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This is the total variation y. Makes sense, if you divide this by n, you actually will get the, I should say this is the total variation in y, if you divide this by n, you're going to get what we typically associate as the variance of y, which is kind of the average square distance. Now we have the total square distance. So what we want to do is how much of this, how much of the total variation y is described by the variation in x? So maybe we can think of it this way, so our denominator, we want what percentage of the total variation in y? So let me write it this way. Let me call this as the squared error from the average. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So what we want to do is how much of this, how much of the total variation y is described by the variation in x? So maybe we can think of it this way, so our denominator, we want what percentage of the total variation in y? So let me write it this way. Let me call this as the squared error from the average. Let me call this, this is equal to the squared error, maybe I'll call this the squared error from the mean of y. And this is really the total variation in y. So let's put that as the denominator. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Let me call this as the squared error from the average. Let me call this, this is equal to the squared error, maybe I'll call this the squared error from the mean of y. And this is really the total variation in y. So let's put that as the denominator. Let's put that as the denominator, the total variation y, which is the squared error from the mean of the y's. Now we want to know what percentage of this is described by the variation in x. Now what is not described by the variation in x? | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So let's put that as the denominator. Let's put that as the denominator, the total variation y, which is the squared error from the mean of the y's. Now we want to know what percentage of this is described by the variation in x. Now what is not described by the variation in x? We want how much is described by the variation in x. But what if we want how much of the total error, how much of the total variation is not described by the line over here, is not described by the regression line. How much of the total data is not? | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Now what is not described by the variation in x? We want how much is described by the variation in x. But what if we want how much of the total error, how much of the total variation is not described by the line over here, is not described by the regression line. How much of the total data is not? Well, we already have a measure for that. We have the squared error of the line. This tells us the square of the distances from each point to our line. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
How much of the total data is not? Well, we already have a measure for that. We have the squared error of the line. This tells us the square of the distances from each point to our line. So it is exactly this measure. It tells us how much of the total variation is not described by the regression line. So if you want to know what percentage of the total variation is not described by the regression line, you would just say, this is the total, it would just be the squared error, the squared error of the line, because this is the total variation not described by the regression line, divided by the total variation. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
This tells us the square of the distances from each point to our line. So it is exactly this measure. It tells us how much of the total variation is not described by the regression line. So if you want to know what percentage of the total variation is not described by the regression line, you would just say, this is the total, it would just be the squared error, the squared error of the line, because this is the total variation not described by the regression line, divided by the total variation. So let me make it clear. This right over here tells us what percentage of variation, of the total variation, is not described by the variation in x, by the variation in x, or by the line, or by the regression line. Regression, by the regression line. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So if you want to know what percentage of the total variation is not described by the regression line, you would just say, this is the total, it would just be the squared error, the squared error of the line, because this is the total variation not described by the regression line, divided by the total variation. So let me make it clear. This right over here tells us what percentage of variation, of the total variation, is not described by the variation in x, by the variation in x, or by the line, or by the regression line. Regression, by the regression line. So to answer our question, what percentage is described by the variation, well, the rest of it has to be described by the variation in x. Because our question is, what percentage of the total variation is described by the variation in x? This is the percentage that is not described. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Regression, by the regression line. So to answer our question, what percentage is described by the variation, well, the rest of it has to be described by the variation in x. Because our question is, what percentage of the total variation is described by the variation in x? This is the percentage that is not described. So if this number right here, if this number is, I don't know, 30%, if 30% of the variation in y is not described by the line, then the remainder will be described by the line. So we can essentially just subtract this from 1. So if we take 1 minus the squared error between our data points and the line, over the squared error between the data points, between the y's and the mean y, we have, we now have a percentage, this actually tells us what percentage of total variation, total variation, is described by the line. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
