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Well, anyway, we don't want to make judgments just staring at the numbers. Let's figure out our chi-squared statistic. And to do that, let's get our statistic, our chi-squared statistic. I'll write it like this, maybe for fun. Or maybe I'll write it as a big X, because this random variable's distribution is approximately a chi-squared distribution. So I'll write it like that. And we'll talk about the degrees of freedom in a second. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
I'll write it like this, maybe for fun. Or maybe I'll write it as a big X, because this random variable's distribution is approximately a chi-squared distribution. So I'll write it like that. And we'll talk about the degrees of freedom in a second. Actually, let me write it with a curly X, just so you see that some people write it with the chi instead of the X. So our chi-squared statistic over here, we're literally just going to find the squared distance between the observed and expected, and then divide it by the expected. So this is going to be 20 minus 25.3 squared over 25.3 plus 30 minus 29.4 squared over 29.4. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And we'll talk about the degrees of freedom in a second. Actually, let me write it with a curly X, just so you see that some people write it with the chi instead of the X. So our chi-squared statistic over here, we're literally just going to find the squared distance between the observed and expected, and then divide it by the expected. So this is going to be 20 minus 25.3 squared over 25.3 plus 30 minus 29.4 squared over 29.4. I'm going to run out of space. Plus 30 minus 25.3 squared over 25.3. And then I'm going to have to do these over here. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So this is going to be 20 minus 25.3 squared over 25.3 plus 30 minus 29.4 squared over 29.4. I'm going to run out of space. Plus 30 minus 25.3 squared over 25.3. And then I'm going to have to do these over here. So let me just continue it. You can ignore this h1 over here. So plus 100 minus 94.7 squared over 94.7 plus, I think you see where this is going, 110 minus 110.6 squared over 110.6. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And then I'm going to have to do these over here. So let me just continue it. You can ignore this h1 over here. So plus 100 minus 94.7 squared over 94.7 plus, I think you see where this is going, 110 minus 110.6 squared over 110.6. And then finally, plus 90 over 94. Sorry, 90 minus 94.7 squared. All of that over 94.7. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So plus 100 minus 94.7 squared over 94.7 plus, I think you see where this is going, 110 minus 110.6 squared over 110.6. And then finally, plus 90 over 94. Sorry, 90 minus 94.7 squared. All of that over 94.7. So let me just get the calculator out to calculate this, take a little bit of time. So we have, I'll have to type it on the calculator for these parentheses. So we have 20 minus 25.3 squared divided by 25.3 plus, open parentheses, 30 minus 29.4 squared divided by 29.4 plus, open parentheses, 30 minus 25.3 squared divided by 25.3, halfway there, plus 100, open parentheses, this is the tedious part, 100 minus 94.7 squared divided by 94.7 plus, 110 minus, well this will, I'll actually type it out, we can do a lot of these in our head, but let me just do it. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
All of that over 94.7. So let me just get the calculator out to calculate this, take a little bit of time. So we have, I'll have to type it on the calculator for these parentheses. So we have 20 minus 25.3 squared divided by 25.3 plus, open parentheses, 30 minus 29.4 squared divided by 29.4 plus, open parentheses, 30 minus 25.3 squared divided by 25.3, halfway there, plus 100, open parentheses, this is the tedious part, 100 minus 94.7 squared divided by 94.7 plus, 110 minus, well this will, I'll actually type it out, we can do a lot of these in our head, but let me just do it. 110 minus 110.6 squared divided by 110.6. And then last one, home stretch, assuming we haven't made any mistakes. We have 90 minus 94.7 squared divided by 94.7. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So we have 20 minus 25.3 squared divided by 25.3 plus, open parentheses, 30 minus 29.4 squared divided by 29.4 plus, open parentheses, 30 minus 25.3 squared divided by 25.3, halfway there, plus 100, open parentheses, this is the tedious part, 100 minus 94.7 squared divided by 94.7 plus, 110 minus, well this will, I'll actually type it out, we can do a lot of these in our head, but let me just do it. 110 minus 110.6 squared divided by 110.6. And then last one, home stretch, assuming we haven't made any mistakes. We have 90 minus 94.7 squared divided by 94.7. And let's see what we get. We get 2.528. So let's just say it's 2.53. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
We have 90 minus 94.7 squared divided by 94.7. And let's see what we get. We get 2.528. So let's just say it's 2.53. So our chi-square statistic, I always have trouble saying that, our chi-square statistic, assuming the null hypothesis is correct, is equal to 2.53. Now, the next thing we have to do is figure out the degrees of freedom that we had in calculating this chi-square statistic. And I'll give you the rule of thumb, and I'll give you a little bit of a sense of why this is the rule of thumb for contingency table like this. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So let's just say it's 2.53. So our chi-square statistic, I always have trouble saying that, our chi-square statistic, assuming the null hypothesis is correct, is equal to 2.53. Now, the next thing we have to do is figure out the degrees of freedom that we had in calculating this chi-square statistic. And I'll give you the rule of thumb, and I'll give you a little bit of a sense of why this is the rule of thumb for contingency table like this. And in the future, we'll talk a little bit more deeply about degrees of freedom. So when you do, the rule of thumb for contingency table is you have the number of rows, so you have rows, and then you have your number of columns. So here we have two rows and we have three columns, you don't count the totals. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
And I'll give you the rule of thumb, and I'll give you a little bit of a sense of why this is the rule of thumb for contingency table like this. And in the future, we'll talk a little bit more deeply about degrees of freedom. So when you do, the rule of thumb for contingency table is you have the number of rows, so you have rows, and then you have your number of columns. So here we have two rows and we have three columns, you don't count the totals. So you have three columns over here. And the degrees of freedom, and this is the rule of thumb, the degrees of freedom for your contingency table is going to be the number of rows minus 1 times the number of columns minus 1. In our situation, we have two rows and three columns, so it's going to be 2 minus 1 times 3 minus 1. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So here we have two rows and we have three columns, you don't count the totals. So you have three columns over here. And the degrees of freedom, and this is the rule of thumb, the degrees of freedom for your contingency table is going to be the number of rows minus 1 times the number of columns minus 1. In our situation, we have two rows and three columns, so it's going to be 2 minus 1 times 3 minus 1. So it's going to be 2 minus 1 times 3 minus 1, which is just 1 times 2, which is 2. We have two degrees of freedom. Now, the reason that that should make a little bit of intuitive sense, we'll talk about this in more depth in the future, is that if you assume that you know the totals, so let's just assume that you know the totals. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
