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Fifth flip, two possibilities. So it's 2 times 2 times 2 times 2 times 2. I hope I said that five times. So 2, or you could use 2 to the fifth power, and that is going to be equal to 32 equally likely possibilities. 32, 2 times 2 is 4, 4 times 2 is 8, 8 times 2 is 16, 16 times 2 is 32 possibilities. And to figure out this probability, we really just have to figure out how many of those possibilities involve getting three heads. We could write out all of the 32 possibilities and literally just count the heads, but let's just use that other technique that we just started to explore in that last video. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
So 2, or you could use 2 to the fifth power, and that is going to be equal to 32 equally likely possibilities. 32, 2 times 2 is 4, 4 times 2 is 8, 8 times 2 is 16, 16 times 2 is 32 possibilities. And to figure out this probability, we really just have to figure out how many of those possibilities involve getting three heads. We could write out all of the 32 possibilities and literally just count the heads, but let's just use that other technique that we just started to explore in that last video. We have five flips here. So let me draw the flips. 1, 2, 3, 4, 5. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
We could write out all of the 32 possibilities and literally just count the heads, but let's just use that other technique that we just started to explore in that last video. We have five flips here. So let me draw the flips. 1, 2, 3, 4, 5. And we want to have exactly three heads. And I'm going to call those three heads, I'm going to call them, let me do it in pink, heads A, heads B, heads C, just to give them a name. Although what we're going to see later in this video is that we don't want to differentiate between them. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
1, 2, 3, 4, 5. And we want to have exactly three heads. And I'm going to call those three heads, I'm going to call them, let me do it in pink, heads A, heads B, heads C, just to give them a name. Although what we're going to see later in this video is that we don't want to differentiate between them. To us, it makes no difference if we get this ordering, heads A, heads B, heads C, tails, tails, or if we get this ordering, heads A, heads C, or heads B, tails, tails. We can't count these as two different orderings. We can only count this as one. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
Although what we're going to see later in this video is that we don't want to differentiate between them. To us, it makes no difference if we get this ordering, heads A, heads B, heads C, tails, tails, or if we get this ordering, heads A, heads C, or heads B, tails, tails. We can't count these as two different orderings. We can only count this as one. So what we're going to do is first come up with all of the different orderings if we cared about the difference between A, B, and C. And then we're going to divide by all of the different ways that you can arrange three different things. So how many ways can we put A, B, and C into these five buckets that we can view as the flips if we cared about A, B, and C? So let's start with A. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
We can only count this as one. So what we're going to do is first come up with all of the different orderings if we cared about the difference between A, B, and C. And then we're going to divide by all of the different ways that you can arrange three different things. So how many ways can we put A, B, and C into these five buckets that we can view as the flips if we cared about A, B, and C? So let's start with A. If we haven't allocated any of these buckets to any of the heads yet, then we could say that A could be in five different buckets. So there's five possibilities where A could be. So let's just say that this is the one that it goes in, although it could be in any one of these five. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
So let's start with A. If we haven't allocated any of these buckets to any of the heads yet, then we could say that A could be in five different buckets. So there's five possibilities where A could be. So let's just say that this is the one that it goes in, although it could be in any one of these five. But if this takes up one of the five, then how many different possibilities can this heads sit in? How many different possibilities are there? Well, then there's only going to be four buckets left. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
So let's just say that this is the one that it goes in, although it could be in any one of these five. But if this takes up one of the five, then how many different possibilities can this heads sit in? How many different possibilities are there? Well, then there's only going to be four buckets left. So then there's only four possibilities. And so if this was where heads A goes, then heads B could be in any of the other four. If heads A was in this first one, then heads B could have been in any of the four. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
Well, then there's only going to be four buckets left. So then there's only four possibilities. And so if this was where heads A goes, then heads B could be in any of the other four. If heads A was in this first one, then heads B could have been in any of the four. I'll just do a particular example. Maybe heads B shows up right there. So once we've taken two of the slots, how many spaces do we have for heads C? | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
If heads A was in this first one, then heads B could have been in any of the four. I'll just do a particular example. Maybe heads B shows up right there. So once we've taken two of the slots, how many spaces do we have for heads C? Well, we only have three spaces left then for heads C. And so it could be in any of these three spaces. And just to show a particular example, it would look like that. And so if you cared about order, how many different ways can you, out of five different spaces, allocate them to three different heads? | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
So once we've taken two of the slots, how many spaces do we have for heads C? Well, we only have three spaces left then for heads C. And so it could be in any of these three spaces. And just to show a particular example, it would look like that. And so if you cared about order, how many different ways can you, out of five different spaces, allocate them to three different heads? You would say it is 5 times 4 times 3. 5 times 4 is 20, times 3 is equal to 60. So you would say there's 60 different ways to arrange heads A, B, and C in five buckets, or five flips, or if these were people in five chairs. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
