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And what would happen if you multiply everything by five? Well, once again, you still have the same ordering, and so it should just multiply that by five. Yep, the middle number is now gonna be five times larger. So both of these measures of central tendency, if you shift all the data points or if you scale them up, you're going to similarly shift or scale up these measures of central tendency. Now let's think about these measures of spread. See if that's the same with these measures of spread. So standard deviation. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
So both of these measures of central tendency, if you shift all the data points or if you scale them up, you're going to similarly shift or scale up these measures of central tendency. Now let's think about these measures of spread. See if that's the same with these measures of spread. So standard deviation. So S-T-D-E-V, I'm gonna take the population standard deviation. I'm assuming that this is my entire population. So let me, why is it, so let me make sure I'm doing, so standard deviation of all of this is going to be 2.99. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
So standard deviation. So S-T-D-E-V, I'm gonna take the population standard deviation. I'm assuming that this is my entire population. So let me, why is it, so let me make sure I'm doing, so standard deviation of all of this is going to be 2.99. Let's see what happens when I shift everything by five. Actually, pause the video. What do you think is going to happen? | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
So let me, why is it, so let me make sure I'm doing, so standard deviation of all of this is going to be 2.99. Let's see what happens when I shift everything by five. Actually, pause the video. What do you think is going to happen? This is a measure of spread. So if you shift, I'll tell you what I think. If I shift everything by the same amount, the mean shifts, but the distance of everything from the mean should not change. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
What do you think is going to happen? This is a measure of spread. So if you shift, I'll tell you what I think. If I shift everything by the same amount, the mean shifts, but the distance of everything from the mean should not change. So the standard deviation should not change, I don't think, in this example. And indeed, it does not change. So if we shift the data sets, in this case we shifted it up by five, or if we shifted it down by one, your measure of spread, in this case standard deviation, should not change, or at least the standard deviation measure of spread does not change. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
If I shift everything by the same amount, the mean shifts, but the distance of everything from the mean should not change. So the standard deviation should not change, I don't think, in this example. And indeed, it does not change. So if we shift the data sets, in this case we shifted it up by five, or if we shifted it down by one, your measure of spread, in this case standard deviation, should not change, or at least the standard deviation measure of spread does not change. But if we scale it, well, I think it should change, because you can imagine a very simple data set, the things that were a certain amount of distance from the mean are now going to be five times further from the mean. So I think this actually should, we should multiply by five here, and it does look like that is the case. If I multiply this by five. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
So if we shift the data sets, in this case we shifted it up by five, or if we shifted it down by one, your measure of spread, in this case standard deviation, should not change, or at least the standard deviation measure of spread does not change. But if we scale it, well, I think it should change, because you can imagine a very simple data set, the things that were a certain amount of distance from the mean are now going to be five times further from the mean. So I think this actually should, we should multiply by five here, and it does look like that is the case. If I multiply this by five. So scaling the data set will scale the standard deviation in a similar way. What about interquartile range? Where it's essentially we're taking the third quartile and subtracting from that the first quartile to figure out kind of the range of the middle 50%. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
If I multiply this by five. So scaling the data set will scale the standard deviation in a similar way. What about interquartile range? Where it's essentially we're taking the third quartile and subtracting from that the first quartile to figure out kind of the range of the middle 50%. And so let's do that. We can have the quartile function equals quartile, and then we wanna look at our data, and we want the third quartile. So that's gonna calculate the third quartile minus quartile, same data set. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
Where it's essentially we're taking the third quartile and subtracting from that the first quartile to figure out kind of the range of the middle 50%. And so let's do that. We can have the quartile function equals quartile, and then we wanna look at our data, and we want the third quartile. So that's gonna calculate the third quartile minus quartile, same data set. So now we wanna select it again. So same data set, but this is now gonna be the first quartile. So this is gonna give us our interquartile range. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
So that's gonna calculate the third quartile minus quartile, same data set. So now we wanna select it again. So same data set, but this is now gonna be the first quartile. So this is gonna give us our interquartile range. This is the, calculates the third quartile on that data set, and this calculates the first quartile on that data set. And we get 2.75. Now let's think about what the inter, whether the interquartile range should change. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
So this is gonna give us our interquartile range. This is the, calculates the third quartile on that data set, and this calculates the first quartile on that data set. And we get 2.75. Now let's think about what the inter, whether the interquartile range should change. And I don't think it will, because remember, everything shifts. And even though the first quartile is gonna be five more, but the third quartile is gonna be five more as well. So the difference shouldn't change. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
Now let's think about what the inter, whether the interquartile range should change. And I don't think it will, because remember, everything shifts. And even though the first quartile is gonna be five more, but the third quartile is gonna be five more as well. So the difference shouldn't change. And indeed, look, the distance does not change, or the difference does not change. But similarly, if we scale everything up, if we were to scale up the first quartile and the third quartile by five, well then their difference should scale up by five. And we see that right over there. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