In our situation, we have two rows and three columns, so it's going to be 2 minus 1 times 3 minus 1. So it's going to be 2 minus 1 times 3 minus 1, which is just 1 times 2, which is 2. We have two degrees of freedom. Now, the reason that that should make a little bit of intuitive sense, we'll talk about this in more depth in the future, is that if you assume that you know the totals, so let's just assume that you know the totals. So if you know all of this information over here, if you know the total information, or actually if you knew the parameters of the population as well, but if you know the total information and if you know this information, or if you know r minus 1 of the information in the rows, the last one can be figured out just by subtracting from the total. So for example, in this situation, if you know this, you can easily figure out this. This is not new information. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
Now, the reason that that should make a little bit of intuitive sense, we'll talk about this in more depth in the future, is that if you assume that you know the totals, so let's just assume that you know the totals. So if you know all of this information over here, if you know the total information, or actually if you knew the parameters of the population as well, but if you know the total information and if you know this information, or if you know r minus 1 of the information in the rows, the last one can be figured out just by subtracting from the total. So for example, in this situation, if you know this, you can easily figure out this. This is not new information. It's just the total minus 20. Same thing. If you know this one right over here, this one over here is not new information. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
This is not new information. It's just the total minus 20. Same thing. If you know this one right over here, this one over here is not new information. And similarly, if you know these two, this guy over here isn't new information. You can always just calculate him based on the total and everything else. So that's the sense of why our degrees of freedom are the columns minus 1 times the rows minus 1. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
If you know this one right over here, this one over here is not new information. And similarly, if you know these two, this guy over here isn't new information. You can always just calculate him based on the total and everything else. So that's the sense of why our degrees of freedom are the columns minus 1 times the rows minus 1. But anyway, so our chi-square statistic has 2 degrees of freedom. So what we have to do is, remember, our alpha value, let me get it up here. We had it right over here. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So that's the sense of why our degrees of freedom are the columns minus 1 times the rows minus 1. But anyway, so our chi-square statistic has 2 degrees of freedom. So what we have to do is, remember, our alpha value, let me get it up here. We had it right over here. Our significance level that we care about, our alpha value is 10%. Let me rewrite it over here. So our alpha is 10%. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
We had it right over here. Our significance level that we care about, our alpha value is 10%. Let me rewrite it over here. So our alpha is 10%. So what we're going to do is figure out what is our critical chi-square statistic that gives us an alpha of 10%. If this is more extreme than that, if the probability of getting this is even less than that critical statistic, it will be less than 10% and we'll reject the null hypothesis. If it's not more extreme, then we won't reject the null hypothesis. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So our alpha is 10%. So what we're going to do is figure out what is our critical chi-square statistic that gives us an alpha of 10%. If this is more extreme than that, if the probability of getting this is even less than that critical statistic, it will be less than 10% and we'll reject the null hypothesis. If it's not more extreme, then we won't reject the null hypothesis. So what we need to do is to figure out with a chi-square distribution and 2 degrees of freedom, what is our critical chi-square statistic? So let's just go back. So we have 2 degrees of freedom here and we care about a significance level of 10%. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
If it's not more extreme, then we won't reject the null hypothesis. So what we need to do is to figure out with a chi-square distribution and 2 degrees of freedom, what is our critical chi-square statistic? So let's just go back. So we have 2 degrees of freedom here and we care about a significance level of 10%. So our critical chi-square value is 4.60. So another way to visualize this, if we look at the chi- square distribution with 2 degrees of freedom, that's this blue one over here. I'm trying to pick a nice blue to use. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So we have 2 degrees of freedom here and we care about a significance level of 10%. So our critical chi-square value is 4.60. So another way to visualize this, if we look at the chi- square distribution with 2 degrees of freedom, that's this blue one over here. I'm trying to pick a nice blue to use. At a value of a critical value of 4.60, so 4.60, this is 5, so 4.60 will be right around here. At a critical value of 4.60, so this is 4.60, the probability of getting something at least that extreme, so that extreme or more extreme, is 10%. This is what we care about. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
I'm trying to pick a nice blue to use. At a value of a critical value of 4.60, so 4.60, this is 5, so 4.60 will be right around here. At a critical value of 4.60, so this is 4.60, the probability of getting something at least that extreme, so that extreme or more extreme, is 10%. This is what we care about. Now, if the chi-square distribution is less than 10%, the chi-square statistic that we calculated falls into this rejection region, then we're going to reject the null hypothesis. But our chi-square statistic is only 2.53. So it's sitting someplace right over here. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
This is what we care about. Now, if the chi-square distribution is less than 10%, the chi-square statistic that we calculated falls into this rejection region, then we're going to reject the null hypothesis. But our chi-square statistic is only 2.53. So it's sitting someplace right over here. It's actually ours. So it's not that crazy to get it if you assume the null hypothesis. So based on our data that we have right now, we cannot reject the null hypothesis. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
So it's sitting someplace right over here. It's actually ours. So it's not that crazy to get it if you assume the null hypothesis. So based on our data that we have right now, we cannot reject the null hypothesis. So we don't know for a fact that the herbs do nothing, but we can't say that they do something based on this. So we're not going to reject it. We won't say 100% that it's true, but we can't say that we're rejecting it. | Contingency table chi-square test Probability and Statistics Khan Academy.mp3 |
In the last video, we figured out the probability of getting exactly 3 heads when we have 5 flips of a fair coin. What I want to do in this video is think about it in a slightly more general way. We're still going to assume we have a fair coin, although we'll shortly see we don't have to make that assumption. What I want to do is figure out the probability of getting k heads in n flips of the fair coin. The first thing to think about is how many possibilities there are. There's going to be n flips, so literally there's first flip, second flip, third flip, all the way to the nth flip. This is a fair coin. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
What I want to do is figure out the probability of getting k heads in n flips of the fair coin. The first thing to think about is how many possibilities there are. There's going to be n flips, so literally there's first flip, second flip, third flip, all the way to the nth flip. This is a fair coin. Each of these has two equally likely possibilities. The total number of possibilities is going to be 2 times 2 times 2 n times. This is going to be equal to 2 to the nth possibilities. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