And so if you cared about order, how many different ways can you, out of five different spaces, allocate them to three different heads? You would say it is 5 times 4 times 3. 5 times 4 is 20, times 3 is equal to 60. So you would say there's 60 different ways to arrange heads A, B, and C in five buckets, or five flips, or if these were people in five chairs. And obviously, there aren't 60 possibilities of getting three heads. In fact, there's only 32 equally likely possibilities. And the reason why we got such a big number over here is that we are counting this scenario as being fundamentally different than if this was heads B, heads A, and then heads C over here. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
So you would say there's 60 different ways to arrange heads A, B, and C in five buckets, or five flips, or if these were people in five chairs. And obviously, there aren't 60 possibilities of getting three heads. In fact, there's only 32 equally likely possibilities. And the reason why we got such a big number over here is that we are counting this scenario as being fundamentally different than if this was heads B, heads A, and then heads C over here. And what we need to do is say, well, these aren't different possibilities. We don't have to overcount for all of the different ways you arrange this. And so what we need to do is divide this by all of the different ways that you can arrange three things. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
And the reason why we got such a big number over here is that we are counting this scenario as being fundamentally different than if this was heads B, heads A, and then heads C over here. And what we need to do is say, well, these aren't different possibilities. We don't have to overcount for all of the different ways you arrange this. And so what we need to do is divide this by all of the different ways that you can arrange three things. So if I have three things that are in three spaces, so here I have heads in the second flip, third flip, and fifth flip. If I have three things in three spaces like this, how many ways can I arrange them? And so if I have three spaces, how many ways can I arrange an A, B, and C in those three spaces? | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
And so what we need to do is divide this by all of the different ways that you can arrange three things. So if I have three things that are in three spaces, so here I have heads in the second flip, third flip, and fifth flip. If I have three things in three spaces like this, how many ways can I arrange them? And so if I have three spaces, how many ways can I arrange an A, B, and C in those three spaces? Well, A can go into three spaces. It can go into any of the three. A can go into any of the three spaces. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
And so if I have three spaces, how many ways can I arrange an A, B, and C in those three spaces? Well, A can go into three spaces. It can go into any of the three. A can go into any of the three spaces. Then B would have two spaces left once A takes one of them. And then C would have one space left once A and B take two of them. So there's three times two times one way to arrange three different things. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
A can go into any of the three spaces. Then B would have two spaces left once A takes one of them. And then C would have one space left once A and B take two of them. So there's three times two times one way to arrange three different things. So three times two times one is equal to six. So the number of possibilities of getting three heads is actually going to be this 5 times 4 times 3. Let me write this down in another color. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
So there's three times two times one way to arrange three different things. So three times two times one is equal to six. So the number of possibilities of getting three heads is actually going to be this 5 times 4 times 3. Let me write this down in another color. So the number of possibilities, let's write this for short, is equal to this 5 times 4 times 3 over the number of ways that I can rearrange three things. Because we don't want to over count for all of these. Viewing this arrangement is fundamentally different than this arrangement. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
Let me write this down in another color. So the number of possibilities, let's write this for short, is equal to this 5 times 4 times 3 over the number of ways that I can rearrange three things. Because we don't want to over count for all of these. Viewing this arrangement is fundamentally different than this arrangement. So then we want to divide it by 3 times 2. I want to do that same orange color. Dividing it by 3 times 2 times 1. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
Viewing this arrangement is fundamentally different than this arrangement. So then we want to divide it by 3 times 2. I want to do that same orange color. Dividing it by 3 times 2 times 1. 3 times 2 times 1. And which gives us, in the numerator, 120 divided by 6. It's 60 divided by 6. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
Dividing it by 3 times 2 times 1. 3 times 2 times 1. And which gives us, in the numerator, 120 divided by 6. It's 60 divided by 6. This is 60. 5 times 4 times 3 is 60. It gives us 60 divided by 6, which gives us 10 possibilities that gives us exactly three heads. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
It's 60 divided by 6. This is 60. 5 times 4 times 3 is 60. It gives us 60 divided by 6, which gives us 10 possibilities that gives us exactly three heads. And that's of 32 equally likely possibilities. So the probability of getting exactly three heads, well you get exactly three heads in 10 of the 32 equally likely possibilities. So you have a 5 over 16 chance of that happening. | Exactly three heads in five flips Probability and Statistics Khan Academy.mp3 |
So let's define the random variable X. So let's say that X is equal to the number of made shots, number of made free throws when taking six free throws. So it's how many of the six do you make? And we're going to assume what we assumed in the first video in this series of these making free throws. So we're gonna assume the 70% free throw probability right over here. So assuming assumptions, assuming 70% free throw free throw percentage. All right, so let's figure out the probabilities of the different values that X could actually take on. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
And we're going to assume what we assumed in the first video in this series of these making free throws. So we're gonna assume the 70% free throw probability right over here. So assuming assumptions, assuming 70% free throw free throw percentage. All right, so let's figure out the probabilities of the different values that X could actually take on. So let's see, what is the probability, what is the probability that X is equal to zero? That even though you have a 70% free throw percentage, that you make none of the shots. And actually you could calculate this through probably some common sense without using all of these fancy things. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