This is the percentage that is not described. So if this number right here, if this number is, I don't know, 30%, if 30% of the variation in y is not described by the line, then the remainder will be described by the line. So we can essentially just subtract this from 1. So if we take 1 minus the squared error between our data points and the line, over the squared error between the data points, between the y's and the mean y, we have, we now have a percentage, this actually tells us what percentage of total variation, total variation, is described by the line. Is described, is described, you can either view it as described by the line, or by the variation in x. Is described by the variation, by the variation in x. And this number right here, this is called the coefficient of determination. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So if we take 1 minus the squared error between our data points and the line, over the squared error between the data points, between the y's and the mean y, we have, we now have a percentage, this actually tells us what percentage of total variation, total variation, is described by the line. Is described, is described, you can either view it as described by the line, or by the variation in x. Is described by the variation, by the variation in x. And this number right here, this is called the coefficient of determination. This is called the coefficient of determination. It's just what statisticians have decided to name it. Coefficient, coefficient of determination. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
And this number right here, this is called the coefficient of determination. This is called the coefficient of determination. It's just what statisticians have decided to name it. Coefficient, coefficient of determination. Of determination. Determination. And it's also called r squared. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Coefficient, coefficient of determination. Of determination. Determination. And it's also called r squared. You might have even heard that term when people talk about regression. Now, let's think about it. If the standard, if the squared error of the line, if the squared error is really small, if the squared error is really small, what does that mean? | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
And it's also called r squared. You might have even heard that term when people talk about regression. Now, let's think about it. If the standard, if the squared error of the line, if the squared error is really small, if the squared error is really small, what does that mean? It means that these errors, it means that these errors right over here are really small, are really small, which means that the line is a really good fit. Which means that the line, this line over here, it tells us that the line is a really good fit. So if the, let me write it over here. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
If the standard, if the squared error of the line, if the squared error is really small, if the squared error is really small, what does that mean? It means that these errors, it means that these errors right over here are really small, are really small, which means that the line is a really good fit. Which means that the line, this line over here, it tells us that the line is a really good fit. So if the, let me write it over here. If the squared error of the line is small, is small, it tells us that the line is a good fit. Line is a good, it tells us it's a good fit. Now, what would happen over here? | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So if the, let me write it over here. If the squared error of the line is small, is small, it tells us that the line is a good fit. Line is a good, it tells us it's a good fit. Now, what would happen over here? Well, if this number is really small, this is going to be a very small fraction over here. One minus a very small fraction is going to be a pretty large, it's going to be a number close to one. So then, so then we're going to have our r squared will be close, close to one, which tells us that a lot of the variation in y is described by the variation in x, which makes sense because the line is a good fit. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
Now, what would happen over here? Well, if this number is really small, this is going to be a very small fraction over here. One minus a very small fraction is going to be a pretty large, it's going to be a number close to one. So then, so then we're going to have our r squared will be close, close to one, which tells us that a lot of the variation in y is described by the variation in x, which makes sense because the line is a good fit. You take the opposite case. If the squared error of the line is huge, if this number over here is huge, if this number over here is huge, then that means there's a lot of error between the data points and the line. And so if this number is huge, then this number over here is going to be huge. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
So then, so then we're going to have our r squared will be close, close to one, which tells us that a lot of the variation in y is described by the variation in x, which makes sense because the line is a good fit. You take the opposite case. If the squared error of the line is huge, if this number over here is huge, if this number over here is huge, then that means there's a lot of error between the data points and the line. And so if this number is huge, then this number over here is going to be huge. One minus, or it's going to be a percentage close to one, and one minus that is going to be close to zero. And so if this, if the squared error of the line is large, is large, is large, if this is large, this whole thing is going to be close to one. And if this whole thing is close to one, the whole coefficient of determination, the whole r squared is going to be close to zero, which makes sense. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
And so if this number is huge, then this number over here is going to be huge. One minus, or it's going to be a percentage close to one, and one minus that is going to be close to zero. And so if this, if the squared error of the line is large, is large, is large, if this is large, this whole thing is going to be close to one. And if this whole thing is close to one, the whole coefficient of determination, the whole r squared is going to be close to zero, which makes sense. R squared will be close to zero, which makes sense. That tells us that very little of the total variation in y is described by the variation in x, or described by the line. Well, anyway, everything I've been dealing with so far has been a little bit in the abstract. | R-squared or coefficient of determination Regression Probability and Statistics Khan Academy.mp3 |
But he only has enough money to buy at most four packs. Suppose that each pack has probability 0.2 of containing the card Hugo is hoping for. Let the random variable X be the number of packs of cards Hugo buys. Here is the probability distribution for X. So it looks like there is a 0.2 probability that he buys one pack, and that makes sense because that first pack, there is a 0.2 probability that it contains his favorite player's card. And if it does, at that point, he'll just stop. He won't buy any more packs. | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
