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She also got a hundred so Efra Efra and then finally Farah got an 80 so 60 70 80 so Farah got Got an 80 so this is Farah score right over here, so this is another way of representing the data and Here we see it in visual form, but it has the same information you can look up someone's name And then figure out their score Amy scored a 90 bill scored a 95 cam scored 100 effort Also scored 100 Farah scored an 80 and there's even other ways you can have some of this information In fact sometimes you might not even know their names and so then it would be less information But it might just a list of scores the professor must say here are the five scores that were That people got on the exam and they were list 90 95 95 100 100 100 and 80 now if it was listed and if this was all of the data you got this would be less information than the data That's in this bar graph or this histogram And or the data that's given in this table right over here because here not only do we know the scores But we know who got what score here. We only know the list of scores But there's even other ways and I this is this isn't and this is not an exhaustive video of all of the different ways you Can represent data you could also represent data by looking at the frequency of scores? So the frequency of scores right over here, so instead of writing the people you could write the scores So let's see you could say this is 80 85 90 95 and 100 and then you could record the frequency that people got these scores So how many times do you have a score of an 80? Well Farah got a is the only person with the score of 80 so you put one data point there No one got an 85 one person got a 90 so you put a data point there one person got a 95 So you could put that data point right over there, and then two people got a hundred so this is one and two Let's see the other hundred is in this color So I'm just doing the color you wouldn't necessarily have to color code it like this So this is another way to represent, and this is you know in this axis you could just view This is the number so this tells you how many 80s there were how many? 90s there are how many 95s and how many hundreds so this right over here has the same data as this list of numbers? It's just another way of looking at it and once you have your data arranged in any of these ways We can start to ask Interesting questions we can ask ourselves things like well. What is the range of data? | Ways to represent data Data and statistics 6th grade Khan Academy.mp3 |
Well Farah got a is the only person with the score of 80 so you put one data point there No one got an 85 one person got a 90 so you put a data point there one person got a 95 So you could put that data point right over there, and then two people got a hundred so this is one and two Let's see the other hundred is in this color So I'm just doing the color you wouldn't necessarily have to color code it like this So this is another way to represent, and this is you know in this axis you could just view This is the number so this tells you how many 80s there were how many? 90s there are how many 95s and how many hundreds so this right over here has the same data as this list of numbers? It's just another way of looking at it and once you have your data arranged in any of these ways We can start to ask Interesting questions we can ask ourselves things like well. What is the range of data? What is the range in? The data and the range is just the spread between the lowest point and the highest point so the range in this data It's going to be the difference between the highest score and the highest scores are hundred and the lowest score and 80 So the range is going to be the difference between the max Minus the min the maximum score minus the minimum score so it's going to be 100 Minus 80 is equal to 20 so that gives you a sense of things it kind of gives you a sense of spread You could also ask yourself. Well. | Ways to represent data Data and statistics 6th grade Khan Academy.mp3 |
What is the range of data? What is the range in? The data and the range is just the spread between the lowest point and the highest point so the range in this data It's going to be the difference between the highest score and the highest scores are hundred and the lowest score and 80 So the range is going to be the difference between the max Minus the min the maximum score minus the minimum score so it's going to be 100 Minus 80 is equal to 20 so that gives you a sense of things it kind of gives you a sense of spread You could also ask yourself. Well. How many how many people scored below 100 these are just interesting questions below 100 and you can actually answer that question well actually you could have answered either of these questions with any of these different ways of Looking at the data if you say how many people scored below 100 well one two three how many people scored below 100 well 100 is up here So it's going to be one two three how many people scored below 100? One two three how many people scored below 100? One two three and so anyway you look at it you would have gotten three And you could also ask yourself. | Ways to represent data Data and statistics 6th grade Khan Academy.mp3 |
We hopefully now have a respectable working knowledge of the sampling distribution of the sample mean. And what I want to do in this video is explore a little bit more on how that distribution changes as we change our sample size n. I'll write n down right here, our sample size n. So just as a bit of review, we saw before, we could just start off with any crazy distribution. Maybe it looks something like this. I'll do discrete distribution. Really, to model anything, at some point you have to make it discrete. It could be a very granular discrete distribution. But let's say something crazy that looks like this. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