All right, so let's figure out the probabilities of the different values that X could actually take on. So let's see, what is the probability, what is the probability that X is equal to zero? That even though you have a 70% free throw percentage, that you make none of the shots. And actually you could calculate this through probably some common sense without using all of these fancy things. But just to make things consistent, I'm gonna write it out. So this is going to be, it's going to be equal to six choose zero times 0.7 to the zeroth power times 0.3 to the sixth power. And this right over here is gonna end up being one. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
And actually you could calculate this through probably some common sense without using all of these fancy things. But just to make things consistent, I'm gonna write it out. So this is going to be, it's going to be equal to six choose zero times 0.7 to the zeroth power times 0.3 to the sixth power. And this right over here is gonna end up being one. This over here is going to end up being one. And so you're just gonna be left with 0.3 to the sixth power. And I calculated it ahead of time. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
And this right over here is gonna end up being one. This over here is going to end up being one. And so you're just gonna be left with 0.3 to the sixth power. And I calculated it ahead of time. So if we just round to the nearest, if we round our percentages to the nearest tenth, this is going to give you approximately, approximately, well if we round the decimal to the nearest thousandth, you're gonna get something like that, which is approximately equal to 0.1% chance of you missing all of them. So roughly, I'm speaking roughly here, one in a thousand, one in a thousand chance of that happening, of missing all six free throws. Now let's keep going. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
And I calculated it ahead of time. So if we just round to the nearest, if we round our percentages to the nearest tenth, this is going to give you approximately, approximately, well if we round the decimal to the nearest thousandth, you're gonna get something like that, which is approximately equal to 0.1% chance of you missing all of them. So roughly, I'm speaking roughly here, one in a thousand, one in a thousand chance of that happening, of missing all six free throws. Now let's keep going. This is fun. So what is the probability that our random variable is equal to one? Well this is going to be six choose one times 0.7 to the first power times 0.3 to the six minus first power, so that's the fifth power. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
Now let's keep going. This is fun. So what is the probability that our random variable is equal to one? Well this is going to be six choose one times 0.7 to the first power times 0.3 to the six minus first power, so that's the fifth power. And I calculated this, and this is approximately 0.01, or we could say 1%. So still a fairly low probability, 10 times more likely than this, roughly, but still a fairly low probability. Let's keep going. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
Well this is going to be six choose one times 0.7 to the first power times 0.3 to the six minus first power, so that's the fifth power. And I calculated this, and this is approximately 0.01, or we could say 1%. So still a fairly low probability, 10 times more likely than this, roughly, but still a fairly low probability. Let's keep going. So the probability that X is equal to two, well that's what our first video was, essentially. So this is going to be six choose two times 0.7 squared times 0.3 to the fourth power. And we saw that this is approximately going to be 0.06, or we could say 6%. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
Let's keep going. So the probability that X is equal to two, well that's what our first video was, essentially. So this is going to be six choose two times 0.7 squared times 0.3 to the fourth power. And we saw that this is approximately going to be 0.06, or we could say 6%. And obviously you could type these things in a calculator and get a much more precise answer, but just for the sake of just getting a sense of what these probabilities look like, that's why I'm giving these rough estimates. Kind of, I guess you could say to the closest, maybe tenth of a percent. And actually if you round to the closest tenth of a percent, you actually get to 6.0%, and this is 1.0%, because this we actually went to a tenth of a percent here. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
And we saw that this is approximately going to be 0.06, or we could say 6%. And obviously you could type these things in a calculator and get a much more precise answer, but just for the sake of just getting a sense of what these probabilities look like, that's why I'm giving these rough estimates. Kind of, I guess you could say to the closest, maybe tenth of a percent. And actually if you round to the closest tenth of a percent, you actually get to 6.0%, and this is 1.0%, because this we actually went to a tenth of a percent here. But let's keep going. We're obviously going to have to do a few more of these. So let me just make sure I have enough real estate. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
And actually if you round to the closest tenth of a percent, you actually get to 6.0%, and this is 1.0%, because this we actually went to a tenth of a percent here. But let's keep going. We're obviously going to have to do a few more of these. So let me just make sure I have enough real estate. All right, so the probability that our random variable is equal to three is going to be six choose three, and I'm sure you could probably fill this out on your own, but I'm going to do it. 0.7 to the third power times 0.3 to the six minus three, which is the third power, which is approximately equal to, well, it's going to be 0.185 or 18.5, 18.5%. So yeah, you know, that's definitely within the realm of possibility. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
So let me just make sure I have enough real estate. All right, so the probability that our random variable is equal to three is going to be six choose three, and I'm sure you could probably fill this out on your own, but I'm going to do it. 0.7 to the third power times 0.3 to the six minus three, which is the third power, which is approximately equal to, well, it's going to be 0.185 or 18.5, 18.5%. So yeah, you know, that's definitely within the realm of possibility. I mean, all of these are in the realm of possibility, but it's starting to be a non-insignificant probability. So now let's do the probability that our random variable is equal to four. Well, it's going to be six choose four times 0.7 to the fourth power times 0.3 to the six minus four, or second power, which is equal to, this is going to get equal to, or approximately, because I am taking away a little bit of the precision when I write these things down, 0.324, so approximately 32.4% chance of making exactly four out of the six free throws. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
