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And what I want to do is think about the outliers. And to help us with that, let's actually visualize this, the distribution of actual numbers. So let us do that. So here on a number line, I have all the numbers from one to 19. And let's see. We have two ones. So I could say that's one one, and then two ones. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So here on a number line, I have all the numbers from one to 19. And let's see. We have two ones. So I could say that's one one, and then two ones. We have one six, so let's put that six there. We have got two 13s. So we're going to go up here. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So I could say that's one one, and then two ones. We have one six, so let's put that six there. We have got two 13s. So we're going to go up here. One 13 and two 13s. Let's see. We have three 14s. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So we're going to go up here. One 13 and two 13s. Let's see. We have three 14s. So 14, 14, and 14. We have a couple of 15s. 15, 15. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
We have three 14s. So 14, 14, and 14. We have a couple of 15s. 15, 15. So 15, 15. We have one 16. So that's our 16 there. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
15, 15. So 15, 15. We have one 16. So that's our 16 there. We have three 18s. One, two, three. So one, two, and then three. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So that's our 16 there. We have three 18s. One, two, three. So one, two, and then three. And then we have a 19. Then we have a 19. So when you look visually at the distribution of numbers, it looks like the meat of the distribution, so to speak, is in this area right over here. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So one, two, and then three. And then we have a 19. Then we have a 19. So when you look visually at the distribution of numbers, it looks like the meat of the distribution, so to speak, is in this area right over here. And so some people might say, OK, we have three outliers. These two ones and the six. Some people might say, well, the six is kind of close enough. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So when you look visually at the distribution of numbers, it looks like the meat of the distribution, so to speak, is in this area right over here. And so some people might say, OK, we have three outliers. These two ones and the six. Some people might say, well, the six is kind of close enough. Maybe only these two ones are outliers. And those would actually be both reasonable things to say. Now to get on the same page, statisticians will use a rule sometimes. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
Some people might say, well, the six is kind of close enough. Maybe only these two ones are outliers. And those would actually be both reasonable things to say. Now to get on the same page, statisticians will use a rule sometimes. We say, well, anything that is more than 1 and 1 1 times the interquartile range from below Q1 or above Q3, well, those are going to be outliers. Well, what am I talking about? Well, actually, let's figure out the median, Q1 and Q3 here. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
Now to get on the same page, statisticians will use a rule sometimes. We say, well, anything that is more than 1 and 1 1 times the interquartile range from below Q1 or above Q3, well, those are going to be outliers. Well, what am I talking about? Well, actually, let's figure out the median, Q1 and Q3 here. Then we can figure out the interquartile range. And then we can figure out by that definition what is going to be an outlier. And if that all made sense to you so far, I encourage you to pause this video and try to work through it on your own. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
Well, actually, let's figure out the median, Q1 and Q3 here. Then we can figure out the interquartile range. And then we can figure out by that definition what is going to be an outlier. And if that all made sense to you so far, I encourage you to pause this video and try to work through it on your own. Or I'll do it for you right now. All right, so what's the median here? Well, the median is the middle number. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
And if that all made sense to you so far, I encourage you to pause this video and try to work through it on your own. Or I'll do it for you right now. All right, so what's the median here? Well, the median is the middle number. We have 15 numbers. So the middle number is going to be whatever number has seven on either side. So that's going to be the eighth number. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
Well, the median is the middle number. We have 15 numbers. So the middle number is going to be whatever number has seven on either side. So that's going to be the eighth number. 1, 2, 3, 4, 5, 6, 7. Is that right? Yep, 6, 7. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So that's going to be the eighth number. 1, 2, 3, 4, 5, 6, 7. Is that right? Yep, 6, 7. So that's the median. And you have 1, 2, 3, 4, 5, 6, 7 numbers on the right side, too. So that is the median, sometimes called Q2. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
Yep, 6, 7. So that's the median. And you have 1, 2, 3, 4, 5, 6, 7 numbers on the right side, too. So that is the median, sometimes called Q2. That is our median. Now, what is Q1? Well, Q1 is going to be the middle of this first group. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So that is the median, sometimes called Q2. That is our median. Now, what is Q1? Well, Q1 is going to be the middle of this first group. This first group has seven numbers in it. And so the middle is going to be the fourth number. It has 3 and 3. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