This is a fair coin. Each of these has two equally likely possibilities. The total number of possibilities is going to be 2 times 2 times 2 n times. This is going to be equal to 2 to the nth possibilities. Let's think about how many of those equally likely possibilities involve k heads. Just like we did for the case where we're thinking about 3 heads, we say, well look, the first of those k heads, how many different buckets could it fall into? The first of the k heads could fall into n different buckets. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
This is going to be equal to 2 to the nth possibilities. Let's think about how many of those equally likely possibilities involve k heads. Just like we did for the case where we're thinking about 3 heads, we say, well look, the first of those k heads, how many different buckets could it fall into? The first of the k heads could fall into n different buckets. It could be the first flip, second flip, all the way to the nth flip. Then the second of those k heads, we don't know exactly how many k is, will have n minus 1 possibilities. The third of those k heads will have n minus 2 possibilities, since two of the spots are already taken up. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
The first of the k heads could fall into n different buckets. It could be the first flip, second flip, all the way to the nth flip. Then the second of those k heads, we don't know exactly how many k is, will have n minus 1 possibilities. The third of those k heads will have n minus 2 possibilities, since two of the spots are already taken up. We would do this until we have essentially accounted for all of the k heads. This will go down all the way to, we will multiply the number of things we're multiplying is going to be k. One for each of the k heads. This is 1, 2, 3, and then you're going to go all the way to n minus k minus 1. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
The third of those k heads will have n minus 2 possibilities, since two of the spots are already taken up. We would do this until we have essentially accounted for all of the k heads. This will go down all the way to, we will multiply the number of things we're multiplying is going to be k. One for each of the k heads. This is 1, 2, 3, and then you're going to go all the way to n minus k minus 1. You could try this out in the case of 5. When n was 5 and k was 3, this resulted in 5 times 4 times 3. That was this term right over here. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
This is 1, 2, 3, and then you're going to go all the way to n minus k minus 1. You could try this out in the case of 5. When n was 5 and k was 3, this resulted in 5 times 4 times 3. That was this term right over here. I'm doing a case that is a little bit longer, where k is a slightly larger number, because this right over here is 5 minus 2. That is this one over here. Just not to confuse you, let me write it like this. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
That was this term right over here. I'm doing a case that is a little bit longer, where k is a slightly larger number, because this right over here is 5 minus 2. That is this one over here. Just not to confuse you, let me write it like this. You'll have n spots for that first head, n minus 1 spots for that second head. Then you keep going and you're going to have a total of these k things you're multiplying. That last one is going to have n minus k minus 1 possibilities. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
Just not to confuse you, let me write it like this. You'll have n spots for that first head, n minus 1 spots for that second head. Then you keep going and you're going to have a total of these k things you're multiplying. That last one is going to have n minus k minus 1 possibilities. Now hopefully it will map a little bit better to the one where we had 5 flips and 3 heads. Here there was 5 possibilities for the first head, 4 possibilities for the second head. Then n is 5, 5 minus 2, you had 3 possibilities for the last head. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
That last one is going to have n minus k minus 1 possibilities. Now hopefully it will map a little bit better to the one where we had 5 flips and 3 heads. Here there was 5 possibilities for the first head, 4 possibilities for the second head. Then n is 5, 5 minus 2, you had 3 possibilities for the last head. This actually works. This is the number of spots or the number of ways that we can put 3 heads into 5 different possible buckets. Just like the last video, we don't want to over count things, because we don't care about the order. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
Then n is 5, 5 minus 2, you had 3 possibilities for the last head. This actually works. This is the number of spots or the number of ways that we can put 3 heads into 5 different possible buckets. Just like the last video, we don't want to over count things, because we don't care about the order. We don't want to differentiate one ordering of heads. I'm just going to use these letters to differentiate the different heads. We don't want to differentiate this from this. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
Just like the last video, we don't want to over count things, because we don't care about the order. We don't want to differentiate one ordering of heads. I'm just going to use these letters to differentiate the different heads. We don't want to differentiate this from this. Heads A, heads B, or any of the other orderings of this. What we need to do is divide this number so that we don't count all of those different orderings. We need to divide this by the different ways that you can order k things. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
We don't want to differentiate this from this. Heads A, heads B, or any of the other orderings of this. What we need to do is divide this number so that we don't count all of those different orderings. We need to divide this by the different ways that you can order k things. If you have k things, how many ways can you order it? The first thing can be in k different positions. Maybe I'll do it like this. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
We need to divide this by the different ways that you can order k things. If you have k things, how many ways can you order it? The first thing can be in k different positions. Maybe I'll do it like this. Maybe I'll do it T for thing. Thing 1, thing 2, thing 3, all the way to thing k. How many different ways can you order it? Thing 1 can be in k different positions. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
Maybe I'll do it like this. Maybe I'll do it T for thing. Thing 1, thing 2, thing 3, all the way to thing k. How many different ways can you order it? Thing 1 can be in k different positions. Thing 2 will be in k minus 1 positions. Then all the way down to the last one is only going to have one position. This is going to be k times k minus 1 times k minus 2, all the way down to 1. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
Thing 1 can be in k different positions. Thing 2 will be in k minus 1 positions. Then all the way down to the last one is only going to have one position. This is going to be k times k minus 1 times k minus 2, all the way down to 1. In the example where we had 3 heads in 5 flips, this was 3 times 2 all the way down to 1. 3 times 2 times 1. Is there a simpler way that we can write this? | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
This is going to be k times k minus 1 times k minus 2, all the way down to 1. In the example where we had 3 heads in 5 flips, this was 3 times 2 all the way down to 1. 3 times 2 times 1. Is there a simpler way that we can write this? This expression right over here is the same thing as k factorial. If you haven't ever heard of what a factorial is, it's exactly this thing right over here. k factorial literally means k times k minus 1 times k minus 2, all the way down to 1. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