So yeah, you know, that's definitely within the realm of possibility. I mean, all of these are in the realm of possibility, but it's starting to be a non-insignificant probability. So now let's do the probability that our random variable is equal to four. Well, it's going to be six choose four times 0.7 to the fourth power times 0.3 to the six minus four, or second power, which is equal to, this is going to get equal to, or approximately, because I am taking away a little bit of the precision when I write these things down, 0.324, so approximately 32.4% chance of making exactly four out of the six free throws. All right, two more to go. Let's see, I have not used purple as yet. So the probability that our random variable is equal to five, it's going to be six choose five, or times, I should say, 0.7 to the fifth power times 0.3 to the first power, and that is going to be roughly, roughly 0.303, which is 30.3%. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
Well, it's going to be six choose four times 0.7 to the fourth power times 0.3 to the six minus four, or second power, which is equal to, this is going to get equal to, or approximately, because I am taking away a little bit of the precision when I write these things down, 0.324, so approximately 32.4% chance of making exactly four out of the six free throws. All right, two more to go. Let's see, I have not used purple as yet. So the probability that our random variable is equal to five, it's going to be six choose five, or times, I should say, 0.7 to the fifth power times 0.3 to the first power, and that is going to be roughly, roughly 0.303, which is 30.3%. That's interesting, one more left. So the probability that I make all of them, of all six, is going to be equal to, is equal to six choose six, and 0.7 to the sixth power times 0.3 to the zeroth power, which is, this right over here is going to be one, this is going to be one, so it's really just 0.7 to the sixth power, and this is approximately 0.118, I calculated that ahead of time, which is 11.8%, and so there's something interesting that's going on here. The first time we looked at the binomial distribution, we said, hey, you know, there's this symmetry as we kind of got to some type of a peak and went down, but I don't see that symmetry here, and the reason why you're not seeing that symmetry is that you are more likely to make a free throw than not. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
So the probability that our random variable is equal to five, it's going to be six choose five, or times, I should say, 0.7 to the fifth power times 0.3 to the first power, and that is going to be roughly, roughly 0.303, which is 30.3%. That's interesting, one more left. So the probability that I make all of them, of all six, is going to be equal to, is equal to six choose six, and 0.7 to the sixth power times 0.3 to the zeroth power, which is, this right over here is going to be one, this is going to be one, so it's really just 0.7 to the sixth power, and this is approximately 0.118, I calculated that ahead of time, which is 11.8%, and so there's something interesting that's going on here. The first time we looked at the binomial distribution, we said, hey, you know, there's this symmetry as we kind of got to some type of a peak and went down, but I don't see that symmetry here, and the reason why you're not seeing that symmetry is that you are more likely to make a free throw than not. So you have a 70% free throw probability. This is no longer just flipping a fair coin. Where you will see the symmetry is in these coefficients. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
The first time we looked at the binomial distribution, we said, hey, you know, there's this symmetry as we kind of got to some type of a peak and went down, but I don't see that symmetry here, and the reason why you're not seeing that symmetry is that you are more likely to make a free throw than not. So you have a 70% free throw probability. This is no longer just flipping a fair coin. Where you will see the symmetry is in these coefficients. If you calculate these coefficients, six choose zero is one, six choose six is one. You would see that six choose one is six, and six choose five is six. You'd see six choose two is 15, and six choose four is also 15, and then six choose three is 20. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
Where you will see the symmetry is in these coefficients. If you calculate these coefficients, six choose zero is one, six choose six is one. You would see that six choose one is six, and six choose five is six. You'd see six choose two is 15, and six choose four is also 15, and then six choose three is 20. So you definitely see the symmetry in the coefficients, but then these things are weighted by the fact that you're more likely to make something than miss something. If these were both 0.5, then you would also see the symmetry right over here, and you can plot this to essentially visualize what the probability distribution looks like for this example, and I encourage you to do that, to take these different cases, just like we did in that first example with the fair coin, and plot these. But this essentially does give you the probability distribution for the random variable in question. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
You'd see six choose two is 15, and six choose four is also 15, and then six choose three is 20. So you definitely see the symmetry in the coefficients, but then these things are weighted by the fact that you're more likely to make something than miss something. If these were both 0.5, then you would also see the symmetry right over here, and you can plot this to essentially visualize what the probability distribution looks like for this example, and I encourage you to do that, to take these different cases, just like we did in that first example with the fair coin, and plot these. But this essentially does give you the probability distribution for the random variable in question. This is, I just wrote it out instead of just visualizing it, but it says, okay, well, so these are the different values that this random variable can take on. It can't take on negative one, or it can't be 15.5, or pi, or one million. These are the only seven values that this random variable can take on, and I've just given you the probabilities, or I guess you could say the rough probabilities, of the random variable taking on each of those seven values. | Free throw binomial probability distribution Probability and Statistics Khan Academy.mp3 |