I'll do discrete distribution. Really, to model anything, at some point you have to make it discrete. It could be a very granular discrete distribution. But let's say something crazy that looks like this. This is clearly not a normal distribution. But we saw in the first video, if you take, let's say, sample sizes of four. So if you took four numbers from this distribution, four random numbers where, let's say, this is the probability of a 1, 2, 3, 4, 5, 6, 7, 8, 9. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
But let's say something crazy that looks like this. This is clearly not a normal distribution. But we saw in the first video, if you take, let's say, sample sizes of four. So if you took four numbers from this distribution, four random numbers where, let's say, this is the probability of a 1, 2, 3, 4, 5, 6, 7, 8, 9. If you took four numbers at a time and averaged them, let me do that here. If you took four numbers at a time, let's say we use this distribution to generate four random numbers. We're very likely to get a 9. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
So if you took four numbers from this distribution, four random numbers where, let's say, this is the probability of a 1, 2, 3, 4, 5, 6, 7, 8, 9. If you took four numbers at a time and averaged them, let me do that here. If you took four numbers at a time, let's say we use this distribution to generate four random numbers. We're very likely to get a 9. We're definitely not going to get any 7 or 8's. We're definitely not going to get a 4. We might get a 1 or 2. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
We're very likely to get a 9. We're definitely not going to get any 7 or 8's. We're definitely not going to get a 4. We might get a 1 or 2. 3 is also very likely. 5 is very likely. So we use this function to essentially generate random numbers for us. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
We might get a 1 or 2. 3 is also very likely. 5 is very likely. So we use this function to essentially generate random numbers for us. And we take samples of four, and then we average them up. So let's say our first average is like, I don't know, it's a 9, it's a 5, it's another 9, and then it's a 1. So what is that? | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
So we use this function to essentially generate random numbers for us. And we take samples of four, and then we average them up. So let's say our first average is like, I don't know, it's a 9, it's a 5, it's another 9, and then it's a 1. So what is that? That's 14 plus 10, 24 divided by 4. The average for this first trial, for this first sample of four, is going to be 6. They add up to 24 divided by 4. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
So what is that? That's 14 plus 10, 24 divided by 4. The average for this first trial, for this first sample of four, is going to be 6. They add up to 24 divided by 4. So we would plot it right here. Our average was 6 that time. Just like that, and we'll just keep doing it. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
They add up to 24 divided by 4. So we would plot it right here. Our average was 6 that time. Just like that, and we'll just keep doing it. And we've seen in the past that if you just keep doing this, this is going to start looking something like a normal distribution. So maybe we do it again. The average is 6 again. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Just like that, and we'll just keep doing it. And we've seen in the past that if you just keep doing this, this is going to start looking something like a normal distribution. So maybe we do it again. The average is 6 again. Maybe we do it again. The average is 5. We do it again. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
The average is 6 again. Maybe we do it again. The average is 5. We do it again. The average is 7. We do it again. The average is 6. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
We do it again. The average is 7. We do it again. The average is 6. And then if you just do this a ton, a ton of times, your distribution might look something that looks very much like a normal distribution. So if these boxes are really small, so we just do a bunch of these trials, at some point it might look a lot like a normal distribution. Obviously, there's some average values. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
The average is 6. And then if you just do this a ton, a ton of times, your distribution might look something that looks very much like a normal distribution. So if these boxes are really small, so we just do a bunch of these trials, at some point it might look a lot like a normal distribution. Obviously, there's some average values. It won't be a perfect normal distribution, because you can't ever get anything less than 0, or anything less than 1, really, as an average. You can't get 0 as an average. And you can't get anything more than 9. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Obviously, there's some average values. It won't be a perfect normal distribution, because you can't ever get anything less than 0, or anything less than 1, really, as an average. You can't get 0 as an average. And you can't get anything more than 9. So it's not going to have infinitely long tails. But at least for the middle part of it, a normal distribution might be a good approximation. In this video, what I want to think about is what happens as we change n. So in this case, n was 4. n is our sample size. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
And you can't get anything more than 9. So it's not going to have infinitely long tails. But at least for the middle part of it, a normal distribution might be a good approximation. In this video, what I want to think about is what happens as we change n. So in this case, n was 4. n is our sample size. Every time we do a trial, we took 4. And we took their average, and we plotted it. We could have had n equal 10. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