Well, Q1 is going to be the middle of this first group. This first group has seven numbers in it. And so the middle is going to be the fourth number. It has 3 and 3. 3 to the left, 3 to the right. So that is Q1. And then Q3 is going to be the middle of this upper group. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
It has 3 and 3. 3 to the left, 3 to the right. So that is Q1. And then Q3 is going to be the middle of this upper group. Well, that also has seven numbers in it. So the middle is going to be right over there. It has 3 on either side. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
And then Q3 is going to be the middle of this upper group. Well, that also has seven numbers in it. So the middle is going to be right over there. It has 3 on either side. So that is Q3. Now, what is the interquartile range going to be? Interquartile range is going to be equal to Q3 minus Q1. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
It has 3 on either side. So that is Q3. Now, what is the interquartile range going to be? Interquartile range is going to be equal to Q3 minus Q1. The difference between 18 and 13. Between 18 and 13, well, that is going to be 18 minus 13, which is equal to 5. Now, to figure out outliers, well, outliers are going to be anything that is below. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
Interquartile range is going to be equal to Q3 minus Q1. The difference between 18 and 13. Between 18 and 13, well, that is going to be 18 minus 13, which is equal to 5. Now, to figure out outliers, well, outliers are going to be anything that is below. So outliers are going to be less than our Q1 minus 1.5 times our interquartile range. And once again, this isn't some rule of the universe. This is something that statisticians have kind of said, well, if we want to have a better definition for outliers, let's just agree that it's something that's more than 1.5 times the interquartile range below Q1. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
Now, to figure out outliers, well, outliers are going to be anything that is below. So outliers are going to be less than our Q1 minus 1.5 times our interquartile range. And once again, this isn't some rule of the universe. This is something that statisticians have kind of said, well, if we want to have a better definition for outliers, let's just agree that it's something that's more than 1.5 times the interquartile range below Q1. Or an outlier could be greater than Q3 plus 1.5 times the interquartile range. And once again, this is somewhat, people just decided it felt right. One could argue it should be 1.6. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
This is something that statisticians have kind of said, well, if we want to have a better definition for outliers, let's just agree that it's something that's more than 1.5 times the interquartile range below Q1. Or an outlier could be greater than Q3 plus 1.5 times the interquartile range. And once again, this is somewhat, people just decided it felt right. One could argue it should be 1.6. Or one could argue it should be 1 or 2 or whatever. But this is what people have tended to agree on. So let's think about what these numbers are. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
One could argue it should be 1.6. Or one could argue it should be 1 or 2 or whatever. But this is what people have tended to agree on. So let's think about what these numbers are. Q1, we already know. So this is going to be 13 minus 1.5 times our interquartile range. Our interquartile range here is 5. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So let's think about what these numbers are. Q1, we already know. So this is going to be 13 minus 1.5 times our interquartile range. Our interquartile range here is 5. So it's 1.5 times 5, which is 7.5. So this is 7.5. 13 minus 7.5 is what? | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
Our interquartile range here is 5. So it's 1.5 times 5, which is 7.5. So this is 7.5. 13 minus 7.5 is what? 13 minus 7 is 6. And then you subtract another 0.5 is 5.5. So we have outliers would be less than 5.5. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
13 minus 7.5 is what? 13 minus 7 is 6. And then you subtract another 0.5 is 5.5. So we have outliers would be less than 5.5. Or Q3 is 18. This is, once again, 7.5. 18 plus 7.5 is 25.5. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So we have outliers would be less than 5.5. Or Q3 is 18. This is, once again, 7.5. 18 plus 7.5 is 25.5. Or outliers greater than 25.5. So based on this, we have a numerical definition for what's an outlier. We're not just subjectively saying, oh, this feels right or that feels right. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
18 plus 7.5 is 25.5. Or outliers greater than 25.5. So based on this, we have a numerical definition for what's an outlier. We're not just subjectively saying, oh, this feels right or that feels right. And based on this, we only have two outliers, that only these two ones are less than 5.5. This is the cutoff right over here. So this dot just happened to make it. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