So if someone just says the mean, they're really referring to what we typically, in everyday language, call the average. Sometimes it's called the arithmetic mean because you'll learn that there's other ways of actually calculating a mean. But it's really you just sum up all of the numbers and you divide by the numbers there are. And so it's one way of measuring the central tendency, or the average, I guess we could say. So this is our mean. We want to average 23 plus 29. Or we want to sum 23 plus 29 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And so it's one way of measuring the central tendency, or the average, I guess we could say. So this is our mean. We want to average 23 plus 29. Or we want to sum 23 plus 29 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25. And then divide that by the number of numbers. So we have 1, 2, 3, 4, 5, 6, 7, 8 numbers. So you want to divide that by 8. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Or we want to sum 23 plus 29 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25. And then divide that by the number of numbers. So we have 1, 2, 3, 4, 5, 6, 7, 8 numbers. So you want to divide that by 8. So let's figure out what that actually is. Actually, I'll just get the calculator out for this part. I could do it by hand, but we'll save some time over here. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So you want to divide that by 8. So let's figure out what that actually is. Actually, I'll just get the calculator out for this part. I could do it by hand, but we'll save some time over here. So we have 23 plus 29 plus 20 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25. So the sum of all the numbers is 206. And then we want to divide 206 by 8. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
I could do it by hand, but we'll save some time over here. So we have 23 plus 29 plus 20 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25. So the sum of all the numbers is 206. And then we want to divide 206 by 8. So if I say 206 divided by 8 gets us 25.75. So the mean is equal to 25.75. So this is one way to kind of measure the center, the central tendency. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And then we want to divide 206 by 8. So if I say 206 divided by 8 gets us 25.75. So the mean is equal to 25.75. So this is one way to kind of measure the center, the central tendency. Another way is with the median. And this is to pick out the middle number, the median. And to figure out the median, what we want to do is order these numbers from least to greatest. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So this is one way to kind of measure the center, the central tendency. Another way is with the median. And this is to pick out the middle number, the median. And to figure out the median, what we want to do is order these numbers from least to greatest. So it looks like the smallest number here is 20. Then the next one is 21. Then we go, there's no 22 here. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And to figure out the median, what we want to do is order these numbers from least to greatest. So it looks like the smallest number here is 20. Then the next one is 21. Then we go, there's no 22 here. Let's see, there's two 23s, 23 and a 23. So 23 and a 23. And no 24s. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Then we go, there's no 22 here. Let's see, there's two 23s, 23 and a 23. So 23 and a 23. And no 24s. There's a 25. There's no 26, 27, 28. There is a 29. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And no 24s. There's a 25. There's no 26, 27, 28. There is a 29. And then you have your 32, 32. And then you have your 33, 33. So what's the middle number now that we've ordered it? | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
There is a 29. And then you have your 32, 32. And then you have your 33, 33. So what's the middle number now that we've ordered it? So we have 1, 2, 3, 4, 5, 6, 7, 8 numbers. We already knew that. And so there's actually going to be two middles. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So what's the middle number now that we've ordered it? So we have 1, 2, 3, 4, 5, 6, 7, 8 numbers. We already knew that. And so there's actually going to be two middles. If you have an even number, there's actually two numbers that kind of qualify close to the middle. And to actually get the median, we're going to average them. So 23 will be one of them. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
And so there's actually going to be two middles. If you have an even number, there's actually two numbers that kind of qualify close to the middle. And to actually get the median, we're going to average them. So 23 will be one of them. That by itself can't be the median because there's three less than it and there's four greater than it. Five by itself can't be the median because there's three larger than it and four less than it. So what we do is we take the mean of these two numbers and we pick that as the median. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So 23 will be one of them. That by itself can't be the median because there's three less than it and there's four greater than it. Five by itself can't be the median because there's three larger than it and four less than it. So what we do is we take the mean of these two numbers and we pick that as the median. So if you take 23 plus 25 divided by 2, that's 48 over 2, which is equal to 24. So even though 24 isn't one of these numbers, the median is 24. So this is the middle number. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So what we do is we take the mean of these two numbers and we pick that as the median. So if you take 23 plus 25 divided by 2, that's 48 over 2, which is equal to 24. So even though 24 isn't one of these numbers, the median is 24. So this is the middle number. So once again, this is one way of thinking about central tendency. If you wanted a number that could somehow represent the middle, and I want to be clear, there's no one way of doing it. This is one way of measuring the middle. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
So this is the middle number. So once again, this is one way of thinking about central tendency. If you wanted a number that could somehow represent the middle, and I want to be clear, there's no one way of doing it. This is one way of measuring the middle. Let me put that in quotes. The middle, if you had to represent this data with one number, and this is another way of representing the middle. Then finally, we can think about the mode. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
This is one way of measuring the middle. Let me put that in quotes. The middle, if you had to represent this data with one number, and this is another way of representing the middle. Then finally, we can think about the mode. And the mode is just the number that shows up the most in this data set. And all of these numbers show up once except we have the 23, it shows up twice. And so because 23 shows up the most, it shows up twice, every other number only shows up once, 23 is our mode. | Finding mean, median, and mode Descriptive statistics Probability and Statistics Khan Academy.mp3 |