So the difference shouldn't change. And indeed, look, the distance does not change, or the difference does not change. But similarly, if we scale everything up, if we were to scale up the first quartile and the third quartile by five, well then their difference should scale up by five. And we see that right over there. So the big takeaway here, and I just use the example of shifting up by five and scaling up by five, but you could subtract by any number, and you could divide by a number as well. The typical measures of central tendency, mean and median, they both shift and scale as you shift and scale the data. But your typical measures of spread, standard deviation and interquartile range, they don't change if you shift the data, but they do change and they scale as you scale the data. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
Now, the first thing I want to do in this video is calculate the total sum of squares. So I'll call that SST. S-S, sum of squares, total. And you could view it as really the numerator when you calculate variance. So you're just going to take the distance between each of these data points and the mean of all of these data points squared. I mean, just take that sum. We're not going to divide by the degree of freedom, which you'd normally do if you were calculating sample variance. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
And you could view it as really the numerator when you calculate variance. So you're just going to take the distance between each of these data points and the mean of all of these data points squared. I mean, just take that sum. We're not going to divide by the degree of freedom, which you'd normally do if you were calculating sample variance. Now, what is this going to be? Well, the first thing we need to do, we have to figure out the mean of all of this stuff over here. And I'm actually going to call that the grand mean. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
We're not going to divide by the degree of freedom, which you'd normally do if you were calculating sample variance. Now, what is this going to be? Well, the first thing we need to do, we have to figure out the mean of all of this stuff over here. And I'm actually going to call that the grand mean. I'm going to call that the grand mean. And I'm going to show you in a second that it's the same thing as the mean of the means of each of these data sets. So let's calculate the grand mean. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
And I'm actually going to call that the grand mean. I'm going to call that the grand mean. And I'm going to show you in a second that it's the same thing as the mean of the means of each of these data sets. So let's calculate the grand mean. So it's going to be 3 plus 2 plus 1, 3 plus 2 plus 1, plus 5 plus 3 plus 4, plus 5 plus 3 plus 4, plus 5 plus 6 plus 7. Plus 5 plus 6 plus 7. And then we have 9 data points here. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
So let's calculate the grand mean. So it's going to be 3 plus 2 plus 1, 3 plus 2 plus 1, plus 5 plus 3 plus 4, plus 5 plus 3 plus 4, plus 5 plus 6 plus 7. Plus 5 plus 6 plus 7. And then we have 9 data points here. We have 9 data points, so we'll divide by 9. And what is this going to be equal to? 3 plus 2 plus 1 is 6. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
And then we have 9 data points here. We have 9 data points, so we'll divide by 9. And what is this going to be equal to? 3 plus 2 plus 1 is 6. 6 plus, let me just add, so these are 6. 5 plus 3 plus 4 is, that's 12. And then 5 plus 6 plus 7 is 18. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
3 plus 2 plus 1 is 6. 6 plus, let me just add, so these are 6. 5 plus 3 plus 4 is, that's 12. And then 5 plus 6 plus 7 is 18. And then 6 plus 12 is 18, plus another 18 is 36. Divided by 9 is equal to 4. Let me show you that that's the exact same thing as the mean of the means. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
And then 5 plus 6 plus 7 is 18. And then 6 plus 12 is 18, plus another 18 is 36. Divided by 9 is equal to 4. Let me show you that that's the exact same thing as the mean of the means. So the mean of this group 1 over here, let me do it in that same green. The mean of group 1 over here is 3 plus 2 plus 1, that's that 6 right over here, divided by 3 data points. So that will be equal to 2. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
Let me show you that that's the exact same thing as the mean of the means. So the mean of this group 1 over here, let me do it in that same green. The mean of group 1 over here is 3 plus 2 plus 1, that's that 6 right over here, divided by 3 data points. So that will be equal to 2. The mean of group 2, the sum here is 12, we saw that right over here. 5 plus 3 plus 4 is 12, divided by 3 is 4, because we have 3 data points. And then the mean of group 3, 5 plus 6 plus 7 is 18, divided by 3 is 6. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
So that will be equal to 2. The mean of group 2, the sum here is 12, we saw that right over here. 5 plus 3 plus 4 is 12, divided by 3 is 4, because we have 3 data points. And then the mean of group 3, 5 plus 6 plus 7 is 18, divided by 3 is 6. So if you were to take the mean of the means, which is another way of viewing this grand mean, you have 2 plus 4 plus 6, which is 12, divided by 3 means here, and once again you would get 4. So you could view this as the mean of all of the data in all of the groups, or the mean of the means of each of these groups. But either way, now that we've calculated it, we can actually figure out the total sum of squares. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
And then the mean of group 3, 5 plus 6 plus 7 is 18, divided by 3 is 6. So if you were to take the mean of the means, which is another way of viewing this grand mean, you have 2 plus 4 plus 6, which is 12, divided by 3 means here, and once again you would get 4. So you could view this as the mean of all of the data in all of the groups, or the mean of the means of each of these groups. But either way, now that we've calculated it, we can actually figure out the total sum of squares. So let's do that. So it's going to be equal to 3 minus 4, the 4 is this 4 right over here, squared, plus 2 minus 4 squared, plus 1 minus 4 squared, now I'll do these guys over here in purple, plus 5 minus 4 squared, plus 3 minus 4 squared, plus 4 minus 4 squared, let me scroll over a little bit, plus 4 minus 4 squared, now we only have 3 left, plus 5 minus 4 squared, plus 6 minus 4 squared, plus 7 minus 4 squared. And what does this give us? | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
But either way, now that we've calculated it, we can actually figure out the total sum of squares. So let's do that. So it's going to be equal to 3 minus 4, the 4 is this 4 right over here, squared, plus 2 minus 4 squared, plus 1 minus 4 squared, now I'll do these guys over here in purple, plus 5 minus 4 squared, plus 3 minus 4 squared, plus 4 minus 4 squared, let me scroll over a little bit, plus 4 minus 4 squared, now we only have 3 left, plus 5 minus 4 squared, plus 6 minus 4 squared, plus 7 minus 4 squared. And what does this give us? Well up here, this first, so this is going to be equal to 3 minus 4, difference is 1, you square it, it's actually negative 1, but you square it, you get 1, plus, you get negative 2 squared is 4, plus negative 3 squared, negative 3 squared is 9. And then we have here in the magenta, 5 minus 4 is 1, squared is still 1, 3 minus 4 squared is 1, you square it again, you still get 1, 5 minus 4 is just a 0, well I'll just write the 0 there just to show you that we actually calculated that, and then we have these last 3 data points, 5 minus 4 squared, that's 1, 6 minus 4 squared, that is 4, that's 2 squared, and then plus 7 minus 4 is 3, squared is 9. So what's this going to be equal to? | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