Is there a simpler way that we can write this? This expression right over here is the same thing as k factorial. If you haven't ever heard of what a factorial is, it's exactly this thing right over here. k factorial literally means k times k minus 1 times k minus 2, all the way down to 1. For example, 2 factorial is equal to 2 times 1. 3 factorial is equal to 3 times 2 times 1. 4 factorial is equal to 4 times 3 times 2 times 1. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
k factorial literally means k times k minus 1 times k minus 2, all the way down to 1. For example, 2 factorial is equal to 2 times 1. 3 factorial is equal to 3 times 2 times 1. 4 factorial is equal to 4 times 3 times 2 times 1. It's actually a fun thing to play with. Factorials get large very, very, very, very fast. Anyway, this denominator right over here can be rewritten as k factorial. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
4 factorial is equal to 4 times 3 times 2 times 1. It's actually a fun thing to play with. Factorials get large very, very, very, very fast. Anyway, this denominator right over here can be rewritten as k factorial. This right over here can be rewritten as k factorial. Is there any way to rewrite this numerator in terms of factorials? If we were to write n factorial, let me see how we can write this. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
Anyway, this denominator right over here can be rewritten as k factorial. This right over here can be rewritten as k factorial. Is there any way to rewrite this numerator in terms of factorials? If we were to write n factorial, let me see how we can write this. If we were to write n factorial, n factorial would be equal to n times n minus 1 times n minus 2, all the way down to 1. That's kind of what we want. We just want the first k terms of it. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
If we were to write n factorial, let me see how we can write this. If we were to write n factorial, n factorial would be equal to n times n minus 1 times n minus 2, all the way down to 1. That's kind of what we want. We just want the first k terms of it. What if we divide this by n minus k factorial? Let's think about what that is going to do. If we have n minus k factorial, that is the same thing as n minus k times n minus k minus 1 times n minus k times n minus k minus 2, all the way down to 1. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
We just want the first k terms of it. What if we divide this by n minus k factorial? Let's think about what that is going to do. If we have n minus k factorial, that is the same thing as n minus k times n minus k minus 1 times n minus k times n minus k minus 2, all the way down to 1. When you divide these, the 1's are going to cancel out. What you may or may not realize, and you can work out the math, is everything is going to cancel out here until you're just left with up here, everything from n times n minus 1 to n minus k minus 1. If you expand this out or if you distribute this negative number, this is the same thing as n minus k plus 1. n minus k plus 1 is the integer that's 1 larger than this right over here. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
If we have n minus k factorial, that is the same thing as n minus k times n minus k minus 1 times n minus k times n minus k minus 2, all the way down to 1. When you divide these, the 1's are going to cancel out. What you may or may not realize, and you can work out the math, is everything is going to cancel out here until you're just left with up here, everything from n times n minus 1 to n minus k minus 1. If you expand this out or if you distribute this negative number, this is the same thing as n minus k plus 1. n minus k plus 1 is the integer that's 1 larger than this right over here. If you were to divide it out, this would cancel with something up here, this would cancel with something up here, this would cancel with something up here. What you're going to be left with is exactly this thing over here. If you don't believe me, we can actually try it out. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
If you expand this out or if you distribute this negative number, this is the same thing as n minus k plus 1. n minus k plus 1 is the integer that's 1 larger than this right over here. If you were to divide it out, this would cancel with something up here, this would cancel with something up here, this would cancel with something up here. What you're going to be left with is exactly this thing over here. If you don't believe me, we can actually try it out. Let's think about what 5 factorial over 5 minus 3 factorial is going to be. This is going to be 5 times 4 times 3 times 2 times 1. All of that stuff up there, all the way down to 1. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
If you don't believe me, we can actually try it out. Let's think about what 5 factorial over 5 minus 3 factorial is going to be. This is going to be 5 times 4 times 3 times 2 times 1. All of that stuff up there, all the way down to 1. 5 minus 3 is 2 over 2 factorial, 2 times 1. 2 cancels with 2, 1 cancels with 1. You don't have to worry about that. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
All of that stuff up there, all the way down to 1. 5 minus 3 is 2 over 2 factorial, 2 times 1. 2 cancels with 2, 1 cancels with 1. You don't have to worry about that. You're just left with 5 times 4 times 3. Exactly what we had up here, 5 times 4 times 3. In general, if you wanted to figure out the number of ways to stick 2 things in 5 chairs and you don't care about differentiating between those 2 things, you're going to have this expression right over here, which is the same thing as this right over here. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
You don't have to worry about that. You're just left with 5 times 4 times 3. Exactly what we had up here, 5 times 4 times 3. In general, if you wanted to figure out the number of ways to stick 2 things in 5 chairs and you don't care about differentiating between those 2 things, you're going to have this expression right over here, which is the same thing as this right over here. You're going to have n factorial over n minus k factorial. Then you're going to divide it by this expression right over here, which we already said is the same thing as k factorial. You're also going to divide it by k factorial. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
In general, if you wanted to figure out the number of ways to stick 2 things in 5 chairs and you don't care about differentiating between those 2 things, you're going to have this expression right over here, which is the same thing as this right over here. You're going to have n factorial over n minus k factorial. Then you're going to divide it by this expression right over here, which we already said is the same thing as k factorial. You're also going to divide it by k factorial. Then you have a generalized way of figuring out the number of ways you can stick 2 things, or the number of ways you can stick k things in n different buckets, k heads in n different flips. This is actually a generalized formula for binomial coefficients. Another way to write this is the number of ways, given that you have n buckets, you can put k things in them without having to differentiate it. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
You're also going to divide it by k factorial. Then you have a generalized way of figuring out the number of ways you can stick 2 things, or the number of ways you can stick k things in n different buckets, k heads in n different flips. This is actually a generalized formula for binomial coefficients. Another way to write this is the number of ways, given that you have n buckets, you can put k things in them without having to differentiate it. Another way to think about it is if you have n buckets or n flips and you want to choose k of them to be heads, or you want to choose k of them in some way but you don't want to differentiate. All of these are generalized ways for binomial coefficients. Going back to the original problem, what is the probability of getting k heads in n flips of the fair coin? | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