In this video, what I want to think about is what happens as we change n. So in this case, n was 4. n is our sample size. Every time we do a trial, we took 4. And we took their average, and we plotted it. We could have had n equal 10. We could have taken 10 samples from this population, you could say, or from this random variable, averaged them, and then plotted them here. And in the last video, we ran the simulation. I'm going to go back to that simulation in a second. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
We could have had n equal 10. We could have taken 10 samples from this population, you could say, or from this random variable, averaged them, and then plotted them here. And in the last video, we ran the simulation. I'm going to go back to that simulation in a second. We saw a couple of things. And I'll show it to you in a little bit more depth this time. When n is pretty small, it doesn't approach a normal distribution that well. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
I'm going to go back to that simulation in a second. We saw a couple of things. And I'll show it to you in a little bit more depth this time. When n is pretty small, it doesn't approach a normal distribution that well. So when n is small, I mean, let's take the extreme case. What happens when n is equal to 1? That literally just means I take one instance of this random variable and average it. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
When n is pretty small, it doesn't approach a normal distribution that well. So when n is small, I mean, let's take the extreme case. What happens when n is equal to 1? That literally just means I take one instance of this random variable and average it. Well, it's just going to be that thing. So if I just take a bunch of trials from this thing and plot it over time, what's it going to look like? Well, it's definitely not going to look like a normal distribution. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
That literally just means I take one instance of this random variable and average it. Well, it's just going to be that thing. So if I just take a bunch of trials from this thing and plot it over time, what's it going to look like? Well, it's definitely not going to look like a normal distribution. It's going to look as if you're going to have a couple of ones, you're going to have a couple of twos, you're going to have more threes like that, you're going to have no fours, you're going to have a bunch of fives, you're going to have some sixes that look like that, and then you're going to have a bunch of nines. So there your sampling distribution of the sample mean for an n of 1 is going to look, I don't care how many trials you do, it's not going to look like a normal distribution. So the central limit theorem, although, I you do a bunch of trials that look like a normal distribution, it definitely doesn't work for n equals 1. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Well, it's definitely not going to look like a normal distribution. It's going to look as if you're going to have a couple of ones, you're going to have a couple of twos, you're going to have more threes like that, you're going to have no fours, you're going to have a bunch of fives, you're going to have some sixes that look like that, and then you're going to have a bunch of nines. So there your sampling distribution of the sample mean for an n of 1 is going to look, I don't care how many trials you do, it's not going to look like a normal distribution. So the central limit theorem, although, I you do a bunch of trials that look like a normal distribution, it definitely doesn't work for n equals 1. As n gets larger, though, it starts to make sense. Let's say if we got n equals 2, and I'm all just doing this in my head, I don't know what the actual distributions would look like, but then it still would be difficult for it to become an exact normal distribution, but then you can get more instances, you could get more, you know, you might get things from all of the above, but you can only get 2 in each of your baskets that you're averaging, you're only going to get 2 numbers, right? So you're never going to, let's see, I mean, for example, you're never going to get a 7.5 in your sampling distribution of the sample mean for n is equal to 2, because it's impossible to get a 7, and it's impossible to get an 8. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
So the central limit theorem, although, I you do a bunch of trials that look like a normal distribution, it definitely doesn't work for n equals 1. As n gets larger, though, it starts to make sense. Let's say if we got n equals 2, and I'm all just doing this in my head, I don't know what the actual distributions would look like, but then it still would be difficult for it to become an exact normal distribution, but then you can get more instances, you could get more, you know, you might get things from all of the above, but you can only get 2 in each of your baskets that you're averaging, you're only going to get 2 numbers, right? So you're never going to, let's see, I mean, for example, you're never going to get a 7.5 in your sampling distribution of the sample mean for n is equal to 2, because it's impossible to get a 7, and it's impossible to get an 8. So you're never going to get 7.5 as, so maybe when you plot, when you plot it, maybe, you know, maybe it looks like this, but there'll be a gap at 7.5, because that's impossible, and, you know, maybe it looks something like that. So it still won't be a normal distribution when n is equal to 2. So there's a couple of interesting things here. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
So you're never going to, let's see, I mean, for example, you're never going to get a 7.5 in your sampling distribution of the sample mean for n is equal to 2, because it's impossible to get a 7, and it's impossible to get an 8. So you're never going to get 7.5 as, so maybe when you plot, when you plot it, maybe, you know, maybe it looks like this, but there'll be a gap at 7.5, because that's impossible, and, you know, maybe it looks something like that. So it still won't be a normal distribution when n is equal to 2. So there's a couple of interesting things here. So one thing, and I didn't mention this the first time, just because I really wanted to get the gut sense of what the central limit theorem is. The central limit theorem says as n approaches, really as it approaches infinity, then is when you get the real normal distribution. Normal distribution. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