Here is the probability distribution for X. So it looks like there is a 0.2 probability that he buys one pack, and that makes sense because that first pack, there is a 0.2 probability that it contains his favorite player's card. And if it does, at that point, he'll just stop. He won't buy any more packs. Now what about the probability that he buys two packs? Well, over here, they give it a 0.16, and that makes sense. There is a 0.8 probability that he does not get the card he wants on the first one, and then there's another 0.2 that he gets it on the second one. | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
He won't buy any more packs. Now what about the probability that he buys two packs? Well, over here, they give it a 0.16, and that makes sense. There is a 0.8 probability that he does not get the card he wants on the first one, and then there's another 0.2 that he gets it on the second one. So 0.8 times 0.2 does indeed equal 0.16. But they're not asking us to calculate that. They give it to us. | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
There is a 0.8 probability that he does not get the card he wants on the first one, and then there's another 0.2 that he gets it on the second one. So 0.8 times 0.2 does indeed equal 0.16. But they're not asking us to calculate that. They give it to us. Then the probability that he gets three packs is 0.128, and then they've left blank the probability that he gets four packs. But this is the entire discrete probability distribution because Hugo has to stop at four. Even if he doesn't get the card he wants at four on the fourth pack, he's just going to stop over there. | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
They give it to us. Then the probability that he gets three packs is 0.128, and then they've left blank the probability that he gets four packs. But this is the entire discrete probability distribution because Hugo has to stop at four. Even if he doesn't get the card he wants at four on the fourth pack, he's just going to stop over there. So we could actually figure out this question mark by just realizing that these four probabilities have to add up to one. But let's just first answer the question. Find the indicated probability. | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
Even if he doesn't get the card he wants at four on the fourth pack, he's just going to stop over there. So we could actually figure out this question mark by just realizing that these four probabilities have to add up to one. But let's just first answer the question. Find the indicated probability. What is the probability that X is greater than or equal to two? What is the probability? Remember, X is the number of packs of cards Hugo buys. | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
Find the indicated probability. What is the probability that X is greater than or equal to two? What is the probability? Remember, X is the number of packs of cards Hugo buys. I encourage you to pause the video and try to figure it out. So let's look at the scenarios we're talking about. Probability that our discrete random variable X is greater than or equal to two. | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
Remember, X is the number of packs of cards Hugo buys. I encourage you to pause the video and try to figure it out. So let's look at the scenarios we're talking about. Probability that our discrete random variable X is greater than or equal to two. Well, that's these three scenarios right over here. And so what is their combined probability? Well, you might want to say, hey, we need to figure out what the probability of getting exactly four packs are. | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
Probability that our discrete random variable X is greater than or equal to two. Well, that's these three scenarios right over here. And so what is their combined probability? Well, you might want to say, hey, we need to figure out what the probability of getting exactly four packs are. But we have to remember that these all add up to 100%. And so this right over here is 0.2. And so this is 0.2, the other three combined have to add up to 0.8. | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
Well, you might want to say, hey, we need to figure out what the probability of getting exactly four packs are. But we have to remember that these all add up to 100%. And so this right over here is 0.2. And so this is 0.2, the other three combined have to add up to 0.8. 0.8 plus 0.2 is one, or 100%. So just like that, we know that this is 0.8. If for kicks, we wanted to figure out this question mark right over here, we could just say that, look, have to add up to one. | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
And so this is 0.2, the other three combined have to add up to 0.8. 0.8 plus 0.2 is one, or 100%. So just like that, we know that this is 0.8. If for kicks, we wanted to figure out this question mark right over here, we could just say that, look, have to add up to one. So we could say the probability of exactly four is going to be equal to one minus 0.2 minus 0.16 minus 0.128. I get one minus 0.2 minus 0.16 minus 0.128 is equal to 0.512, is equal to 0.512. 0.512. | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
If for kicks, we wanted to figure out this question mark right over here, we could just say that, look, have to add up to one. So we could say the probability of exactly four is going to be equal to one minus 0.2 minus 0.16 minus 0.128. I get one minus 0.2 minus 0.16 minus 0.128 is equal to 0.512, is equal to 0.512. 0.512. You might immediately say, wait, wait, this seems like a very high probability. There's more than a 50% chance that he buys four packs. And you have to remember, he has to stop at four. | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
0.512. You might immediately say, wait, wait, this seems like a very high probability. There's more than a 50% chance that he buys four packs. And you have to remember, he has to stop at four. Even if on the fourth, he doesn't get the card he wants, he still has to stop there. So there's a high probability that that's where we end up. There is a little less than 50% chance that he gets the card he's looking for before that point. | Probability with discrete random variable example Random variables AP Statistics Khan Academy.mp3 |