Another way to write this is the number of ways, given that you have n buckets, you can put k things in them without having to differentiate it. Another way to think about it is if you have n buckets or n flips and you want to choose k of them to be heads, or you want to choose k of them in some way but you don't want to differentiate. All of these are generalized ways for binomial coefficients. Going back to the original problem, what is the probability of getting k heads in n flips of the fair coin? There's 2 to the n equally likely possibilities. Let's write this down. The probability of 2 to the n equally likely possibilities, and how many of those possibilities result in exactly k heads? | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
Going back to the original problem, what is the probability of getting k heads in n flips of the fair coin? There's 2 to the n equally likely possibilities. Let's write this down. The probability of 2 to the n equally likely possibilities, and how many of those possibilities result in exactly k heads? We just figured that out during this video. That's the number of possibilities. n factorial over k factorial times n minus k factorial. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
The probability of 2 to the n equally likely possibilities, and how many of those possibilities result in exactly k heads? We just figured that out during this video. That's the number of possibilities. n factorial over k factorial times n minus k factorial. It probably is an okay idea to memorize this, but I'll just tell you frankly, the only reason why I still know how to do this 20 years after first seeing it, or whenever I first saw it, is that I actually just like to reason through it every time. I like to just reason through, okay, I've got 5 flips, 3 of them need to be heads, the first of those heads can be in 5 different buckets, the next in 4 different buckets, the next one in 3 different buckets, and then of course I don't want to differentiate between all of the different ways that I can rearrange 3 different things, so I have to make sure that I divide it by 3 factorial, by 3 times 2 times 1. I want to make sure that I divide it by all of the different ways that I can arrange 3 different things. | Generalizing with binomial coefficients (bit advanced) Probability and Statistics Khan Academy.mp3 |
All right, so here we have the different names of the drinks, and then here we have the type of the drink, and it looks like they're either hot or cold. Here we have the calories for each of those drinks. Here we have the sugar content in grams for each of those drinks, and here we have the caffeine in milligrams for each of those drinks. And then we are asked, the individuals in this data set are, and we have three choices, Ben's Beans customers, Ben's Beans drinks, or the caffeine contents. Now, we have to be careful. When someone says the individuals in a data set, they don't necessarily mean that they have to be people. They could be things, and the individuals in this data set, each of these rows, they're referring to a certain type of drink at Ben's Beans Coffee Shops. | Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3 |
And then we are asked, the individuals in this data set are, and we have three choices, Ben's Beans customers, Ben's Beans drinks, or the caffeine contents. Now, we have to be careful. When someone says the individuals in a data set, they don't necessarily mean that they have to be people. They could be things, and the individuals in this data set, each of these rows, they're referring to a certain type of drink at Ben's Beans Coffee Shops. So the different types of drinks that Ben's Beans offers, those are the individuals in this data set. So they're Ben's Beans drinks. Next, they ask us, the data set contains, and they say how many variables and how many of those variables are categorical. | Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3 |
They could be things, and the individuals in this data set, each of these rows, they're referring to a certain type of drink at Ben's Beans Coffee Shops. So the different types of drinks that Ben's Beans offers, those are the individuals in this data set. So they're Ben's Beans drinks. Next, they ask us, the data set contains, and they say how many variables and how many of those variables are categorical. So if we look up here, let's look at the variables. So this first column that is essentially giving us the type of drink, this wouldn't be a variable. This would be more of an identifier, but all of these other columns are representing variables. | Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3 |
Next, they ask us, the data set contains, and they say how many variables and how many of those variables are categorical. So if we look up here, let's look at the variables. So this first column that is essentially giving us the type of drink, this wouldn't be a variable. This would be more of an identifier, but all of these other columns are representing variables. So for example, type is a variable. It can either be hot or cold. And because it can only take on one of, kind of a number of buckets, it's either going to be hot or cold, it's going to fit in one category or another. | Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3 |
This would be more of an identifier, but all of these other columns are representing variables. So for example, type is a variable. It can either be hot or cold. And because it can only take on one of, kind of a number of buckets, it's either going to be hot or cold, it's going to fit in one category or another. And you don't just have two categories. You could have more than two categories. But it isn't just some type of variable number that can take on a bunch of different values. | Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3 |
And because it can only take on one of, kind of a number of buckets, it's either going to be hot or cold, it's going to fit in one category or another. And you don't just have two categories. You could have more than two categories. But it isn't just some type of variable number that can take on a bunch of different values. So this right over here is a categorical variable. Calories is not a categorical variable. You could have something with 4.1 calories. | Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3 |
But it isn't just some type of variable number that can take on a bunch of different values. So this right over here is a categorical variable. Calories is not a categorical variable. You could have something with 4.1 calories. You could have something with 178. Things aren't fitting into nice buckets. Same thing for sugars and for the caffeine, that these are quantitative variables that don't just fit into a category. | Identifying individuals, variables and categorical variables in a data set Khan Academy.mp3 |
What we are going to do in this video is talk about the idea of power when we are dealing with significance tests. And power is an idea that you might encounter in a first-year statistics course. It turns out that it's fairly difficult to calculate, but it's interesting to know what it means and what are the levers that might increase the power or decrease the power in a significance test. So just to cut to the chase, power is a probability. You can view it as the probability that you're doing the right thing when the null hypothesis is not true. And the right thing is you should reject the null hypothesis if it's not true. So it's the probability of rejecting, rejecting your null hypothesis given that the null hypothesis is false. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