We're not just subjectively saying, oh, this feels right or that feels right. And based on this, we only have two outliers, that only these two ones are less than 5.5. This is the cutoff right over here. So this dot just happened to make it. And we don't have any outliers on the high side. Now, another thing to think about is drawing box and whiskers plots based on Q1, our median, our range, all the range of numbers. And you could do it either taking in consideration your outliers or not taking into consideration your outliers. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So this dot just happened to make it. And we don't have any outliers on the high side. Now, another thing to think about is drawing box and whiskers plots based on Q1, our median, our range, all the range of numbers. And you could do it either taking in consideration your outliers or not taking into consideration your outliers. So there's a couple of ways that we can do it. So let me actually clear all of this. We've figured out all of this stuff. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
And you could do it either taking in consideration your outliers or not taking into consideration your outliers. So there's a couple of ways that we can do it. So let me actually clear all of this. We've figured out all of this stuff. So let me clear all of that out. And let's actually draw a box and whiskers plot. So I'll put another, actually, let me do two here. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
We've figured out all of this stuff. So let me clear all of that out. And let's actually draw a box and whiskers plot. So I'll put another, actually, let me do two here. That's one. And then let me put another one down there. This is another. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So I'll put another, actually, let me do two here. That's one. And then let me put another one down there. This is another. Now, if we were to just draw a classic box and whiskers plot here, we would say, all right, our median's at 14. And actually, I'll do it both ways. Median's at 14. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
This is another. Now, if we were to just draw a classic box and whiskers plot here, we would say, all right, our median's at 14. And actually, I'll do it both ways. Median's at 14. Q1's at 13. Q3 is at 18. So that's the box part. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
Median's at 14. Q1's at 13. Q3 is at 18. So that's the box part. And let me draw that as an actual, let me actually draw that as a box. So my best attempt. There you go. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So that's the box part. And let me draw that as an actual, let me actually draw that as a box. So my best attempt. There you go. That's the box. And this is also a box. So far, I'm doing the exact same thing. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
There you go. That's the box. And this is also a box. So far, I'm doing the exact same thing. Now, if we don't want to consider outliers, we would say, well, what's the entire range here? Well, we have things that go from 1 all the way to 19. So one way to do it is to say, hey, we start at 1. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So far, I'm doing the exact same thing. Now, if we don't want to consider outliers, we would say, well, what's the entire range here? Well, we have things that go from 1 all the way to 19. So one way to do it is to say, hey, we start at 1. And so our entire range, we go, actually, let me draw it a little bit better than that. We're going all the way from 1 to 19. Now, in this one, we're including everything. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So one way to do it is to say, hey, we start at 1. And so our entire range, we go, actually, let me draw it a little bit better than that. We're going all the way from 1 to 19. Now, in this one, we're including everything. We're including even these two outliers. But if we don't want to include those outliers, we want to make it clear that they're outliers, well, let's not include them. And what we can do instead is say, all right, including our non-outliers, we would start at 6. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
Now, in this one, we're including everything. We're including even these two outliers. But if we don't want to include those outliers, we want to make it clear that they're outliers, well, let's not include them. And what we can do instead is say, all right, including our non-outliers, we would start at 6. Because 6, we're saying, is in our data set. But it is not an outlier. Let me make this look better. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
And what we can do instead is say, all right, including our non-outliers, we would start at 6. Because 6, we're saying, is in our data set. But it is not an outlier. Let me make this look better. So we are going to start at 6 and go all the way to 19. And then to say that we have these outliers, we would put this, we have outliers over there. So once again, this is a box and whiskers plot of the same data set without outliers. | Judging outliers in a dataset Summarizing quantitative data AP Statistics Khan Academy.mp3 |
So a set is really just a collection of distinct objects. So for example, I could have a set. Let's call this set x. And I'll deal with numbers right now, but a set could contain anything. It could contain colors. It could contain people. It could contain other sets. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
And I'll deal with numbers right now, but a set could contain anything. It could contain colors. It could contain people. It could contain other sets. It could contain cars. It could contain farm animals. But the numbers will be easy to deal with just because, well, they're numbers. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