And what does this give us? Well up here, this first, so this is going to be equal to 3 minus 4, difference is 1, you square it, it's actually negative 1, but you square it, you get 1, plus, you get negative 2 squared is 4, plus negative 3 squared, negative 3 squared is 9. And then we have here in the magenta, 5 minus 4 is 1, squared is still 1, 3 minus 4 squared is 1, you square it again, you still get 1, 5 minus 4 is just a 0, well I'll just write the 0 there just to show you that we actually calculated that, and then we have these last 3 data points, 5 minus 4 squared, that's 1, 6 minus 4 squared, that is 4, that's 2 squared, and then plus 7 minus 4 is 3, squared is 9. So what's this going to be equal to? So I have 1 plus 4 plus 9 right over here, that's 5 plus 9, this right over here is 14, and then we also have another 14 right over here, because we have a 1 plus 4 plus 9, so that right over there is also 14, and then we have 2 over here, so it's going to be 28, 14 times 2, 14 plus 14 is 28, plus 2 is 30, is equal to 30. So our total sum of squares, and actually if we wanted the variance here, we would divide this by the degrees of freedom, and we've learned multiple times the degrees of freedom here, so let's say that we have, so we know that we have m groups over here, so let me just write it as m, and I'm not going to prove things rigorously here, but I want to show you where some of these strange formulas that show up in statistics books actually come from, without proving it rigorously, more to give you the intuition. So we have m groups here, and each group here has n members. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
So what's this going to be equal to? So I have 1 plus 4 plus 9 right over here, that's 5 plus 9, this right over here is 14, and then we also have another 14 right over here, because we have a 1 plus 4 plus 9, so that right over there is also 14, and then we have 2 over here, so it's going to be 28, 14 times 2, 14 plus 14 is 28, plus 2 is 30, is equal to 30. So our total sum of squares, and actually if we wanted the variance here, we would divide this by the degrees of freedom, and we've learned multiple times the degrees of freedom here, so let's say that we have, so we know that we have m groups over here, so let me just write it as m, and I'm not going to prove things rigorously here, but I want to show you where some of these strange formulas that show up in statistics books actually come from, without proving it rigorously, more to give you the intuition. So we have m groups here, and each group here has n members. So how many total members do we have here? Well we had m times n, or 9, 3 times 3 total members. So our degrees of freedom, and remember, you have this many, however many data points you had, minus 1 degrees of freedom, because if you know, if you knew the mean of means, if you know the mean of means, if you assume you knew that, then you only would, then only 9 minus 1, only 8 of these are going to give you new information, because if you know that, you could calculate the last one, or it really doesn't have to be the last one, if you have the other 8, you could calculate this one, if you have 8 of them, you can always calculate the 9th one, using the mean of means. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
So we have m groups here, and each group here has n members. So how many total members do we have here? Well we had m times n, or 9, 3 times 3 total members. So our degrees of freedom, and remember, you have this many, however many data points you had, minus 1 degrees of freedom, because if you know, if you knew the mean of means, if you know the mean of means, if you assume you knew that, then you only would, then only 9 minus 1, only 8 of these are going to give you new information, because if you know that, you could calculate the last one, or it really doesn't have to be the last one, if you have the other 8, you could calculate this one, if you have 8 of them, you can always calculate the 9th one, using the mean of means. So one way to think about it is that there's only 8 independent measurements here, or if we want to talk in terms of general, if we want to talk generally, there are m times n, so that tells us the total number of samples, minus 1 degrees of freedom. So if we were actually calculating the variance here, we would just divide 30 by m times n minus 1, or this is another way of saying 8 degrees of freedom, for this exact example. We would take 30 divided by 8, and we would actually have the variance for this entire group, for the group of 9, when you combine them. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
So our degrees of freedom, and remember, you have this many, however many data points you had, minus 1 degrees of freedom, because if you know, if you knew the mean of means, if you know the mean of means, if you assume you knew that, then you only would, then only 9 minus 1, only 8 of these are going to give you new information, because if you know that, you could calculate the last one, or it really doesn't have to be the last one, if you have the other 8, you could calculate this one, if you have 8 of them, you can always calculate the 9th one, using the mean of means. So one way to think about it is that there's only 8 independent measurements here, or if we want to talk in terms of general, if we want to talk generally, there are m times n, so that tells us the total number of samples, minus 1 degrees of freedom. So if we were actually calculating the variance here, we would just divide 30 by m times n minus 1, or this is another way of saying 8 degrees of freedom, for this exact example. We would take 30 divided by 8, and we would actually have the variance for this entire group, for the group of 9, when you combine them. I'll leave you here in this video, in the next video, we're going to try to figure out how much of this total variance, how much of this total sum of squared, the total squared sum, total variation, comes from the variation within each of these groups, versus the variation between the groups. And I think you get a sense of where this whole analysis of variance is coming from. It's the sense that, look, there's a variance of this entire sample of 9, but some of that variance, if these groups are different in some way, might come from the variation from being in different groups, versus the variation from being within a group. | ANOVA 1 Calculating SST (total sum of squares) Probability and Statistics Khan Academy.mp3 |
So I've defined some sets here. And just to make things interesting, I haven't only put numbers in these sets. I've even put some colors and some little yellow stars here. And what I want you to figure out is what would this set be, this crazy thing that involves relative complements, intersections, unions, absolute complements. So I encourage you to pause it and try to figure out what this set would be. Well, let's give it a shot. And the key here is to really break it down, work on the parentheses, the stuff in the parentheses first, just as you would do if you were trying to parse a traditional mathematical statement. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
And what I want you to figure out is what would this set be, this crazy thing that involves relative complements, intersections, unions, absolute complements. So I encourage you to pause it and try to figure out what this set would be. Well, let's give it a shot. And the key here is to really break it down, work on the parentheses, the stuff in the parentheses first, just as you would do if you were trying to parse a traditional mathematical statement. And then it should hopefully make a little bit of sense. So a good place to start might be to try to figure out what is the relative complement of C in B. Or another way of thinking about it is what is B minus C? | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