So just to cut to the chase, power is a probability. You can view it as the probability that you're doing the right thing when the null hypothesis is not true. And the right thing is you should reject the null hypothesis if it's not true. So it's the probability of rejecting, rejecting your null hypothesis given that the null hypothesis is false. So you could view it as a conditional probability like that. But there's other ways to conceptualize it. We can connect it to type two errors. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
So it's the probability of rejecting, rejecting your null hypothesis given that the null hypothesis is false. So you could view it as a conditional probability like that. But there's other ways to conceptualize it. We can connect it to type two errors. For example, you could say this is equal to one minus the probability of not rejecting, one minus the probability of not rejecting, not rejecting the null hypothesis given that the null hypothesis is false. And this thing that I just described, not rejecting the null hypothesis given that the null hypothesis is false, this is, that's the definition of a type, type two error. So you could view it as just the probability of not making a type two error or one minus the probability of making a type two error. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
We can connect it to type two errors. For example, you could say this is equal to one minus the probability of not rejecting, one minus the probability of not rejecting, not rejecting the null hypothesis given that the null hypothesis is false. And this thing that I just described, not rejecting the null hypothesis given that the null hypothesis is false, this is, that's the definition of a type, type two error. So you could view it as just the probability of not making a type two error or one minus the probability of making a type two error. Hopefully that's not confusing. So let me just write it the other way. So you could say it's the probability of not making, not making a type, type two error. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
So you could view it as just the probability of not making a type two error or one minus the probability of making a type two error. Hopefully that's not confusing. So let me just write it the other way. So you could say it's the probability of not making, not making a type, type two error. So what are the things that would actually drive power? And to help us conceptualize that, I'll draw two sampling distributions. One, if we assume that the null hypothesis is true, and one where we assume that the null hypothesis is false and the true population parameter is something different than the null hypothesis is saying. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
So you could say it's the probability of not making, not making a type, type two error. So what are the things that would actually drive power? And to help us conceptualize that, I'll draw two sampling distributions. One, if we assume that the null hypothesis is true, and one where we assume that the null hypothesis is false and the true population parameter is something different than the null hypothesis is saying. So for example, let's say that we have a null hypothesis that our population mean is equal to, let's just call it mu one, and we have an alternative hypothesis, so H sub A, that says, hey, no, the population mean is not equal to mu one. So if you assumed a world where the null hypothesis is true, so I'll do that in blue. So if we assume the null hypothesis is true, what would be our sampling distribution? | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
One, if we assume that the null hypothesis is true, and one where we assume that the null hypothesis is false and the true population parameter is something different than the null hypothesis is saying. So for example, let's say that we have a null hypothesis that our population mean is equal to, let's just call it mu one, and we have an alternative hypothesis, so H sub A, that says, hey, no, the population mean is not equal to mu one. So if you assumed a world where the null hypothesis is true, so I'll do that in blue. So if we assume the null hypothesis is true, what would be our sampling distribution? Remember, what we do in significance tests is we have some form of a population, let me draw that. You have a population right over here, and our hypotheses are making some statement about a parameter in that population, and to test it, we take a sample of a certain size, we calculate a statistic, in this case, we would be the sample mean, and we say if we assume that our null hypothesis is true, what is the probability of getting that sample statistic? And if that's below a threshold, which we call a significance level, we reject the null hypothesis. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
So if we assume the null hypothesis is true, what would be our sampling distribution? Remember, what we do in significance tests is we have some form of a population, let me draw that. You have a population right over here, and our hypotheses are making some statement about a parameter in that population, and to test it, we take a sample of a certain size, we calculate a statistic, in this case, we would be the sample mean, and we say if we assume that our null hypothesis is true, what is the probability of getting that sample statistic? And if that's below a threshold, which we call a significance level, we reject the null hypothesis. And so that world that we have been living in, one way to think about it, in a world where you assume the null hypothesis is true, you might have a sampling distribution that looks something like this. The null hypothesis is true, then the center of your sampling distribution would be right over here at mu one, and given your sample size, you would get a certain sampling distribution for the sample means. If your sample size increases, this will be narrow, or if it decreases, this thing is going to be wider, and you set a significance level, which is essentially your probability of rejecting the null hypothesis, even if it is true. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
And if that's below a threshold, which we call a significance level, we reject the null hypothesis. And so that world that we have been living in, one way to think about it, in a world where you assume the null hypothesis is true, you might have a sampling distribution that looks something like this. The null hypothesis is true, then the center of your sampling distribution would be right over here at mu one, and given your sample size, you would get a certain sampling distribution for the sample means. If your sample size increases, this will be narrow, or if it decreases, this thing is going to be wider, and you set a significance level, which is essentially your probability of rejecting the null hypothesis, even if it is true. You could even view it as, and we've talked about it, you can view your significance level as a probability of making a type one error. So your significance level is some area, and so let's say it's this area that I'm shading in orange right over here, that would be your significance level. So if you took a sample right over here, and you calculated its sample mean, and you happen to fall in this area, or this area, or this area right over here, then you would reject your null hypothesis. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
If your sample size increases, this will be narrow, or if it decreases, this thing is going to be wider, and you set a significance level, which is essentially your probability of rejecting the null hypothesis, even if it is true. You could even view it as, and we've talked about it, you can view your significance level as a probability of making a type one error. So your significance level is some area, and so let's say it's this area that I'm shading in orange right over here, that would be your significance level. So if you took a sample right over here, and you calculated its sample mean, and you happen to fall in this area, or this area, or this area right over here, then you would reject your null hypothesis. Now, if the null hypothesis actually was true, you would be committing a type one error without knowing about it. But for power, we are concerned with a type two error. So in this one, it's a conditional probability that our null hypothesis is false. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