It could contain other sets. It could contain cars. It could contain farm animals. But the numbers will be easy to deal with just because, well, they're numbers. So let's say I have a set x. And it has the distinct objects in it, the number 3, the number 12, the number 5, and the number 13. That right there is a set. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
But the numbers will be easy to deal with just because, well, they're numbers. So let's say I have a set x. And it has the distinct objects in it, the number 3, the number 12, the number 5, and the number 13. That right there is a set. I could have another set. Let's call that set y. I didn't have to call it y. I could have called it a. I could have called it sal. I could have called it a bunch of different things, but I'll just call it y. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
That right there is a set. I could have another set. Let's call that set y. I didn't have to call it y. I could have called it a. I could have called it sal. I could have called it a bunch of different things, but I'll just call it y. And let's say that set y, it's a collection of the distinct objects, the number 14, the number 15, the number 6, and the number 3. So fair enough. Those are just two set definitions. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
I could have called it a bunch of different things, but I'll just call it y. And let's say that set y, it's a collection of the distinct objects, the number 14, the number 15, the number 6, and the number 3. So fair enough. Those are just two set definitions. The way that we typically do it in mathematics is we put these little curly brackets around the objects that are separated by commas. Now let's do some basic operations on sets. And the first operation that I will do is called intersection. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
Those are just two set definitions. The way that we typically do it in mathematics is we put these little curly brackets around the objects that are separated by commas. Now let's do some basic operations on sets. And the first operation that I will do is called intersection. And so we would say x intersect the intersection of x and y. And the way that I think about this, this is going to yield another set that contains the elements that are in both x and y. So I often view this intersection symbol right here as and. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
And the first operation that I will do is called intersection. And so we would say x intersect the intersection of x and y. And the way that I think about this, this is going to yield another set that contains the elements that are in both x and y. So I often view this intersection symbol right here as and. So all of the things that are in x and in y. So what are those things going to be? Well, let's look at both sets x and y. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
So I often view this intersection symbol right here as and. So all of the things that are in x and in y. So what are those things going to be? Well, let's look at both sets x and y. So the number 3 is in set x. Is it in set y as well? Well, sure. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
Well, let's look at both sets x and y. So the number 3 is in set x. Is it in set y as well? Well, sure. It's in both. So it will be in the intersection of x and y. Now the number 12, that's in set x, but it isn't at y. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
Well, sure. It's in both. So it will be in the intersection of x and y. Now the number 12, that's in set x, but it isn't at y. So we're not going to include that. The number 5, it's in x, but it's not in y. And then we have the number 13 is in x, but it's not in y. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
Now the number 12, that's in set x, but it isn't at y. So we're not going to include that. The number 5, it's in x, but it's not in y. And then we have the number 13 is in x, but it's not in y. And so over here, the intersection of x and y is the set that only has one object in it. It only has the number 3. So we are done. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
And then we have the number 13 is in x, but it's not in y. And so over here, the intersection of x and y is the set that only has one object in it. It only has the number 3. So we are done. The intersection of x and y is 3. Now another common operation on sets is union. So you could have the union of x and y. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
So we are done. The intersection of x and y is 3. Now another common operation on sets is union. So you could have the union of x and y. And the union I often view, or people often view, as or. So we're thinking about all of the elements that are in x or y. So in some ways, you can kind of imagine that we're bringing these two sets together. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
So you could have the union of x and y. And the union I often view, or people often view, as or. So we're thinking about all of the elements that are in x or y. So in some ways, you can kind of imagine that we're bringing these two sets together. So this is going to be. And the key here is that a set is a collection of distinct objects. And the way we're conceptualizing things right here, this is the number 3. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
So in some ways, you can kind of imagine that we're bringing these two sets together. So this is going to be. And the key here is that a set is a collection of distinct objects. And the way we're conceptualizing things right here, this is the number 3. This isn't like somebody's score on a test or the number of apples they have. So there you could have multiple people with the same number of apples. Here we're talking about the object, the number 3. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