And the key here is to really break it down, work on the parentheses, the stuff in the parentheses first, just as you would do if you were trying to parse a traditional mathematical statement. And then it should hopefully make a little bit of sense. So a good place to start might be to try to figure out what is the relative complement of C in B. Or another way of thinking about it is what is B minus C? What is B if you take out all the stuff with C in it? So let me write this down. The relative complement of C in B, or you could call this B minus C, this is all the stuff in B with all the stuff in C taken out of it. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
Or another way of thinking about it is what is B minus C? What is B if you take out all the stuff with C in it? So let me write this down. The relative complement of C in B, or you could call this B minus C, this is all the stuff in B with all the stuff in C taken out of it. So let's think about what this would be. B has a 0. Does C have a 0? | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
The relative complement of C in B, or you could call this B minus C, this is all the stuff in B with all the stuff in C taken out of it. So let's think about what this would be. B has a 0. Does C have a 0? No, so we don't have to take out the 0. B has a 17. Does C have a 17? | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
Does C have a 0? No, so we don't have to take out the 0. B has a 17. Does C have a 17? Yes, it does. So we take out the 17. B has a 3. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
Does C have a 17? Yes, it does. So we take out the 17. B has a 3. But C has a 3, so we take that out. B has a blue. C does not have a blue, so we leave the blue in. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
B has a 3. But C has a 3, so we take that out. B has a blue. C does not have a blue, so we leave the blue in. So let me write down, we leave the blue in. And then B has a gold star. C also has a gold star, so we take the gold star out. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
C does not have a blue, so we leave the blue in. So let me write down, we leave the blue in. And then B has a gold star. C also has a gold star, so we take the gold star out. So the relative complement of C in B is just the set of 0 and this blue written in blue. So let me write this down. Now it gets interesting. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
C also has a gold star, so we take the gold star out. So the relative complement of C in B is just the set of 0 and this blue written in blue. So let me write this down. Now it gets interesting. We're going to take the absolute complement of that. So let me write this down. So B, the absolute complement of this business is going to be all things, let me write this, the set of all things in our universe that are neither a 0 or a blue. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
Now it gets interesting. We're going to take the absolute complement of that. So let me write this down. So B, the absolute complement of this business is going to be all things, let me write this, the set of all things in our universe that are neither a 0 or a blue. That's the only way I can describe it right now. I haven't really defined the universe well. We already see that our universe definitely contains some integers. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
So B, the absolute complement of this business is going to be all things, let me write this, the set of all things in our universe that are neither a 0 or a blue. That's the only way I can describe it right now. I haven't really defined the universe well. We already see that our universe definitely contains some integers. It contains colors. It contains some stars. So this is all I can really say. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
We already see that our universe definitely contains some integers. It contains colors. It contains some stars. So this is all I can really say. This is a set of all things in the universe that are neither a 0 or a blue. So fair enough. So far we've figured out all of this stuff. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
So this is all I can really say. This is a set of all things in the universe that are neither a 0 or a blue. So fair enough. So far we've figured out all of this stuff. Let me box this off. So that is that right over there. And now we want to find the intersection. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
So far we've figured out all of this stuff. Let me box this off. So that is that right over there. And now we want to find the intersection. We need to find the intersection of A and this business. Let me write that down. So it's going to be A intersected with the relative complement of C in B and the absolute complement of that. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
And now we want to find the intersection. We need to find the intersection of A and this business. Let me write that down. So it's going to be A intersected with the relative complement of C in B and the absolute complement of that. So this is going to be the intersection of the set A and the set of all things in the universe that are neither a 0 or a blue. So it's essentially the things that satisfy both of these, that it has to be in set A and it has to be in the set of all things in the universe that are neither a 0 or a blue. So let's think about what this is. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
So it's going to be A intersected with the relative complement of C in B and the absolute complement of that. So this is going to be the intersection of the set A and the set of all things in the universe that are neither a 0 or a blue. So it's essentially the things that satisfy both of these, that it has to be in set A and it has to be in the set of all things in the universe that are neither a 0 or a blue. So let's think about what this is. So the number 3 is in set A and it's in the set of all things in the universe that are neither a 0 or blue. So let's throw a 3 in there. The number 7, it's an A and it's in the set of all things in the universe that are neither a 0 or a blue. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
So let's think about what this is. So the number 3 is in set A and it's in the set of all things in the universe that are neither a 0 or blue. So let's throw a 3 in there. The number 7, it's an A and it's in the set of all things in the universe that are neither a 0 or a blue. So let's put a 7 there. Negative 5 also meets that constraint. A 0 does not meet that constraint. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
The number 7, it's an A and it's in the set of all things in the universe that are neither a 0 or a blue. So let's put a 7 there. Negative 5 also meets that constraint. A 0 does not meet that constraint. A 0 is an A, but it's not in the set of all things in the universe that are neither a 0 or a blue, because it is a 0. So we're not going to throw a 0 in there. And then a 13 is an A, and it's in the set of all things in the universe that are neither a 0 or a blue. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