We have a whole video on it on Khan Academy, but it is an average measure of your blood sugar over roughly a three-month period. So that's the explanatory variable, whether or not you're taking the pill, and the response variable is, well, what does it do to your hemoglobin A1c? We constructed a somewhat classic experiment where we had a control group and a treatment group, and we randomly assigned folks into either the control or the treatment group. And to ensure that one group or the other, or I guess both of them, don't end up with an imbalance of, in the case of the last video, an imbalance of men or women, we did what we call block design, where we took our 100 people, and we just happened to have 60 women and 40 men, and we said, okay, well, let's split the 60 women randomly between the two groups, and let's split the 40 men between these two groups so that we have at least an even distribution with respect to sex. And so we would measure folks' A1cs before they get the treatment or the placebo. Then we would wait three months of getting either the treatment or the placebo, and then we'll see if there's a statistically significant improvement. Now, this was a pretty good, and it's a bit of a classic experimental design. | Matched pairs experiment design Study design AP Statistics Khan Academy.mp3 |
And to ensure that one group or the other, or I guess both of them, don't end up with an imbalance of, in the case of the last video, an imbalance of men or women, we did what we call block design, where we took our 100 people, and we just happened to have 60 women and 40 men, and we said, okay, well, let's split the 60 women randomly between the two groups, and let's split the 40 men between these two groups so that we have at least an even distribution with respect to sex. And so we would measure folks' A1cs before they get the treatment or the placebo. Then we would wait three months of getting either the treatment or the placebo, and then we'll see if there's a statistically significant improvement. Now, this was a pretty good, and it's a bit of a classic experimental design. We would also do it so that the patients don't know which one they're getting, placebo or the actual treatment, so it's a blind experiment. And it would probably be good if even the nurses or the doctors who are administering the pills, who are giving the pills, also don't know which one they're giving, so it would be a double blind experiment. But this doesn't mean that it's a perfect experiment, and there seldom is a perfect experiment, and that's why it should be able to be replicated. | Matched pairs experiment design Study design AP Statistics Khan Academy.mp3 |
Now, this was a pretty good, and it's a bit of a classic experimental design. We would also do it so that the patients don't know which one they're getting, placebo or the actual treatment, so it's a blind experiment. And it would probably be good if even the nurses or the doctors who are administering the pills, who are giving the pills, also don't know which one they're giving, so it would be a double blind experiment. But this doesn't mean that it's a perfect experiment, and there seldom is a perfect experiment, and that's why it should be able to be replicated. Other people should try to prove the same thing, maybe in different ways. But even the way that we designed it, there's still a possibility that there are some lurking variables in here. Maybe, you know, we took care to make sure that our distribution of men and women was roughly even across both of these groups, but maybe by, through that random sampling, we got a disproportionate number of young people in the treatment group, and maybe young people responded better to taking a pill. | Matched pairs experiment design Study design AP Statistics Khan Academy.mp3 |
But this doesn't mean that it's a perfect experiment, and there seldom is a perfect experiment, and that's why it should be able to be replicated. Other people should try to prove the same thing, maybe in different ways. But even the way that we designed it, there's still a possibility that there are some lurking variables in here. Maybe, you know, we took care to make sure that our distribution of men and women was roughly even across both of these groups, but maybe by, through that random sampling, we got a disproportionate number of young people in the treatment group, and maybe young people responded better to taking a pill. Maybe it changes their behaviors in other ways, or maybe older people, when they take a pill, they decide to eat worse because they say, oh, this pill's gonna solve all my problems. And so you could have these other lurking variables, like age, or where in the country they live, or other types of things, that just by the random process, you might have things get uneven in one way or another. Now, one technique to help control for this a little bit, and I shouldn't use the word control too much, another technique to help mitigate this is something called matched pairs design. | Matched pairs experiment design Study design AP Statistics Khan Academy.mp3 |
Maybe, you know, we took care to make sure that our distribution of men and women was roughly even across both of these groups, but maybe by, through that random sampling, we got a disproportionate number of young people in the treatment group, and maybe young people responded better to taking a pill. Maybe it changes their behaviors in other ways, or maybe older people, when they take a pill, they decide to eat worse because they say, oh, this pill's gonna solve all my problems. And so you could have these other lurking variables, like age, or where in the country they live, or other types of things, that just by the random process, you might have things get uneven in one way or another. Now, one technique to help control for this a little bit, and I shouldn't use the word control too much, another technique to help mitigate this is something called matched pairs design. Matched, matched pairs, pairs design of an experiment, and it's essentially, instead of going through all of this trouble saying, oh boy, maybe we do block design, all this random sampling, instead, you randomly put people first into either the control or the treatment group, and then we do another round, you measure, and then you do another round where you switch, where the people who are in the treatment go into the control, and the people who are in the control go into the treatment. So we could even extend from what we have here, we could imagine a world where the first three months, we have the 50 people in this treatment group, we have another 50 people in this control group that are taking the placebo, we see what happens to the A1Cs, and then we switch, where this group over here, then, and they don't know, they don't know, first of all, ideally, it's a blind experiment, so they don't even know they were in the treatment groups, and hopefully the pills look identical, so now, that same group, for the next three months, is now going to be the control group, and so they got the medicine for the first three months, and we saw what happens to their A1C, and now they're gonna get the placebo, they're going to get the placebo for the second three months, and then we are going to see what happens to their A1C, and likewise, the other group is going to be switched around. The thing that, the folks that used to be getting the placebo could now get, could now get the treatment. | Matched pairs experiment design Study design AP Statistics Khan Academy.mp3 |