So there's a couple of interesting things here. So one thing, and I didn't mention this the first time, just because I really wanted to get the gut sense of what the central limit theorem is. The central limit theorem says as n approaches, really as it approaches infinity, then is when you get the real normal distribution. Normal distribution. But in kind of everyday practice, you don't have to get that much beyond n equals 2. If you get to n equals 10 or n equals 15, you're getting very close to a normal distribution. So this converges to a normal distribution very quickly. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Normal distribution. But in kind of everyday practice, you don't have to get that much beyond n equals 2. If you get to n equals 10 or n equals 15, you're getting very close to a normal distribution. So this converges to a normal distribution very quickly. Distribution. Now the other thing is you obviously want many, many trials. So this is your sample size. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
So this converges to a normal distribution very quickly. Distribution. Now the other thing is you obviously want many, many trials. So this is your sample size. That is your sample size. That's the size of each of your baskets. In the very first video I did on this, I took a sample as a size of 4. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
So this is your sample size. That is your sample size. That's the size of each of your baskets. In the very first video I did on this, I took a sample as a size of 4. In the simulation I did in the last video, we did sample sizes of 4 and 10 and whatever else. This is a sample size of 1. So that's our sample size. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
In the very first video I did on this, I took a sample as a size of 4. In the simulation I did in the last video, we did sample sizes of 4 and 10 and whatever else. This is a sample size of 1. So that's our sample size. So as that approaches infinity, your actual sampling distribution of the sample mean will approach a normal distribution. Now in order to actually see that normal distribution and actually prove it to yourself, you would have to do this many, many times. Remember, the normal distribution happens... | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
So that's our sample size. So as that approaches infinity, your actual sampling distribution of the sample mean will approach a normal distribution. Now in order to actually see that normal distribution and actually prove it to yourself, you would have to do this many, many times. Remember, the normal distribution happens... This is essentially... This is kind of the population or this is the random variable. That tells you all of the possibilities. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Remember, the normal distribution happens... This is essentially... This is kind of the population or this is the random variable. That tells you all of the possibilities. In real life, we seldom know all of the possibilities. In fact, in real life, we seldom know the pure probability generating function, only if we're kind of writing it or if we're writing a computer program. Normally, we're doing samples and we're trying to estimate things. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
That tells you all of the possibilities. In real life, we seldom know all of the possibilities. In fact, in real life, we seldom know the pure probability generating function, only if we're kind of writing it or if we're writing a computer program. Normally, we're doing samples and we're trying to estimate things. So normally, there's some random variable and then maybe we'll do a bunch of... We take a bunch of samples, we take their means and we plot them and then we're going to get some type of normal distribution. Let's say we take samples of 100 and we average them. We're going to get some normal distribution. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Normally, we're doing samples and we're trying to estimate things. So normally, there's some random variable and then maybe we'll do a bunch of... We take a bunch of samples, we take their means and we plot them and then we're going to get some type of normal distribution. Let's say we take samples of 100 and we average them. We're going to get some normal distribution. And in theory, as we take those averages hundreds or thousands of times, our data set is going to more closely approximate that pure sampling distribution of the sample mean. This thing is a real distribution. It's a real distribution with a real mean. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
We're going to get some normal distribution. And in theory, as we take those averages hundreds or thousands of times, our data set is going to more closely approximate that pure sampling distribution of the sample mean. This thing is a real distribution. It's a real distribution with a real mean. Its mean, it has a pure mean. Its mean, so the mean of the sampling distribution of the sample mean, we'll write it like that. Notice I didn't write it as just the X. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