A 0 does not meet that constraint. A 0 is an A, but it's not in the set of all things in the universe that are neither a 0 or a blue, because it is a 0. So we're not going to throw a 0 in there. And then a 13 is an A, and it's in the set of all things in the universe that are neither a 0 or a blue. So we could throw a 13 in there. So we've simplified things a good bit. This whole crazy business, all of this crazy business, has simplified to this set right over here. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
And then a 13 is an A, and it's in the set of all things in the universe that are neither a 0 or a blue. So we could throw a 13 in there. So we've simplified things a good bit. This whole crazy business, all of this crazy business, has simplified to this set right over here. Now, we want to find the relative complement of this business in A. So let me pick another color here. So we want to find the relative complement of this business in A. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
This whole crazy business, all of this crazy business, has simplified to this set right over here. Now, we want to find the relative complement of this business in A. So let me pick another color here. So we want to find the relative complement of this business in A. So now I'll just actually write out the set. 3, 7, negative 5, 13. Actually, let me write out both of them, just to make it, just so that we can really just visualize them both right over here. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
So we want to find the relative complement of this business in A. So now I'll just actually write out the set. 3, 7, negative 5, 13. Actually, let me write out both of them, just to make it, just so that we can really just visualize them both right over here. So A is this. It is 3, 7, negative 5, 0, and 13. And I could write the relative complement sign. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
Actually, let me write out both of them, just to make it, just so that we can really just visualize them both right over here. So A is this. It is 3, 7, negative 5, 0, and 13. And I could write the relative complement sign. Or actually, let me just write relative complement. I was going to write minus. And so in all of this business, we already figured out is a 3, a 7, a negative 5, and a 13. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
And I could write the relative complement sign. Or actually, let me just write relative complement. I was going to write minus. And so in all of this business, we already figured out is a 3, a 7, a negative 5, and a 13. So it's essentially, start with this set and take out all the stuff that are in this set. So this is going to be equal to, so you see, we're going to have to take out a 3 out of this set. We're going to take out a 7. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
And so in all of this business, we already figured out is a 3, a 7, a negative 5, and a 13. So it's essentially, start with this set and take out all the stuff that are in this set. So this is going to be equal to, so you see, we're going to have to take out a 3 out of this set. We're going to take out a 7. We're going to take out a negative 5. And we're going to take out a 13. So we're just left with the set that contains a 0. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
We're going to take out a 7. We're going to take out a negative 5. And we're going to take out a 13. So we're just left with the set that contains a 0. So all of this business right over here has simplified to a set that only contains 0. Now let's think about what B intersect C is. These are all of the things that are in both B and C. So this is going to be B intersect C. Let's see. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
So we're just left with the set that contains a 0. So all of this business right over here has simplified to a set that only contains 0. Now let's think about what B intersect C is. These are all of the things that are in both B and C. So this is going to be B intersect C. Let's see. 0 is not in both of them. 17 is in both of them. So we'll throw a 17 there. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
These are all of the things that are in both B and C. So this is going to be B intersect C. Let's see. 0 is not in both of them. 17 is in both of them. So we'll throw a 17 there. The number 3 is in both of them. Blue is not in both of them. The star is in both of them. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
So we'll throw a 17 there. The number 3 is in both of them. Blue is not in both of them. The star is in both of them. So I'll put the little gold star right over there. And so that's B intersect C. And so we're essentially going to take the union of this crazy thing, which ended up just being a set with a 0 in it. We're taking the union of that and B intersect C. And we deserve a drum roll now. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
The star is in both of them. So I'll put the little gold star right over there. And so that's B intersect C. And so we're essentially going to take the union of this crazy thing, which ended up just being a set with a 0 in it. We're taking the union of that and B intersect C. And we deserve a drum roll now. This is all going to be equal to, we're just going to combine these two sets. It's going to be the set with a 0, a 17, a 3, and our gold star. I should make the brackets in a different color. | Bringing the set operations together Probability and Statistics Khan Academy.mp3 |
Let A represent the event that he eats a bagel for breakfast and let B represent the event that he eats pizza for lunch. Fair enough. On a randomly selected day, the probability that Rahul will eat a bagel for breakfast, probability of A, is.6. Let me write that down. The probability that he eats a bagel for breakfast is 0.6. The probability that he will eat a pizza for lunch, probability of event B, so the probability of, let me do that in that red color, the probability of event B, that he eats a pizza for lunch, is 0.5. And the conditional probability that he eats a bagel for breakfast, given that he eats a pizza for lunch, so probability of event A happening, that he eats a bagel for breakfast, given that he's had a pizza for lunch, is equal to 0.7, which is interesting. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
Let me write that down. The probability that he eats a bagel for breakfast is 0.6. The probability that he will eat a pizza for lunch, probability of event B, so the probability of, let me do that in that red color, the probability of event B, that he eats a pizza for lunch, is 0.5. And the conditional probability that he eats a bagel for breakfast, given that he eats a pizza for lunch, so probability of event A happening, that he eats a bagel for breakfast, given that he's had a pizza for lunch, is equal to 0.7, which is interesting. Let me write this down. The probability of A, given that B is true, given B, is not 0.6. It's equal to 0.7. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
And the conditional probability that he eats a bagel for breakfast, given that he eats a pizza for lunch, so probability of event A happening, that he eats a bagel for breakfast, given that he's had a pizza for lunch, is equal to 0.7, which is interesting. Let me write this down. The probability of A, given that B is true, given B, is not 0.6. It's equal to 0.7. And because these two things are not the same, because probability of A by itself is different than the probability of A, given that B is true, this tells us that these two events are not independent, that we're dealing with dependent probability. This shows us the fact that B being true has changed the probability of A being true. This tells us that A and B are dependent. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