Let's say we define the random variable capital X as the number of heads we get after three flips of a fair coin. So given that definition of a random variable, what we're going to try to do in this video is think about the probability distribution. So what's the probability of the different possible outcomes or the different possible values for this random variable? And we'll plot them to see how that distribution is spread out amongst those possible outcomes. So let's think about all of the different values that you could get when you flip a fair coin three times. So you could get all heads. Heads, heads, heads. | Constructing a probability distribution for random variable Khan Academy.mp3 |
And we'll plot them to see how that distribution is spread out amongst those possible outcomes. So let's think about all of the different values that you could get when you flip a fair coin three times. So you could get all heads. Heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. | Constructing a probability distribution for random variable Khan Academy.mp3 |
Heads, heads, heads. You could get heads, heads, tails. You could get heads, tails, heads. You could get heads, tails, tails. You could have tails, heads, head. You could have tails, head, tails. You could have tails, tails, heads. | Constructing a probability distribution for random variable Khan Academy.mp3 |
You could get heads, tails, tails. You could have tails, heads, head. You could have tails, head, tails. You could have tails, tails, heads. And then you could have all tails. So when you do the actual experiment, there's eight equally likely outcomes here. But which of them, how would these relate to the value of this random variable? | Constructing a probability distribution for random variable Khan Academy.mp3 |
You could have tails, tails, heads. And then you could have all tails. So when you do the actual experiment, there's eight equally likely outcomes here. But which of them, how would these relate to the value of this random variable? So let's think about what's the probability. There is a situation where you have zero heads. So we could say, what's the probability that our random variable X is equal to 0? | Constructing a probability distribution for random variable Khan Academy.mp3 |
But which of them, how would these relate to the value of this random variable? So let's think about what's the probability. There is a situation where you have zero heads. So we could say, what's the probability that our random variable X is equal to 0? Well, that's this situation right over here where you have zero heads. It is one out of the eight equally likely outcomes. So that's going to be 1 over 8. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So we could say, what's the probability that our random variable X is equal to 0? Well, that's this situation right over here where you have zero heads. It is one out of the eight equally likely outcomes. So that's going to be 1 over 8. What's the probability that our random variable capital X is equal to 1? Well, let's see. Which of these outcomes gets us exactly one head? | Constructing a probability distribution for random variable Khan Academy.mp3 |
So that's going to be 1 over 8. What's the probability that our random variable capital X is equal to 1? Well, let's see. Which of these outcomes gets us exactly one head? We have this one right over here. We have that one right over there. We have this one right over there. | Constructing a probability distribution for random variable Khan Academy.mp3 |
Which of these outcomes gets us exactly one head? We have this one right over here. We have that one right over there. We have this one right over there. And I think that's all of them. So three out of the eight equally likely outcomes get us to one head, which is the same thing as saying that our random variable equals 1. So this has a 3 8's probability. | Constructing a probability distribution for random variable Khan Academy.mp3 |
We have this one right over there. And I think that's all of them. So three out of the eight equally likely outcomes get us to one head, which is the same thing as saying that our random variable equals 1. So this has a 3 8's probability. Now, what's the probability? I think you're maybe getting the hang for it at this point. What's the probability that our random variable X is going to be equal to 2? | Constructing a probability distribution for random variable Khan Academy.mp3 |
So this has a 3 8's probability. Now, what's the probability? I think you're maybe getting the hang for it at this point. What's the probability that our random variable X is going to be equal to 2? Well, for X to be equal to 2, that means we got two heads when we flipped the coin three times. So this outcome meets that constraint. This outcome would get our random variable to be equal to 2. | Constructing a probability distribution for random variable Khan Academy.mp3 |
What's the probability that our random variable X is going to be equal to 2? Well, for X to be equal to 2, that means we got two heads when we flipped the coin three times. So this outcome meets that constraint. This outcome would get our random variable to be equal to 2. And this outcome would make our random variable equal to 2. And this is three out of the eight equally likely outcomes. So this has a 3 8's probability. | Constructing a probability distribution for random variable Khan Academy.mp3 |
This outcome would get our random variable to be equal to 2. And this outcome would make our random variable equal to 2. And this is three out of the eight equally likely outcomes. So this has a 3 8's probability. And then finally, we could say, what is the probability that our random variable X is equal to 3? Well, how does our random variable X equal 3? Well, we would have to get three heads when we flip the coin. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So this has a 3 8's probability. And then finally, we could say, what is the probability that our random variable X is equal to 3? Well, how does our random variable X equal 3? Well, we would have to get three heads when we flip the coin. So there's only one out of the eight equally likely outcomes that meets that constraint. So it's a 1 8 probability. So now we just have to think about how we plot this to really see how it's distributed. | Constructing a probability distribution for random variable Khan Academy.mp3 |