And the way we're conceptualizing things right here, this is the number 3. This isn't like somebody's score on a test or the number of apples they have. So there you could have multiple people with the same number of apples. Here we're talking about the object, the number 3. So we can only have a 3 once. But a 3 is in set is in x or y. So I'll put a 3 there. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
Here we're talking about the object, the number 3. So we can only have a 3 once. But a 3 is in set is in x or y. So I'll put a 3 there. A 12 is in x or y. A 5 is in x or y. The 13 is in x or y. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
So I'll put a 3 there. A 12 is in x or y. A 5 is in x or y. The 13 is in x or y. And just to simplify things, we really don't care about order if we're just talking about a set. I've just put all the things that are in set x here. And now let's see what we have to add from set y. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
The 13 is in x or y. And just to simplify things, we really don't care about order if we're just talking about a set. I've just put all the things that are in set x here. And now let's see what we have to add from set y. So we haven't put a 14 yet. So let's put a 14. We haven't put a 15 yet. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
And now let's see what we have to add from set y. So we haven't put a 14 yet. So let's put a 14. We haven't put a 15 yet. We haven't put the 6 yet. And we already have a 3 in our set. So there you go. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
We haven't put a 15 yet. We haven't put the 6 yet. And we already have a 3 in our set. So there you go. You have the union of x and y. And one way to visualize sets and visualize intersections and unions and more complicated things is using a Venn diagram. So let's say this whole box is a set of all numbers. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
So there you go. You have the union of x and y. And one way to visualize sets and visualize intersections and unions and more complicated things is using a Venn diagram. So let's say this whole box is a set of all numbers. So that's all the numbers right over there. We have set x. I'll just draw a circle right over here. And I could even draw the elements of set x. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
So let's say this whole box is a set of all numbers. So that's all the numbers right over there. We have set x. I'll just draw a circle right over here. And I could even draw the elements of set x. So you have 3 and 5 and 12, 3, 5, 12, and 13. And then we can draw a set y. And notice, I drew a little overlapping here because they overlap at 3. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
And I could even draw the elements of set x. So you have 3 and 5 and 12, 3, 5, 12, and 13. And then we can draw a set y. And notice, I drew a little overlapping here because they overlap at 3. 3 is an element in both set x and set y. But set y also has the numbers 14, 15, and 6. And so when we're talking about x intersect y, we're talking about where the two sets overlap. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
And notice, I drew a little overlapping here because they overlap at 3. 3 is an element in both set x and set y. But set y also has the numbers 14, 15, and 6. And so when we're talking about x intersect y, we're talking about where the two sets overlap. So we're talking about this region right over here. And the only place that they overlap the way I've drawn it is at the number 3. So this is x intersect y. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
And so when we're talking about x intersect y, we're talking about where the two sets overlap. So we're talking about this region right over here. And the only place that they overlap the way I've drawn it is at the number 3. So this is x intersect y. And then x union y is the combination of these two sets. So x union y is literally everything right here that we are combining. Let's do one more example just so that we make sure we understand intersection and union. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
So this is x intersect y. And then x union y is the combination of these two sets. So x union y is literally everything right here that we are combining. Let's do one more example just so that we make sure we understand intersection and union. So let's say that I have set A. And set A has the numbers 11, 4, 12, and 7 in it. And I have set B. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
Let's do one more example just so that we make sure we understand intersection and union. So let's say that I have set A. And set A has the numbers 11, 4, 12, and 7 in it. And I have set B. And it has the numbers 13, 4, 12, 10, and 3 in it. So first of all, let's think about what A intersect B is going to be equal to. Well, it's the things that are in both sets. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
And I have set B. And it has the numbers 13, 4, 12, 10, and 3 in it. So first of all, let's think about what A intersect B is going to be equal to. Well, it's the things that are in both sets. So I have 11 here. I don't have an 11 there. So that doesn't make the intersection. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
Well, it's the things that are in both sets. So I have 11 here. I don't have an 11 there. So that doesn't make the intersection. I have a 4 here. I also have a 4 here. So 4 is in A and B. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