It's equal to 0.7. And because these two things are not the same, because probability of A by itself is different than the probability of A, given that B is true, this tells us that these two events are not independent, that we're dealing with dependent probability. This shows us the fact that B being true has changed the probability of A being true. This tells us that A and B are dependent. Before I start going on my little soapbox about dependent events, let's just think about what they actually want us to figure out. The probability of A given B is equal to 0.7. That's what we wrote right over here. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
This tells us that A and B are dependent. Before I start going on my little soapbox about dependent events, let's just think about what they actually want us to figure out. The probability of A given B is equal to 0.7. That's what we wrote right over here. Based on this information, what is the probability of B given A? They want us to figure out the probability of B given A. The conditional probability that Rahul eats pizza for lunch, given that he eats a bagel for breakfast, rounded to the nearest hundredth. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
That's what we wrote right over here. Based on this information, what is the probability of B given A? They want us to figure out the probability of B given A. The conditional probability that Rahul eats pizza for lunch, given that he eats a bagel for breakfast, rounded to the nearest hundredth. How would we think about this? I encourage you to pause this video before I work through it. I'm assuming you've given a go at it. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
The conditional probability that Rahul eats pizza for lunch, given that he eats a bagel for breakfast, rounded to the nearest hundredth. How would we think about this? I encourage you to pause this video before I work through it. I'm assuming you've given a go at it. The best way to tackle this is to just think about what's the probability that both A and B are going to happen? The probability of A and B happening, let me do this in a new color, the probability of A and B happening, I want to stay true to the colors, is equal to, there's a couple of ways you could write this out. This is equivalent to the probability of A given B times the probability of B. Hopefully that makes sense. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
I'm assuming you've given a go at it. The best way to tackle this is to just think about what's the probability that both A and B are going to happen? The probability of A and B happening, let me do this in a new color, the probability of A and B happening, I want to stay true to the colors, is equal to, there's a couple of ways you could write this out. This is equivalent to the probability of A given B times the probability of B. Hopefully that makes sense. The probability that B happens, and that given that B has happened, the probability that A happens. That would also be equal to, obviously this is A and B is happening, or you could do it the other way around. You could do it as the probability that B, the probability that B given A happens, times the probability of A. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
This is equivalent to the probability of A given B times the probability of B. Hopefully that makes sense. The probability that B happens, and that given that B has happened, the probability that A happens. That would also be equal to, obviously this is A and B is happening, or you could do it the other way around. You could do it as the probability that B, the probability that B given A happens, times the probability of A. This also makes sense. What's the probability that A happened? And that given A happened, what's the probability of B? | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
You could do it as the probability that B, the probability that B given A happens, times the probability of A. This also makes sense. What's the probability that A happened? And that given A happened, what's the probability of B? You multiply those together, you get the probability that both happened. Why is this helpful for us? We know a lot of this information. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
And that given A happened, what's the probability of B? You multiply those together, you get the probability that both happened. Why is this helpful for us? We know a lot of this information. We know the probability of A given B is 0.7. Let me write that, I'll scroll down a little bit. This is 0.7. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
We know a lot of this information. We know the probability of A given B is 0.7. Let me write that, I'll scroll down a little bit. This is 0.7. We know that the probability of B is 0.5. This is 0.5. We know that the probability of A and B is the product of these two things. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
This is 0.7. We know that the probability of B is 0.5. This is 0.5. We know that the probability of A and B is the product of these two things. That's going to be 0.35. Seven times five is 35, or I guess you could say half of 0.7 is 0.35, 0.5 of 0.7. That is going to be equal to what we need to figure out, the probability of B given A times the probability of A. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
We know that the probability of A and B is the product of these two things. That's going to be 0.35. Seven times five is 35, or I guess you could say half of 0.7 is 0.35, 0.5 of 0.7. That is going to be equal to what we need to figure out, the probability of B given A times the probability of A. But we know the probability of A. We know that that is 0.6. We know that this is 0.6. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
That is going to be equal to what we need to figure out, the probability of B given A times the probability of A. But we know the probability of A. We know that that is 0.6. We know that this is 0.6. Just like that, we've set up a situation, an equation where we can solve for the probability of B given A. The probability of B given A, notice, let me just rewrite it right over here. Actually, I'll write this part first. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
We know that this is 0.6. Just like that, we've set up a situation, an equation where we can solve for the probability of B given A. The probability of B given A, notice, let me just rewrite it right over here. Actually, I'll write this part first. 0.6 times the probability of B given A, times that right over there, and I'll just copy and paste it so I don't have to keep changing colors. That over there is equal to 0.35. To solve for the probability of B given A, we can just divide both sides by 0.6. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
Actually, I'll write this part first. 0.6 times the probability of B given A, times that right over there, and I'll just copy and paste it so I don't have to keep changing colors. That over there is equal to 0.35. To solve for the probability of B given A, we can just divide both sides by 0.6. We get the probability of B given A is equal to, let me get our calculator out, 0.35 divided by 0.6, and we deserve a little bit of a drum roll here, is 0.5833, keeps going. They tell us to round to the nearest hundredth. It's 0.58. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