Well, we would have to get three heads when we flip the coin. So there's only one out of the eight equally likely outcomes that meets that constraint. So it's a 1 8 probability. So now we just have to think about how we plot this to really see how it's distributed. So let me draw over here on the vertical axis. I'll draw this will be the probability. And it's going to be between 0 and 1. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So now we just have to think about how we plot this to really see how it's distributed. So let me draw over here on the vertical axis. I'll draw this will be the probability. And it's going to be between 0 and 1. You can have a probability larger than 1. So just like this. So let's see. | Constructing a probability distribution for random variable Khan Academy.mp3 |
And it's going to be between 0 and 1. You can have a probability larger than 1. So just like this. So let's see. If this is 1 right over here. And let's see. Everything here, it looks like it's an eighth. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So let's see. If this is 1 right over here. And let's see. Everything here, it looks like it's an eighth. So let's put everything in terms of eighths. So that's half. This is a fourth. | Constructing a probability distribution for random variable Khan Academy.mp3 |
Everything here, it looks like it's an eighth. So let's put everything in terms of eighths. So that's half. This is a fourth. That's a fourth. That's not quite a fourth. This is a fourth right over here. | Constructing a probability distribution for random variable Khan Academy.mp3 |
This is a fourth. That's a fourth. That's not quite a fourth. This is a fourth right over here. And then we can do it in terms of eighths. So that's a pretty good rough approximation. And then over here, we could have the outcomes. | Constructing a probability distribution for random variable Khan Academy.mp3 |
This is a fourth right over here. And then we can do it in terms of eighths. So that's a pretty good rough approximation. And then over here, we could have the outcomes. And so outcomes, I'll say outcomes for, or let's write this so value. So value for X. So X could be 0, 1. | Constructing a probability distribution for random variable Khan Academy.mp3 |
And then over here, we could have the outcomes. And so outcomes, I'll say outcomes for, or let's write this so value. So value for X. So X could be 0, 1. Actually, let me do those same colors. X could be 0. X could be 1. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So X could be 0, 1. Actually, let me do those same colors. X could be 0. X could be 1. X could be 2. X could be equal to 2. And X could be equal to 3. | Constructing a probability distribution for random variable Khan Academy.mp3 |
X could be 1. X could be 2. X could be equal to 2. And X could be equal to 3. These are the possible values for X. And now we're just going to plot the probability. The probability that X has a value of 0 is 1 eighth. | Constructing a probability distribution for random variable Khan Academy.mp3 |
And X could be equal to 3. These are the possible values for X. And now we're just going to plot the probability. The probability that X has a value of 0 is 1 eighth. So I'll make a little bar right over here that goes up to 1 eighth. So actually, let me draw it like this. So this is 1 eighth right over here. | Constructing a probability distribution for random variable Khan Academy.mp3 |
The probability that X has a value of 0 is 1 eighth. So I'll make a little bar right over here that goes up to 1 eighth. So actually, let me draw it like this. So this is 1 eighth right over here. The probability that X equals 1 is 3 eighths. So that's 2 eighths, 3 eighths. Gets us right over. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So this is 1 eighth right over here. The probability that X equals 1 is 3 eighths. So that's 2 eighths, 3 eighths. Gets us right over. Let me do that in that purple color. So probability of 1, that's 3 eighths. That's right over there. | Constructing a probability distribution for random variable Khan Academy.mp3 |
Gets us right over. Let me do that in that purple color. So probability of 1, that's 3 eighths. That's right over there. That's 3 eighths. So let me draw that bar. Just like that. | Constructing a probability distribution for random variable Khan Academy.mp3 |
That's right over there. That's 3 eighths. So let me draw that bar. Just like that. The probability that X equals 2 is also 3 eighths. So that's going to be that same level, just like that. And then the probability that X equals 3, well, that's 1 eighth. | Constructing a probability distribution for random variable Khan Academy.mp3 |
Just like that. The probability that X equals 2 is also 3 eighths. So that's going to be that same level, just like that. And then the probability that X equals 3, well, that's 1 eighth. So it's going to be the same height as this thing right over here. So actually, I'm using the wrong color. So it's going to look like this. | Constructing a probability distribution for random variable Khan Academy.mp3 |
And then the probability that X equals 3, well, that's 1 eighth. So it's going to be the same height as this thing right over here. So actually, I'm using the wrong color. So it's going to look like this. It's going to look like this. And actually, let me just write this a little bit neater. I can move that 3. | Constructing a probability distribution for random variable Khan Academy.mp3 |
So it's going to look like this. It's going to look like this. And actually, let me just write this a little bit neater. I can move that 3. So cut and paste. Let me move that 3 a little bit closer in, just so it looks a little bit neater. And I can move that 2 in, actually, as well. | Constructing a probability distribution for random variable Khan Academy.mp3 |
I can move that 3. So cut and paste. Let me move that 3 a little bit closer in, just so it looks a little bit neater. And I can move that 2 in, actually, as well. So cut and paste. So I can move that 2. And there you have it. | Constructing a probability distribution for random variable Khan Academy.mp3 |
And I can move that 2 in, actually, as well. So cut and paste. So I can move that 2. And there you have it. We have made a probability distribution for the random variable X. And the random variable X can only take on these discrete values. It can't take on the value half or the value pi or anything like that. | Constructing a probability distribution for random variable Khan Academy.mp3 |