So that doesn't make the intersection. I have a 4 here. I also have a 4 here. So 4 is in A and B. It's in A and B. So I'll put a 4 here. The number 12, it's in A and B. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
So 4 is in A and B. It's in A and B. So I'll put a 4 here. The number 12, it's in A and B. So I'll put a 12 here. Number 7's only in A. And then number 13, 10, and 3 is only in B. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
The number 12, it's in A and B. So I'll put a 12 here. Number 7's only in A. And then number 13, 10, and 3 is only in B. So we're done. 4 and 12, the set of 4 and 12 is the intersection of sets A and B. And if we want to, we could even label this as a new set. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
And then number 13, 10, and 3 is only in B. So we're done. 4 and 12, the set of 4 and 12 is the intersection of sets A and B. And if we want to, we could even label this as a new set. We could say set C is the intersection of A and B. And it's this set right over here. Now let's think about union. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
And if we want to, we could even label this as a new set. We could say set C is the intersection of A and B. And it's this set right over here. Now let's think about union. Let's think about A. I want to do that in orange. Let's think about A union B. What are all the elements that are in A or B? | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
Now let's think about union. Let's think about A. I want to do that in orange. Let's think about A union B. What are all the elements that are in A or B? Well, we can just literally put all the elements in A. 11, 4, 12, 7. And then put the things in B that aren't already in A. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
What are all the elements that are in A or B? Well, we can just literally put all the elements in A. 11, 4, 12, 7. And then put the things in B that aren't already in A. So let's see, 13, we already put the 4 and the 12, a 10, and a 3. And I could write this in any order I want. We don't care about order if we're thinking about a set. | Intersection and union of sets Probability and Statistics Khan Academy.mp3 |
Each person snapped their fingers with their dominant hand for 10 seconds and their non-dominant hand for 10 seconds, where if you're right-handed, right hand would be your dominant hand. If you were left-handed, left hand would be your dominant hand. Each participant flipped a coin to determine which hand they would use first, because if you always used your dominant hand first, maybe you're tired by the time you're doing your non-dominant hand or there's something else. So here, it's random which one you use first. Here are the data for how many snaps they performed with each hand, the difference for each participant, and summary statistics. And this is actually real data from the Khan Academy content team. And so you see for each of the participants, for Jeff right over here, he was able to do 44 snaps in 10 seconds on his dominant hand, which is impressive, more than I think I could do. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
So here, it's random which one you use first. Here are the data for how many snaps they performed with each hand, the difference for each participant, and summary statistics. And this is actually real data from the Khan Academy content team. And so you see for each of the participants, for Jeff right over here, he was able to do 44 snaps in 10 seconds on his dominant hand, which is impressive, more than I think I could do. And he was even able to do 35 on his non-dominant hand. And so the difference here, the dominant hand minus the non-dominant was nine. And then they tabulated this data for all five members. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
And so you see for each of the participants, for Jeff right over here, he was able to do 44 snaps in 10 seconds on his dominant hand, which is impressive, more than I think I could do. And he was even able to do 35 on his non-dominant hand. And so the difference here, the dominant hand minus the non-dominant was nine. And then they tabulated this data for all five members. Now, they also calculated summary statistics for them. But this is the really interesting thing right over here. This is the difference between the dominant and the non-dominant hand. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
And then they tabulated this data for all five members. Now, they also calculated summary statistics for them. But this is the really interesting thing right over here. This is the difference between the dominant and the non-dominant hand. And so what they did here, the mean difference, what they did is they took this row right over here, and they calculated the mean, which they got to be 6.8. And then they calculated the standard deviation of these differences right over here, which they got to be approximately 1.64. And then we are asked, create and interpret a 95% confidence interval for mean difference in number of snaps for these participants. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
This is the difference between the dominant and the non-dominant hand. And so what they did here, the mean difference, what they did is they took this row right over here, and they calculated the mean, which they got to be 6.8. And then they calculated the standard deviation of these differences right over here, which they got to be approximately 1.64. And then we are asked, create and interpret a 95% confidence interval for mean difference in number of snaps for these participants. So pause this video, see if you can make some headway here. See if you can think about how to approach this. So what's interesting here is we're not trying to construct a confidence interval for just the mean number of snaps for the dominant hand or the mean number of snaps for the non-dominant hand. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