Sheera drew the line below to show the trend in the data. Assuming the line is correct, what does the line slope of 15 mean? So let's see. The horizontal axis is time studying in hours. Vertical axis is scores on the test. And each of these blue dots represent the time and the score for a given student. So this student right over here spent, I don't know, it looks like they spent about.6 hours studying, and they didn't do too well on the exam. | Interpreting a trend line Data and modeling 8th grade Khan Academy.mp3 |
The horizontal axis is time studying in hours. Vertical axis is scores on the test. And each of these blue dots represent the time and the score for a given student. So this student right over here spent, I don't know, it looks like they spent about.6 hours studying, and they didn't do too well on the exam. They look like they got below a 45, looks like a 43 or 44 on the exam. This student over here spent almost 4 1 1 1 studying and got, looks like a 94, close to a 95 on the exam. And what Sheera did is try to draw a line that tries to fit this data. | Interpreting a trend line Data and modeling 8th grade Khan Academy.mp3 |
So this student right over here spent, I don't know, it looks like they spent about.6 hours studying, and they didn't do too well on the exam. They look like they got below a 45, looks like a 43 or 44 on the exam. This student over here spent almost 4 1 1 1 studying and got, looks like a 94, close to a 95 on the exam. And what Sheera did is try to draw a line that tries to fit this data. It seems like it does a pretty good job of at least showing the trend in the data. Now, slope of 15 means that if I'm on the line, so let's say I'm here, and if I increase in the horizontal direction by one, so there I increased by the horizontal direction by one, I should be increasing in the vertical direction by 15. And you see that. | Interpreting a trend line Data and modeling 8th grade Khan Academy.mp3 |
And what Sheera did is try to draw a line that tries to fit this data. It seems like it does a pretty good job of at least showing the trend in the data. Now, slope of 15 means that if I'm on the line, so let's say I'm here, and if I increase in the horizontal direction by one, so there I increased by the horizontal direction by one, I should be increasing in the vertical direction by 15. And you see that. We increased by one hour here, we increased by 15% on the test. Now, what that means is that the trend it shows is that in general along this trend, if someone studies an extra hour, then if we're going with that trend, then hey, you know, it seems reasonable that they might expect to see a 15% gain on their test. Now let's see which of these are consistent. | Interpreting a trend line Data and modeling 8th grade Khan Academy.mp3 |
And you see that. We increased by one hour here, we increased by 15% on the test. Now, what that means is that the trend it shows is that in general along this trend, if someone studies an extra hour, then if we're going with that trend, then hey, you know, it seems reasonable that they might expect to see a 15% gain on their test. Now let's see which of these are consistent. In general, students who didn't study at all got scores of about 15 on the test. Well, let's see, this is neither true, we don't see that these are the people who didn't study at all and they didn't get 15 on the test, and that's definitely not what this 15 implies. This doesn't say what the people who didn't study at all get. | Interpreting a trend line Data and modeling 8th grade Khan Academy.mp3 |
Now let's see which of these are consistent. In general, students who didn't study at all got scores of about 15 on the test. Well, let's see, this is neither true, we don't see that these are the people who didn't study at all and they didn't get 15 on the test, and that's definitely not what this 15 implies. This doesn't say what the people who didn't study at all get. So this one is not true. That one is not true. Let's try this one. | Interpreting a trend line Data and modeling 8th grade Khan Academy.mp3 |
This doesn't say what the people who didn't study at all get. So this one is not true. That one is not true. Let's try this one. If one student studied for one hour more than another student, the student who studied more got exactly 15 more points on the test. Well, this is getting closer to the spirit of what the slope means, but this word exactly is what, at least in my mind, messes this choice up. Because this isn't saying that it's guaranteed that if you study an hour extra that you'll get 15% more on the test. | Interpreting a trend line Data and modeling 8th grade Khan Academy.mp3 |
Let's try this one. If one student studied for one hour more than another student, the student who studied more got exactly 15 more points on the test. Well, this is getting closer to the spirit of what the slope means, but this word exactly is what, at least in my mind, messes this choice up. Because this isn't saying that it's guaranteed that if you study an hour extra that you'll get 15% more on the test. This is just saying that this is the general trend that this line is seeing. So it's not guaranteed. For example, we could find this student here who studied exactly two hours, and if we look at the students who studied for three hours, well, there's no one exactly at three hours, but some of them, so this was, let's see, the student who was at two hours, you go to three hours, there's no one exactly there, but there's gonna be students who got better than what would be expected and students who might get a little bit worse. | Interpreting a trend line Data and modeling 8th grade Khan Academy.mp3 |
Because this isn't saying that it's guaranteed that if you study an hour extra that you'll get 15% more on the test. This is just saying that this is the general trend that this line is seeing. So it's not guaranteed. For example, we could find this student here who studied exactly two hours, and if we look at the students who studied for three hours, well, there's no one exactly at three hours, but some of them, so this was, let's see, the student who was at two hours, you go to three hours, there's no one exactly there, but there's gonna be students who got better than what would be expected and students who might get a little bit worse. Notice, there's points above the trend line and there's points below the trend line. So this exactly, you can't say it's guaranteed an hour more turns into 15%. Let's try this choice. | Interpreting a trend line Data and modeling 8th grade Khan Academy.mp3 |