And there you have it. We have made a probability distribution for the random variable X. And the random variable X can only take on these discrete values. It can't take on the value half or the value pi or anything like that. And so what we've just done here is we've just constructed a discrete probability distribution. Let me write that down. So this right over here is a discrete. | Constructing a probability distribution for random variable Khan Academy.mp3 |
It can't take on the value half or the value pi or anything like that. And so what we've just done here is we've just constructed a discrete probability distribution. Let me write that down. So this right over here is a discrete. The random variable only takes on discrete values. It can't take on any value in between these things. So discrete probability distribution for our random variable X. | Constructing a probability distribution for random variable Khan Academy.mp3 |
And you don't want to cut open every watermelon in your watermelon farm or patch or whatever it might be called, because you want to sell most of them. You just want to sample a few watermelons and then take samples of those watermelons to figure out how dense the seeds are, and hope that you can calculate statistics on those samples that are decent estimates of the parameters for the population. So let's start doing that. So let's say that you take these little cubic inch chunks out of a random sample of your watermelons, and then you count the number of seeds in them. And you have eight samples like this. So in one of them, you found four seeds. In the next, you found three, five, seven, two, nine, 11, and seven. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So let's say that you take these little cubic inch chunks out of a random sample of your watermelons, and then you count the number of seeds in them. And you have eight samples like this. So in one of them, you found four seeds. In the next, you found three, five, seven, two, nine, 11, and seven. So this is a sample just to make sure we're visualizing it right. If this is a population of all of the chunks, these little cubic, I guess if we view this as a cubic inch, the cubic inch chunks in my entire watermelon farm, I'm sampling a very small sample of them. So I'm sampling a very small sample. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
In the next, you found three, five, seven, two, nine, 11, and seven. So this is a sample just to make sure we're visualizing it right. If this is a population of all of the chunks, these little cubic, I guess if we view this as a cubic inch, the cubic inch chunks in my entire watermelon farm, I'm sampling a very small sample of them. So I'm sampling a very small sample. Maybe I could have had a million over here. A million chunks of watermelon could have been produced from my farm, but I'm only sampling. So capital N would be 1 million, lowercase n is equal to 8. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So I'm sampling a very small sample. Maybe I could have had a million over here. A million chunks of watermelon could have been produced from my farm, but I'm only sampling. So capital N would be 1 million, lowercase n is equal to 8. And once again, you might want to have more samples, but this will make our math easy. Now, let's think about what statistics we can measure. Well, the first one that we often do is a measure of central tendency, and that's the arithmetic mean. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So capital N would be 1 million, lowercase n is equal to 8. And once again, you might want to have more samples, but this will make our math easy. Now, let's think about what statistics we can measure. Well, the first one that we often do is a measure of central tendency, and that's the arithmetic mean. But here, we were trying to estimate the population mean by coming up with the sample mean. So what is the sample mean going to be? Well, all we have to do is add up these points, add up these measurements, and then divide by the number of measurements we have. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Well, the first one that we often do is a measure of central tendency, and that's the arithmetic mean. But here, we were trying to estimate the population mean by coming up with the sample mean. So what is the sample mean going to be? Well, all we have to do is add up these points, add up these measurements, and then divide by the number of measurements we have. So let's get our calculator out for that. Actually, maybe I don't need my calculator. Let's see. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Well, all we have to do is add up these points, add up these measurements, and then divide by the number of measurements we have. So let's get our calculator out for that. Actually, maybe I don't need my calculator. Let's see. So 4 plus 3 is 7. 7 plus 5 is 12. 12 plus 7 is 19. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Let's see. So 4 plus 3 is 7. 7 plus 5 is 12. 12 plus 7 is 19. 19 plus 2 is 21. Plus 9 is 30. Plus 11 is 41. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
12 plus 7 is 19. 19 plus 2 is 21. Plus 9 is 30. Plus 11 is 41. Plus 7 is 48. So I'm going to get 48 over 8 data points. So this worked out quite well. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Plus 11 is 41. Plus 7 is 48. So I'm going to get 48 over 8 data points. So this worked out quite well. 48 divided by 8 is equal to 6. So our sample mean is 6. It's our estimate of what the population mean might be. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So this worked out quite well. 48 divided by 8 is equal to 6. So our sample mean is 6. It's our estimate of what the population mean might be. But we also want to think about how much in our population, we want to estimate how much in our population, how much spread is there? How much do our measurements vary from this mean? So there we say, well, we can try to estimate the population variance by calculating the sample variance. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