To solve for the probability of B given A, we can just divide both sides by 0.6. We get the probability of B given A is equal to, let me get our calculator out, 0.35 divided by 0.6, and we deserve a little bit of a drum roll here, is 0.5833, keeps going. They tell us to round to the nearest hundredth. It's 0.58. It is approximately 0.58. Notice, this is approximately equal to 0.58. Once again, verifying that these are dependent. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
It's 0.58. It is approximately 0.58. Notice, this is approximately equal to 0.58. Once again, verifying that these are dependent. The probability that B happens given A is true is higher than just the probability that B by itself, or without knowing anything else. Just the probability of B is lower than the probability of B given that you know A has happened, or event A is true. And we're done. | Calculating conditional probability Probability and Statistics Khan Academy.mp3 |
Let's say I go to 16 students, and I ask them to measure how many glasses of water they drink per day for the last 30 days, and then to average it. And so this data point right over here tells us one student drank an average of 0.5 glasses of water per day. That person is probably very dehydrated. This person drank 8.1 glasses of water per day on average for the last 30 days. They are better hydrated. If we want to visualize that, we can set up a frequency histogram, where we can create some categories. So this first category would be for data points that are greater than or equal to zero and less than one. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
This person drank 8.1 glasses of water per day on average for the last 30 days. They are better hydrated. If we want to visualize that, we can set up a frequency histogram, where we can create some categories. So this first category would be for data points that are greater than or equal to zero and less than one. And we can see that two data points fall into that category, and that's why the bar right over here for that category is up to two. This category right over here is greater than or equal to three and less than four. Notice there are four data points in that category, and on this frequency histogram, the height of the bar is indeed four. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
So this first category would be for data points that are greater than or equal to zero and less than one. And we can see that two data points fall into that category, and that's why the bar right over here for that category is up to two. This category right over here is greater than or equal to three and less than four. Notice there are four data points in that category, and on this frequency histogram, the height of the bar is indeed four. So this is a nice way of looking at a distribution, but you might be more concerned with what percentage of my data falls into each of these categories. And that becomes especially interesting if we have many, many, many, many data points. And if we had 1,600,432,507 data points, well, just knowing the absolute number that fit into each category isn't so useful. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
Notice there are four data points in that category, and on this frequency histogram, the height of the bar is indeed four. So this is a nice way of looking at a distribution, but you might be more concerned with what percentage of my data falls into each of these categories. And that becomes especially interesting if we have many, many, many, many data points. And if we had 1,600,432,507 data points, well, just knowing the absolute number that fit into each category isn't so useful. The percent that fits into each category is a lot more useful. And so for that, we could set up a relative frequency histogram. So notice, this is representing the same data. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
And if we had 1,600,432,507 data points, well, just knowing the absolute number that fit into each category isn't so useful. The percent that fits into each category is a lot more useful. And so for that, we could set up a relative frequency histogram. So notice, this is representing the same data. But in that first category, instead of the bar height being two, the bar height is now 12.5%. Why is that? Because two of the 16 data points fall into this category. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
So notice, this is representing the same data. But in that first category, instead of the bar height being two, the bar height is now 12.5%. Why is that? Because two of the 16 data points fall into this category. 2 16ths is 1 8th, which is 12.5%. And this one right over here, notice, instead of the height being four, for four data points, it's now 25%. But these are saying the same thing. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
Because two of the 16 data points fall into this category. 2 16ths is 1 8th, which is 12.5%. And this one right over here, notice, instead of the height being four, for four data points, it's now 25%. But these are saying the same thing. Four out of the 16 data points fall into this category. 4 16ths is 1 4th, which is 25%. So both of these types of histograms are really useful, and you will see them used all of the time. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
But these are saying the same thing. Four out of the 16 data points fall into this category. 4 16ths is 1 4th, which is 25%. So both of these types of histograms are really useful, and you will see them used all of the time. But there are also cases where you have many, many, many more data points, and you want more granular categories. So what you could do is, well, let's just make our categories a little more granular. So for example, instead of them being one glass of water wide, maybe you make them half a glass of water wide. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
So both of these types of histograms are really useful, and you will see them used all of the time. But there are also cases where you have many, many, many more data points, and you want more granular categories. So what you could do is, well, let's just make our categories a little more granular. So for example, instead of them being one glass of water wide, maybe you make them half a glass of water wide. So this first category could be greater than or equal to zero, and less than 0.5. And that will give you a clearer picture. And I'm now assuming a world where we have more than 16 data points. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
So for example, instead of them being one glass of water wide, maybe you make them half a glass of water wide. So this first category could be greater than or equal to zero, and less than 0.5. And that will give you a clearer picture. And I'm now assuming a world where we have more than 16 data points. Maybe we have 16 million data points. This would be percentages on the left-hand side. But maybe that isn't good enough for you. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
And I'm now assuming a world where we have more than 16 data points. Maybe we have 16 million data points. This would be percentages on the left-hand side. But maybe that isn't good enough for you. Maybe you wanna get even more granular. So you make everything, each category, a quarter of a glass. But maybe that doesn't satisfy you. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
But maybe that isn't good enough for you. Maybe you wanna get even more granular. So you make everything, each category, a quarter of a glass. But maybe that doesn't satisfy you. You wanna get more and more and more granular. Well, you could imagine where this is going. You could get to a point where you're approaching an infinite number of categories, and each category is infinitely thin, is super, super thin, to a point that if you just connect the tops of the bars, that you will actually get a curve. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
