sampling. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
And each of these classrooms have an even distribution of males and females, or pretty close to even distributions. And so what we do is we sample the actual classrooms. So that's why it's called cluster, or cluster technique, or clustered random sample, because we're going to randomly sample our classrooms, each of which have a close, or maybe an exact balance of males and females. So we know that we're gonna get good representation, but we are still sampling. We are sampling from the clusters, but then we're gonna survey every single person in each of these clusters, every single person in one of these classrooms. So once again, these are all forms of random surveys, or random samples. You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
So we know that we're gonna get good representation, but we are still sampling. We are sampling from the clusters, but then we're gonna survey every single person in each of these clusters, every single person in one of these classrooms. So once again, these are all forms of random surveys, or random samples. You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster. Now, if these are all random samples, what are the non-random things like? Well, one case of non-random, you could have a voluntary, voluntary survey, or voluntary sample. This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
You have the simple random sample, you can stratify, or you can cluster, and then randomly pick the clusters, and then survey everyone in that cluster. Now, if these are all random samples, what are the non-random things like? Well, one case of non-random, you could have a voluntary, voluntary survey, or voluntary sample. This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey. And that's likely to introduce bias, because you might have, maybe, the students who really like the math instruction at their school, more likely to fill it out. Maybe the students who really don't like it, more likely to fill it out. Maybe it's just the kids who have more time, more likely to fill it out. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
This might just be, you tell every student at the school, hey, here's a web address, if you're interested, come and fill out this survey. And that's likely to introduce bias, because you might have, maybe, the students who really like the math instruction at their school, more likely to fill it out. Maybe the students who really don't like it, more likely to fill it out. Maybe it's just the kids who have more time, more likely to fill it out. So this has a good chance of introducing bias. The students who fill out the survey might be just more skewed one way or the other, because they volunteered for it. Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
Maybe it's just the kids who have more time, more likely to fill it out. So this has a good chance of introducing bias. The students who fill out the survey might be just more skewed one way or the other, because they volunteered for it. Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used. And this might say, well, let's just sample the hundred first students who show up in school. And that's just convenient for me, because I didn't have to do these random numbers, or do the stratification, or doing any of this clustering. But you can understand how this also would introduce bias. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
Another non-random sample would be called a, a, you're introducing bias because of convenience, is a term that's often used. And this might say, well, let's just sample the hundred first students who show up in school. And that's just convenient for me, because I didn't have to do these random numbers, or do the stratification, or doing any of this clustering. But you can understand how this also would introduce bias. Because the first hundred students who show up at school, maybe those are the most diligent students, maybe they all take an early math class that has a very good instructor, or they're all happy about it. Or it might go the other way. The instructor there isn't the best one, and so it might introduce bias the other way. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
But you can understand how this also would introduce bias. Because the first hundred students who show up at school, maybe those are the most diligent students, maybe they all take an early math class that has a very good instructor, or they're all happy about it. Or it might go the other way. The instructor there isn't the best one, and so it might introduce bias the other way. So if you let people volunteer, or you just say, oh, let me go to the first N students, or you say, hey, let me just talk to all the students who happen to be in front of me right now. They might be in front of you out of convenience, but they might not be a true random sample. Now, there's other reasons why you might introduce bias, and it might not be because of the sampling. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
The instructor there isn't the best one, and so it might introduce bias the other way. So if you let people volunteer, or you just say, oh, let me go to the first N students, or you say, hey, let me just talk to all the students who happen to be in front of me right now. They might be in front of you out of convenience, but they might not be a true random sample. Now, there's other reasons why you might introduce bias, and it might not be because of the sampling. You might introduce bias because of the wording of your survey. You could imagine a survey that says, do you consider yourself lucky to get a math education that very few other people in the world have access to? Well, that might bias you to say, well, yeah, I guess I feel lucky. | Techniques for random sampling and avoiding bias Study design AP Statistics Khan Academy.mp3 |
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