For example, we could find this student here who studied exactly two hours, and if we look at the students who studied for three hours, well, there's no one exactly at three hours, but some of them, so this was, let's see, the student who was at two hours, you go to three hours, there's no one exactly there, but there's gonna be students who got better than what would be expected and students who might get a little bit worse. Notice, there's points above the trend line and there's points below the trend line. So this exactly, you can't say it's guaranteed an hour more turns into 15%. Let's try this choice. In general, studying for one extra hour was associated with a 15-point improvement in test score. That feels about right. In general, studying for 15 extra hours was associated with a one-point improvement in test score. | Interpreting a trend line Data and modeling 8th grade Khan Academy.mp3 |
Let's try this choice. In general, studying for one extra hour was associated with a 15-point improvement in test score. That feels about right. In general, studying for 15 extra hours was associated with a one-point improvement in test score. Now, that would get the slope the other way around, so that's definitely not the case. Let's check our answer. We got it right. | Interpreting a trend line Data and modeling 8th grade Khan Academy.mp3 |
So for example, this one over here in the top left, it's made out of chocolate on the outside, but it doesn't have coconut on the inside. While this one right over here does, is chocolate on the outside, and has coconut on the inside. While this one, whoops, I didn't want to do that. While this one, while this one right over here does not have chocolate on the outside, but it does have coconut on the inside. And this one right over here has neither chocolate nor coconut. And what I want to think about is ways to represent this information that we are looking at. And one way to do it is using a Venn diagram. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
While this one, while this one right over here does not have chocolate on the outside, but it does have coconut on the inside. And this one right over here has neither chocolate nor coconut. And what I want to think about is ways to represent this information that we are looking at. And one way to do it is using a Venn diagram. So let me draw a Venn diagram. So Venn diagram is one way to represent it. And the way it's typically done, my intention is that you would make a rectangle to represent the universe that you care about. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
And one way to do it is using a Venn diagram. So let me draw a Venn diagram. So Venn diagram is one way to represent it. And the way it's typically done, my intention is that you would make a rectangle to represent the universe that you care about. In this case, it would be all the chocolates. So all the numbers inside of this should add up to the number of chocolates I have. So it should add up to 12. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
And the way it's typically done, my intention is that you would make a rectangle to represent the universe that you care about. In this case, it would be all the chocolates. So all the numbers inside of this should add up to the number of chocolates I have. So it should add up to 12. So that's our universe right over here. And then I'll draw circles to represent the sets that I care about. So say for this one, I care about the set of the things that have chocolate. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
So it should add up to 12. So that's our universe right over here. And then I'll draw circles to represent the sets that I care about. So say for this one, I care about the set of the things that have chocolate. So I'll draw that with a circle. Oftentimes, you could draw them to scale, but I'm not going to draw them to scale. So that is my chocolate set. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
So say for this one, I care about the set of the things that have chocolate. So I'll draw that with a circle. Oftentimes, you could draw them to scale, but I'm not going to draw them to scale. So that is my chocolate set. And then I'll have a coconut set. So coconut, once again, not drawn to scale. I drew them roughly the same size, but you can see the chocolate set is bigger than the coconut set in reality. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
So that is my chocolate set. And then I'll have a coconut set. So coconut, once again, not drawn to scale. I drew them roughly the same size, but you can see the chocolate set is bigger than the coconut set in reality. Coconut set. And now we can fill in the different sections. So how many of these things have chocolate but no coconut? | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
I drew them roughly the same size, but you can see the chocolate set is bigger than the coconut set in reality. Coconut set. And now we can fill in the different sections. So how many of these things have chocolate but no coconut? Let's see. We have one, two, three, four, five, six have chocolate but no coconut. So that's going to be the... Actually, let me do that in a different color because I think the colors are important. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
So how many of these things have chocolate but no coconut? Let's see. We have one, two, three, four, five, six have chocolate but no coconut. So that's going to be the... Actually, let me do that in a different color because I think the colors are important. So let me do it in green. So one, two, three, four, five, and six. So this section right over here is six. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
So that's going to be the... Actually, let me do that in a different color because I think the colors are important. So let me do it in green. So one, two, three, four, five, and six. So this section right over here is six. And once again, I'm not talking about the whole brown thing. I'm talking about just this area that I've shaded in green. Now how many have chocolate and coconut? | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
So this section right over here is six. And once again, I'm not talking about the whole brown thing. I'm talking about just this area that I've shaded in green. Now how many have chocolate and coconut? Well, that's going to be one, two, three. So three of them have chocolate and coconut. And notice that's this section here that's in the overlap between. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
Now how many have chocolate and coconut? Well, that's going to be one, two, three. So three of them have chocolate and coconut. And notice that's this section here that's in the overlap between. Three of them go into both sets, both categories. These three have coconut and they have chocolate. How many total have chocolate? | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