So if you took a sample right over here, and you calculated its sample mean, and you happen to fall in this area, or this area, or this area right over here, then you would reject your null hypothesis. Now, if the null hypothesis actually was true, you would be committing a type one error without knowing about it. But for power, we are concerned with a type two error. So in this one, it's a conditional probability that our null hypothesis is false. And so let's construct another sampling distribution in the case where our null hypothesis is false. So let me just continue this line right over here, and I'll do that. So let's imagine a world where our null hypothesis is false, and it's actually the case that our mean is mu two. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
So in this one, it's a conditional probability that our null hypothesis is false. And so let's construct another sampling distribution in the case where our null hypothesis is false. So let me just continue this line right over here, and I'll do that. So let's imagine a world where our null hypothesis is false, and it's actually the case that our mean is mu two. And let's say that mu two is right over here. And in this reality, our sampling distribution might look something like this. Once again, it'll be for a given sample size. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
So let's imagine a world where our null hypothesis is false, and it's actually the case that our mean is mu two. And let's say that mu two is right over here. And in this reality, our sampling distribution might look something like this. Once again, it'll be for a given sample size. The larger the sample size, the narrower this bell curve would be. And so it might look something like this. So in which situation? | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
Once again, it'll be for a given sample size. The larger the sample size, the narrower this bell curve would be. And so it might look something like this. So in which situation? So in this world, we should be rejecting the null hypothesis. But what are the samples in which case we are not rejecting the null hypothesis even though we should? Well, we're not going to reject the null hypothesis if we get samples in, if we get a sample here or a sample here or a sample here. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
So in which situation? So in this world, we should be rejecting the null hypothesis. But what are the samples in which case we are not rejecting the null hypothesis even though we should? Well, we're not going to reject the null hypothesis if we get samples in, if we get a sample here or a sample here or a sample here. A sample where if you assume the null hypothesis is true, the probability isn't that unlikely. And so the probability of making a type two error when we should reject the null hypothesis but we don't is actually this area right over here. And the power, the probability of rejecting the null hypothesis given that it's false, so given that it's false would be this red distribution, that would be the rest of this area right over here. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
Well, we're not going to reject the null hypothesis if we get samples in, if we get a sample here or a sample here or a sample here. A sample where if you assume the null hypothesis is true, the probability isn't that unlikely. And so the probability of making a type two error when we should reject the null hypothesis but we don't is actually this area right over here. And the power, the probability of rejecting the null hypothesis given that it's false, so given that it's false would be this red distribution, that would be the rest of this area right over here. So how can we increase the power? Well, one way is to increase our alpha, increase our significance level. If we increase our significance level, say from that, remember the significance level is an area. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
And the power, the probability of rejecting the null hypothesis given that it's false, so given that it's false would be this red distribution, that would be the rest of this area right over here. So how can we increase the power? Well, one way is to increase our alpha, increase our significance level. If we increase our significance level, say from that, remember the significance level is an area. So if we want it to go up, if we increase the area, and it looked something like that, now by expanding that significance area, we have increased the power because now this yellow area is larger, we've pushed this boundary to the left of it. Now you might say, oh, well, hey, if we wanna increase the power, power sounds like a good thing. Why don't we just always increase alpha? | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
If we increase our significance level, say from that, remember the significance level is an area. So if we want it to go up, if we increase the area, and it looked something like that, now by expanding that significance area, we have increased the power because now this yellow area is larger, we've pushed this boundary to the left of it. Now you might say, oh, well, hey, if we wanna increase the power, power sounds like a good thing. Why don't we just always increase alpha? Well, the problem with that is if you increase alpha, so let me write this down. So if you take alpha, your significance level, and you increase it, that will increase the power, that will increase the power, but it's also going to increase your probability of a type one error. Because remember, that's one way to conceptualize what alpha is, what your significance level is. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
Why don't we just always increase alpha? Well, the problem with that is if you increase alpha, so let me write this down. So if you take alpha, your significance level, and you increase it, that will increase the power, that will increase the power, but it's also going to increase your probability of a type one error. Because remember, that's one way to conceptualize what alpha is, what your significance level is. It's a probability of a type one error. Now what are other ways to increase your power? Well, if you increase your sample size, then both of these distributions will, these sampling distributions are going to get narrower. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
Because remember, that's one way to conceptualize what alpha is, what your significance level is. It's a probability of a type one error. Now what are other ways to increase your power? Well, if you increase your sample size, then both of these distributions will, these sampling distributions are going to get narrower. And so if these sampling distributions, if both of these sampling distributions get narrower, then that situation where you are not rejecting your null hypothesis, even though you should, is going to have a lot less area. There's gonna be, one way to think about it, there's going to be a lot less overlap between these two sampling distributions. And so let me write that down. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
Well, if you increase your sample size, then both of these distributions will, these sampling distributions are going to get narrower. And so if these sampling distributions, if both of these sampling distributions get narrower, then that situation where you are not rejecting your null hypothesis, even though you should, is going to have a lot less area. There's gonna be, one way to think about it, there's going to be a lot less overlap between these two sampling distributions. And so let me write that down. So another way is to, if you increase n, your sample size, that's going to increase your power. And this, in general, is always a good thing if you can do it. Now other things that may or may not be under your control is, well, the less variability there is in the data set, that would also make these sampling distributions narrower, and that would also increase the power. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