It's our estimate of what the population mean might be. But we also want to think about how much in our population, we want to estimate how much in our population, how much spread is there? How much do our measurements vary from this mean? So there we say, well, we can try to estimate the population variance by calculating the sample variance. And we're going to calculate the unbiased sample variance. Hopefully we're fairly convinced at this point why we divide by n minus 1. So we're going to calculate the unbiased sample variance. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So there we say, well, we can try to estimate the population variance by calculating the sample variance. And we're going to calculate the unbiased sample variance. Hopefully we're fairly convinced at this point why we divide by n minus 1. So we're going to calculate the unbiased sample variance. And if we do that, what do we get? Well, it's just going to be four minus 6 squared plus 3 minus 6 squared plus 5 minus 6 squared plus 7 minus 6 squared plus 2 minus 6 squared plus 9 minus 6 squared plus 11 minus 6 squared plus 7 minus 6 squared, All of that divided by not by 8. Remember, we want the unbiased sample variance. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So we're going to calculate the unbiased sample variance. And if we do that, what do we get? Well, it's just going to be four minus 6 squared plus 3 minus 6 squared plus 5 minus 6 squared plus 7 minus 6 squared plus 2 minus 6 squared plus 9 minus 6 squared plus 11 minus 6 squared plus 7 minus 6 squared, All of that divided by not by 8. Remember, we want the unbiased sample variance. We're going to divide it by 8 minus 1. So we're going to divide by 7. And so this is going to be equal to, let me give myself a little bit more real estate, the unbiased sample variance. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Remember, we want the unbiased sample variance. We're going to divide it by 8 minus 1. So we're going to divide by 7. And so this is going to be equal to, let me give myself a little bit more real estate, the unbiased sample variance. And I could even denote it by this to make it clear that we're dividing by lowercase n minus 1 is going to be equal to, let's see, 4 minus 6 is negative 2. That squared is positive 4. So I did that one. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
And so this is going to be equal to, let me give myself a little bit more real estate, the unbiased sample variance. And I could even denote it by this to make it clear that we're dividing by lowercase n minus 1 is going to be equal to, let's see, 4 minus 6 is negative 2. That squared is positive 4. So I did that one. 3 minus 6 is negative 3. That squared is going to be 9. 5 minus 6 squared is 1 squared, which is 1. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So I did that one. 3 minus 6 is negative 3. That squared is going to be 9. 5 minus 6 squared is 1 squared, which is 1. 7 minus 6 is, once again, 1 squared, which is 1. 2 minus 6, negative 4 squared. Negative 4 squared is 16. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
5 minus 6 squared is 1 squared, which is 1. 7 minus 6 is, once again, 1 squared, which is 1. 2 minus 6, negative 4 squared. Negative 4 squared is 16. 9 minus 6 squared, well, that's going to be 9. 11 minus 6 squared, that is 25. And then finally, 7 minus 6 squared, that's another 1. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Negative 4 squared is 16. 9 minus 6 squared, well, that's going to be 9. 11 minus 6 squared, that is 25. And then finally, 7 minus 6 squared, that's another 1. And we're going to divide it by 7. Now let's see if we can add this up in our heads. 4 plus 9 is 13. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
And then finally, 7 minus 6 squared, that's another 1. And we're going to divide it by 7. Now let's see if we can add this up in our heads. 4 plus 9 is 13. Plus 1 is 14, 15, 31, 40, 65, 66. So this is going to be equal to 66 over 7. And we could either divide. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
4 plus 9 is 13. Plus 1 is 14, 15, 31, 40, 65, 66. So this is going to be equal to 66 over 7. And we could either divide. That's 9 and 3 sevenths. We could write that as 9 and 3 sevenths. Or if we want to write that as a decimal, I can just take 66. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
And we could either divide. That's 9 and 3 sevenths. We could write that as 9 and 3 sevenths. Or if we want to write that as a decimal, I can just take 66. 66 divided by 7 gives us 9 point, I'll just round it. So it's approximately 9.43. So this is approximately 9.43. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
Or if we want to write that as a decimal, I can just take 66. 66 divided by 7 gives us 9 point, I'll just round it. So it's approximately 9.43. So this is approximately 9.43. Now that gave us our unbiased sample variance. Well, how could we calculate a sample standard deviation? We want to somehow get at an estimate of what the population standard deviation might be. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So this is approximately 9.43. Now that gave us our unbiased sample variance. Well, how could we calculate a sample standard deviation? We want to somehow get at an estimate of what the population standard deviation might be. Well, the logic should, I guess, is reasonable to say, well, this is our unbiased sample variance. It's our best estimate of what the true population variance is. When we think about population parameters to get the population standard deviation, we just take the square root of the population variance. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
We want to somehow get at an estimate of what the population standard deviation might be. Well, the logic should, I guess, is reasonable to say, well, this is our unbiased sample variance. It's our best estimate of what the true population variance is. When we think about population parameters to get the population standard deviation, we just take the square root of the population variance. So if we want to get an estimate of the sample standard deviation, why don't we just take the square root of the unbiased sample variance? So that's what we'll do. So we'll define it that way. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
When we think about population parameters to get the population standard deviation, we just take the square root of the population variance. So if we want to get an estimate of the sample standard deviation, why don't we just take the square root of the unbiased sample variance? So that's what we'll do. So we'll define it that way. We'll call the sample standard deviation, we're going to define it to be equal to the square root of the unbiased sample variance. So it's going to be the square root of this quantity. And we could take our calculator out. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
So we'll define it that way. We'll call the sample standard deviation, we're going to define it to be equal to the square root of the unbiased sample variance. So it's going to be the square root of this quantity. And we could take our calculator out. It's going to be the square root of what I just typed in. I could do second answer. It'll be the last entry here. | Sample standard deviation and bias Probability and Statistics Khan Academy.mp3 |
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