And then we are asked, create and interpret a 95% confidence interval for mean difference in number of snaps for these participants. So pause this video, see if you can make some headway here. See if you can think about how to approach this. So what's interesting here is we're not trying to construct a confidence interval for just the mean number of snaps for the dominant hand or the mean number of snaps for the non-dominant hand. We're constructing a 95% confidence interval for a mean difference. Now, you might say, wait, wait, wait. You know, I have two different samples here, and then this third sample is, or this third data is somehow constructed from these other two. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
So what's interesting here is we're not trying to construct a confidence interval for just the mean number of snaps for the dominant hand or the mean number of snaps for the non-dominant hand. We're constructing a 95% confidence interval for a mean difference. Now, you might say, wait, wait, wait. You know, I have two different samples here, and then this third sample is, or this third data is somehow constructed from these other two. But one way to think about it, this is matched pairs design. So in a matched pairs design, what you do is, for each participant, for each member in your sample, you will make them do the control and the treatment. So for example, you could do the control as how many they can do in the dominant hand in 10 seconds, and the treatment is how many they can do in the non-dominant hand. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
You know, I have two different samples here, and then this third sample is, or this third data is somehow constructed from these other two. But one way to think about it, this is matched pairs design. So in a matched pairs design, what you do is, for each participant, for each member in your sample, you will make them do the control and the treatment. So for example, you could do the control as how many they can do in the dominant hand in 10 seconds, and the treatment is how many they can do in the non-dominant hand. And in matched pairs design, you're really concerned about the difference. And so what you can really view this as is you just have one sample size of five for which you are calculating the difference for each member of that sample and the standard deviation across that entire sample. Now, before we calculate the confidence interval, let's just remind ourselves some of our conditions that we like to think about when we are constructing confidence intervals. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
So for example, you could do the control as how many they can do in the dominant hand in 10 seconds, and the treatment is how many they can do in the non-dominant hand. And in matched pairs design, you're really concerned about the difference. And so what you can really view this as is you just have one sample size of five for which you are calculating the difference for each member of that sample and the standard deviation across that entire sample. Now, before we calculate the confidence interval, let's just remind ourselves some of our conditions that we like to think about when we are constructing confidence intervals. The first condition we think about is whether our sample is random. Now, if we were trying to make some type of judgment about all human beings and their snapping ability, this would not be a random sample. These people all work at Khan Academy. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
Now, before we calculate the confidence interval, let's just remind ourselves some of our conditions that we like to think about when we are constructing confidence intervals. The first condition we think about is whether our sample is random. Now, if we were trying to make some type of judgment about all human beings and their snapping ability, this would not be a random sample. These people all work at Khan Academy. Maybe somehow in our interview process, we select for people who snap particularly well. But whatever inferences we make, we can say, hey, this is roughly true about this group of friends. Now, the next condition we wanna think about is the normal condition. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
These people all work at Khan Academy. Maybe somehow in our interview process, we select for people who snap particularly well. But whatever inferences we make, we can say, hey, this is roughly true about this group of friends. Now, the next condition we wanna think about is the normal condition. Now, there's a couple of ways to think about it. If we had sample size of 30 or larger, the central limit theorem says, okay, the distribution, the sampling distribution would be roughly normal, the sampling distribution of the sample means. But obviously, our sample size is much smaller than that. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
Now, the next condition we wanna think about is the normal condition. Now, there's a couple of ways to think about it. If we had sample size of 30 or larger, the central limit theorem says, okay, the distribution, the sampling distribution would be roughly normal, the sampling distribution of the sample means. But obviously, our sample size is much smaller than that. One way to think about it, we could just plot our data points and see whether they seem to be skewed in any way. And if we just do a little dot plot right over here, we could say, let's say make this zero, one, two, three, four, five, six, seven, eight, and nine. So we have one data point where the difference was nine, one data point where the difference is five, one data point where the difference is eight, one data point where the difference is six, and another data point where the difference is six. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