And so let me write that down. So another way is to, if you increase n, your sample size, that's going to increase your power. And this, in general, is always a good thing if you can do it. Now other things that may or may not be under your control is, well, the less variability there is in the data set, that would also make these sampling distributions narrower, and that would also increase the power. So less variability, and you could measure that as by variance or standard deviation of your underlying data set, that would increase your power. Another thing that would increase the power is if the true parameter is further away than what the null hypothesis is saying. So if you say true parameter far from null hypothesis, what it's saying, that also will increase the power. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
Now other things that may or may not be under your control is, well, the less variability there is in the data set, that would also make these sampling distributions narrower, and that would also increase the power. So less variability, and you could measure that as by variance or standard deviation of your underlying data set, that would increase your power. Another thing that would increase the power is if the true parameter is further away than what the null hypothesis is saying. So if you say true parameter far from null hypothesis, what it's saying, that also will increase the power. So these two are not typically under your control, but the sample size is, and the significance level is. Significance level, there's a trade-off, though. If you increase the power through that, you're also increasing the probability of a type one error. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
So if you say true parameter far from null hypothesis, what it's saying, that also will increase the power. So these two are not typically under your control, but the sample size is, and the significance level is. Significance level, there's a trade-off, though. If you increase the power through that, you're also increasing the probability of a type one error. So for a lot of researchers, they might say, hey, if a type two error is worse, I'm willing to make this trade-off. I'll increase the significance level. But if a type one error is actually what I'm afraid of, then I wouldn't wanna use this lever. | Introduction to power in significance tests AP Statistics Khan Academy.mp3 |
And then each of these pairs of bar charts give us for an individual, where the blue bar is, for example, how Ishan did on the midterm. The yellow is how he did on the final. For Emily, the blue is how she did on the midterm. Yellow is how she did on the final. And we have a bunch of interesting questions here. The first question, what was the median score for the final exam? So just as a review, median literally means what was the middle score. | Reading bar charts putting it together with central tendency Pre-Algebra Khan Academy.mp3 |
Yellow is how she did on the final. And we have a bunch of interesting questions here. The first question, what was the median score for the final exam? So just as a review, median literally means what was the middle score. So really, we should list all the scores for the final exam and sort them in order, and then figure out what the middle score actually was. So let's look at all the scores on final exams. So scores on the final exam. | Reading bar charts putting it together with central tendency Pre-Algebra Khan Academy.mp3 |
So just as a review, median literally means what was the middle score. So really, we should list all the scores for the final exam and sort them in order, and then figure out what the middle score actually was. So let's look at all the scores on final exams. So scores on the final exam. So you have 100 here. Ishan got 100 on the final exam. Remember, this yellow bar is the final exam. | Reading bar charts putting it together with central tendency Pre-Algebra Khan Academy.mp3 |
So scores on the final exam. So you have 100 here. Ishan got 100 on the final exam. Remember, this yellow bar is the final exam. So there's 100. Emily also got 100 on the final exam. Looks like it was an easy final exam. | Reading bar charts putting it together with central tendency Pre-Algebra Khan Academy.mp3 |
Remember, this yellow bar is the final exam. So there's 100. Emily also got 100 on the final exam. Looks like it was an easy final exam. Daniel also got 100 on the final exam. And then let's see. Jessica looks like she got a 75. | Reading bar charts putting it together with central tendency Pre-Algebra Khan Academy.mp3 |
Looks like it was an easy final exam. Daniel also got 100 on the final exam. And then let's see. Jessica looks like she got a 75. And then William got an 80. So if we were to sort these in order, and let's say we did it in increasing order, you could write, well, the lowest score was a 75. Then you have an 80. | Reading bar charts putting it together with central tendency Pre-Algebra Khan Academy.mp3 |
Jessica looks like she got a 75. And then William got an 80. So if we were to sort these in order, and let's say we did it in increasing order, you could write, well, the lowest score was a 75. Then you have an 80. And then you have three 100s. 100, another 100, and another 100. So there's five scores right over here. | Reading bar charts putting it together with central tendency Pre-Algebra Khan Academy.mp3 |
Then you have an 80. And then you have three 100s. 100, another 100, and another 100. So there's five scores right over here. So you will have a middle. If you had an even number, then you would take the mean of the two center values. But here you have one center value. | Reading bar charts putting it together with central tendency Pre-Algebra Khan Academy.mp3 |
So there's five scores right over here. So you will have a middle. If you had an even number, then you would take the mean of the two center values. But here you have one center value. And when you order it like this, it's pretty clear that your center value, your middle value, is 100. So the median score for the final exam is 100. And that's because you had so many 100s here that the median, the middle score, was still 100. | Reading bar charts putting it together with central tendency Pre-Algebra Khan Academy.mp3 |
But here you have one center value. And when you order it like this, it's pretty clear that your center value, your middle value, is 100. So the median score for the final exam is 100. And that's because you had so many 100s here that the median, the middle score, was still 100. What is the midrange of the midterm scores? I'll do it in blue in honor of the color of the bars for the midterm. So the midrange is the mean of your highest and lowest scores. | Reading bar charts putting it together with central tendency Pre-Algebra Khan Academy.mp3 |
And that's because you had so many 100s here that the median, the middle score, was still 100. What is the midrange of the midterm scores? I'll do it in blue in honor of the color of the bars for the midterm. So the midrange is the mean of your highest and lowest scores. So let's calculate this. So let's go to the midrange. The midrange, you could view it as the arithmetic mean or the average of your highest and lowest scores. | Reading bar charts putting it together with central tendency Pre-Algebra Khan Academy.mp3 |
So the midrange is the mean of your highest and lowest scores. So let's calculate this. So let's go to the midrange. The midrange, you could view it as the arithmetic mean or the average of your highest and lowest scores. So the midrange of midterm. So let's see. The highest midterm score, looking at the blue, the highest one is right here. | Reading bar charts putting it together with central tendency Pre-Algebra Khan Academy.mp3 |
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