But maybe that doesn't satisfy you. You wanna get more and more and more granular. Well, you could imagine where this is going. You could get to a point where you're approaching an infinite number of categories, and each category is infinitely thin, is super, super thin, to a point that if you just connect the tops of the bars, that you will actually get a curve. And this type of curve is something that we actually use in the statistics. And as promised at the beginning of the video, this is the density curve we talk about. And what's valuable about a density curve, it is a visualization of a distribution where the data points can take on any value in a continuum. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
You could get to a point where you're approaching an infinite number of categories, and each category is infinitely thin, is super, super thin, to a point that if you just connect the tops of the bars, that you will actually get a curve. And this type of curve is something that we actually use in the statistics. And as promised at the beginning of the video, this is the density curve we talk about. And what's valuable about a density curve, it is a visualization of a distribution where the data points can take on any value in a continuum. They're not just thrown into these coarse buckets. So how would you interpret something like this? If you look over the entire interval from zero, let's say, to nine, assuming no one drank more than an average of nine glasses per day, even in our 16 million data points, well then the area under the curve over that interval is going to be 100%, or 1.0. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
And what's valuable about a density curve, it is a visualization of a distribution where the data points can take on any value in a continuum. They're not just thrown into these coarse buckets. So how would you interpret something like this? If you look over the entire interval from zero, let's say, to nine, assuming no one drank more than an average of nine glasses per day, even in our 16 million data points, well then the area under the curve over that interval is going to be 100%, or 1.0. This is going to be true for any density curve, that the entire area of the curve is 100%. It represents all of the data points. A density curve will also never take on a negative value. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
If you look over the entire interval from zero, let's say, to nine, assuming no one drank more than an average of nine glasses per day, even in our 16 million data points, well then the area under the curve over that interval is going to be 100%, or 1.0. This is going to be true for any density curve, that the entire area of the curve is 100%. It represents all of the data points. A density curve will also never take on a negative value. You won't see the curve dip down and do something strange like that. Now with that out of the way, let's think about how we would make use of it. If I wanted to know what percentage of my data falls between two and four glasses, well I would look at that interval. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
A density curve will also never take on a negative value. You won't see the curve dip down and do something strange like that. Now with that out of the way, let's think about how we would make use of it. If I wanted to know what percentage of my data falls between two and four glasses, well I would look at that interval. I'd go from two to four. I would look at this interval right over here, and I would try to figure out the area under the curve here. And this area is going to be greater than or equal to zero and less than or equal to 100%. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
If I wanted to know what percentage of my data falls between two and four glasses, well I would look at that interval. I'd go from two to four. I would look at this interval right over here, and I would try to figure out the area under the curve here. And this area is going to be greater than or equal to zero and less than or equal to 100%. When I eyeball it right over here, it looks like it's about 40% of the entire area under the curve. So just eyeballing it, I would say roughly 40% of my data falls into this interval. If I were to ask you what percentage of the data is greater than three, well then you would be looking at this area, and it looks like it is about 50%, but once again, I am estimating it. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
And this area is going to be greater than or equal to zero and less than or equal to 100%. When I eyeball it right over here, it looks like it's about 40% of the entire area under the curve. So just eyeballing it, I would say roughly 40% of my data falls into this interval. If I were to ask you what percentage of the data is greater than three, well then you would be looking at this area, and it looks like it is about 50%, but once again, I am estimating it. But you can start to see how even with estimation, a density curve could be useful. In the real world, statisticians will often have tables that might represent the information for the density curve. They might have computer programs or some type of automated tool, and there are also well-known density curves, the famous bell curve that we will study later on where there's a lot of precise data and a lot of tools to exactly figure out the areas. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
If I were to ask you what percentage of the data is greater than three, well then you would be looking at this area, and it looks like it is about 50%, but once again, I am estimating it. But you can start to see how even with estimation, a density curve could be useful. In the real world, statisticians will often have tables that might represent the information for the density curve. They might have computer programs or some type of automated tool, and there are also well-known density curves, the famous bell curve that we will study later on where there's a lot of precise data and a lot of tools to exactly figure out the areas. The last thing I'd like to cover is a key misconception for density curves. If I were to ask you approximately what percentage of my data is exactly three glasses of water per day, and when I say exactly, I mean exactly the number 3.000 with zeros just going on and on forever, the exact number three. So you might be tempted to just say, okay, this is three. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
They might have computer programs or some type of automated tool, and there are also well-known density curves, the famous bell curve that we will study later on where there's a lot of precise data and a lot of tools to exactly figure out the areas. The last thing I'd like to cover is a key misconception for density curves. If I were to ask you approximately what percentage of my data is exactly three glasses of water per day, and when I say exactly, I mean exactly the number 3.000 with zeros just going on and on forever, the exact number three. So you might be tempted to just say, okay, this is three. Let me see the corresponding point on the curve. It looks like it is about 0.2 or a little higher than that, so maybe you would say a little bit more than 20% or approximately 20%. And what I would say to you is this is wrong. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
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