But obviously, our sample size is much smaller than that. One way to think about it, we could just plot our data points and see whether they seem to be skewed in any way. And if we just do a little dot plot right over here, we could say, let's say make this zero, one, two, three, four, five, six, seven, eight, and nine. So we have one data point where the difference was nine, one data point where the difference is five, one data point where the difference is eight, one data point where the difference is six, and another data point where the difference is six. And so this doesn't look massively skewed in any way. Our mean difference was right over here, it was about 6.8. It looks roughly symmetric. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
So we have one data point where the difference was nine, one data point where the difference is five, one data point where the difference is eight, one data point where the difference is six, and another data point where the difference is six. And so this doesn't look massively skewed in any way. Our mean difference was right over here, it was about 6.8. It looks roughly symmetric. So we can feel okay about this normal distribution. This isn't the best study that one could conduct. This is obviously a small sample size. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
It looks roughly symmetric. So we can feel okay about this normal distribution. This isn't the best study that one could conduct. This is obviously a small sample size. It's not random of the entire population. But maybe we could go with it. Also, when you think about biological processes, like how well someone snaps, which is a product of a lot of things happening in a human body, and it's the sum of many, many processes, those things also tend to have a roughly normal distribution but I won't go into too much depth there. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
This is obviously a small sample size. It's not random of the entire population. But maybe we could go with it. Also, when you think about biological processes, like how well someone snaps, which is a product of a lot of things happening in a human body, and it's the sum of many, many processes, those things also tend to have a roughly normal distribution but I won't go into too much depth there. But all of these things, once again, this isn't a super robust study, but this is a fun thing for friends to do if they have nothing else to do. All right. Now the third one is independence. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
Also, when you think about biological processes, like how well someone snaps, which is a product of a lot of things happening in a human body, and it's the sum of many, many processes, those things also tend to have a roughly normal distribution but I won't go into too much depth there. But all of these things, once again, this isn't a super robust study, but this is a fun thing for friends to do if they have nothing else to do. All right. Now the third one is independence. And this one actually we can feel pretty good about because Jeff's difference right over here really shouldn't impact David's difference or David's difference really shouldn't impact Kim's difference, especially if they're not observing each other. And let's just say for the sake of argument that they did it all independently in a closed room with a independent observer so they weren't trying to get competitive or something like that. But needless to say, this isn't super robust study, but we can still calculate a 95% confidence interval. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
Now the third one is independence. And this one actually we can feel pretty good about because Jeff's difference right over here really shouldn't impact David's difference or David's difference really shouldn't impact Kim's difference, especially if they're not observing each other. And let's just say for the sake of argument that they did it all independently in a closed room with a independent observer so they weren't trying to get competitive or something like that. But needless to say, this isn't super robust study, but we can still calculate a 95% confidence interval. So how do we do that? Well, we've done this so many times. Our confidence interval would be our sample mean. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
But needless to say, this isn't super robust study, but we can still calculate a 95% confidence interval. So how do we do that? Well, we've done this so many times. Our confidence interval would be our sample mean. So it would be the mean of our difference, the mean of our difference plus or minus. Now we don't know the population standard deviation, so we're going to use our sample standard deviation. And if you're using a sample standard deviation and this confidence interval is all about the mean, and so our critical value here is going to be based on a t-table, on a t-statistic, and then we're gonna multiply that times the sample standard deviation of the differences divided by the square root of our sample size, divided by the square root of five. | Confidence interval for a mean with paired data AP Statistics Khan Academy.mp3 |
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