And notice that's this section here that's in the overlap between. Three of them go into both sets, both categories. These three have coconut and they have chocolate. How many total have chocolate? Well, six plus three, nine. How many total have coconut? Well, we're going to have to figure that out in a second. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
How many total have chocolate? Well, six plus three, nine. How many total have coconut? Well, we're going to have to figure that out in a second. So how many have coconut but no chocolate? Well, there's only one with coconut and no chocolate. So that's that one right over there. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
Well, we're going to have to figure that out in a second. So how many have coconut but no chocolate? Well, there's only one with coconut and no chocolate. So that's that one right over there. And that represents this area that I'm shading in in white. So how many total coconut are there? Well, one plus three or four. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
So that's that one right over there. And that represents this area that I'm shading in in white. So how many total coconut are there? Well, one plus three or four. And you see that, one, two, three, four. And then the last thing we'd want to fill in, because notice six plus three plus one only adds up to ten. What about the other two? | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
Well, one plus three or four. And you see that, one, two, three, four. And then the last thing we'd want to fill in, because notice six plus three plus one only adds up to ten. What about the other two? Well, the other two are neither chocolate nor coconut. Actually, let me color this. So that's one, two. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
What about the other two? Well, the other two are neither chocolate nor coconut. Actually, let me color this. So that's one, two. These are neither chocolate nor coconut. And I could write these two right over here. These are neither chocolate nor coconut. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
So that's one, two. These are neither chocolate nor coconut. And I could write these two right over here. These are neither chocolate nor coconut. So that's one way to represent the information of how many chocolates, how many coconuts, and how many chocolate and coconuts, and how many neither. But there's other ways that we could do it. Another way to do it would be with a two-way table. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
These are neither chocolate nor coconut. So that's one way to represent the information of how many chocolates, how many coconuts, and how many chocolate and coconuts, and how many neither. But there's other ways that we could do it. Another way to do it would be with a two-way table. A two-way table. And on one axis, say the vertical axis, we could say, let me write this. So has chocolate. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
Another way to do it would be with a two-way table. A two-way table. And on one axis, say the vertical axis, we could say, let me write this. So has chocolate. Has chocolate. I'll write chalk for short. And then I'll write no chocolate. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
So has chocolate. Has chocolate. I'll write chalk for short. And then I'll write no chocolate. No chocolate, chalk for short. And then over here, I could write coconut. I want to do that in white. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
And then I'll write no chocolate. No chocolate, chalk for short. And then over here, I could write coconut. I want to do that in white. I got new tools, and sometimes the color changing isn't so easy. So this is coconut. And then over here, I'll write no coconut. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
I want to do that in white. I got new tools, and sometimes the color changing isn't so easy. So this is coconut. And then over here, I'll write no coconut. No coconut. And then let me make a little table. Let me make a table, make it clear what I'm doing here. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
And then over here, I'll write no coconut. No coconut. And then let me make a little table. Let me make a table, make it clear what I'm doing here. So a line there and a line there. And then I'll add a line over here as well. And then I can just fill in the different things. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
Let me make a table, make it clear what I'm doing here. So a line there and a line there. And then I'll add a line over here as well. And then I can just fill in the different things. So how many have this cell right over this square? This is going to represent the number that has coconut and chocolate. Coconut and chocolate. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
And then I can just fill in the different things. So how many have this cell right over this square? This is going to represent the number that has coconut and chocolate. Coconut and chocolate. Well, we already looked into that. That's one, two, three. That's these three right over here. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
Coconut and chocolate. Well, we already looked into that. That's one, two, three. That's these three right over here. So that's those three right over there. This one right over here is it has chocolate, but it doesn't have coconut. Well, that's this six right over here. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
That's these three right over here. So that's those three right over there. This one right over here is it has chocolate, but it doesn't have coconut. Well, that's this six right over here. It has chocolate, but it doesn't have coconut. So let me write this is that six right over there. And then this box would be it has coconut, but no chocolate. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
Well, that's this six right over here. It has chocolate, but it doesn't have coconut. So let me write this is that six right over there. And then this box would be it has coconut, but no chocolate. Well, how many is that? Well, coconut, no chocolate. That's that one there. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
And then this box would be it has coconut, but no chocolate. Well, how many is that? Well, coconut, no chocolate. That's that one there. And this one is going to be no coconut and no chocolate. And we know what that's going to be. No coconut and no chocolate is going to be two. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
That's that one there. And this one is going to be no coconut and no chocolate. And we know what that's going to be. No coconut and no chocolate is going to be two. And if we wanted to, we could even throw in totals over here. We could write, actually let me just do that just for fun. I could write total. | Two-way frequency tables and Venn diagrams Data and modeling 8th grade Khan Academy.mp3 |
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