It's a real distribution with a real mean. Its mean, it has a pure mean. Its mean, so the mean of the sampling distribution of the sample mean, we'll write it like that. Notice I didn't write it as just the X. What this is, this is actually saying that this is a real population mean. This is a real random variable mean. If you looked at every possibility of all of the samples that you can take from your original distribution, from some other random original distribution, and you took all of the possibilities of, let's say sample size, let's say we're dealing with a world where our sample size is 10. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Notice I didn't write it as just the X. What this is, this is actually saying that this is a real population mean. This is a real random variable mean. If you looked at every possibility of all of the samples that you can take from your original distribution, from some other random original distribution, and you took all of the possibilities of, let's say sample size, let's say we're dealing with a world where our sample size is 10. If you took all of the combinations of 10 samples from some original distribution and you averaged them out, this would describe that function. Of course, in reality, if you don't know the original distribution, you can't take an infinite samples from it. So you won't know every combination. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
If you looked at every possibility of all of the samples that you can take from your original distribution, from some other random original distribution, and you took all of the possibilities of, let's say sample size, let's say we're dealing with a world where our sample size is 10. If you took all of the combinations of 10 samples from some original distribution and you averaged them out, this would describe that function. Of course, in reality, if you don't know the original distribution, you can't take an infinite samples from it. So you won't know every combination. But if you did it with a thousand, if you did the trial a thousand times, so a thousand times you took 10 samples from some distribution and took a thousand averages and then plotted them, you're gonna get pretty close. You're gonna get pretty close. Now, the next thing I wanna touch on is, what happens is, we know as n approaches infinity, it becomes more of a normal distribution. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
So you won't know every combination. But if you did it with a thousand, if you did the trial a thousand times, so a thousand times you took 10 samples from some distribution and took a thousand averages and then plotted them, you're gonna get pretty close. You're gonna get pretty close. Now, the next thing I wanna touch on is, what happens is, we know as n approaches infinity, it becomes more of a normal distribution. But as I said already, n equals 10 is pretty good and n equals 20 is even better. But we saw something in the last video that at least I find pretty interesting. Let's say we start with this crazy distribution up here. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Now, the next thing I wanna touch on is, what happens is, we know as n approaches infinity, it becomes more of a normal distribution. But as I said already, n equals 10 is pretty good and n equals 20 is even better. But we saw something in the last video that at least I find pretty interesting. Let's say we start with this crazy distribution up here. It really doesn't matter what distribution we're starting with. We saw in the simulation that when n is equal, let's say n is equal to five, our graph after we try, we take samples of five, average them, and we do it 10,000 times, our graph looks something like this. It's kind of wide like that. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Let's say we start with this crazy distribution up here. It really doesn't matter what distribution we're starting with. We saw in the simulation that when n is equal, let's say n is equal to five, our graph after we try, we take samples of five, average them, and we do it 10,000 times, our graph looks something like this. It's kind of wide like that. And then when we did n is equal to 10, our graph looked a little bit, it was actually a little bit squeezed in like that, a little bit more. So not only was it more normal, that's what the central limit theorem tells us because we're taking larger sample sizes, but it had a smaller standard deviation or a smaller variance, right? The mean is gonna be the same either case. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
It's kind of wide like that. And then when we did n is equal to 10, our graph looked a little bit, it was actually a little bit squeezed in like that, a little bit more. So not only was it more normal, that's what the central limit theorem tells us because we're taking larger sample sizes, but it had a smaller standard deviation or a smaller variance, right? The mean is gonna be the same either case. But when our sample size was larger, our standard deviation became smaller. In fact, our standard deviation became smaller than our original population distribution, so let me, or our original probability density function. Let me show you that with a simulation. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
The mean is gonna be the same either case. But when our sample size was larger, our standard deviation became smaller. In fact, our standard deviation became smaller than our original population distribution, so let me, or our original probability density function. Let me show you that with a simulation. So let me clear everything. And this simulation is as good as any. So the first thing I wanna show, or this distribution is as good as any. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Let me show you that with a simulation. So let me clear everything. And this simulation is as good as any. So the first thing I wanna show, or this distribution is as good as any. The first thing I wanna show you is that n of two is really not that good. So let's compare an n of two to, let's say an n of 16. So when you compare an n of two to an n of 16, you know, let's do it once. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
So the first thing I wanna show, or this distribution is as good as any. The first thing I wanna show you is that n of two is really not that good. So let's compare an n of two to, let's say an n of 16. So when you compare an n of two to an n of 16, you know, let's do it once. So you got one, two trials, you average them, and then it's gonna do it 16. And then it's gonna plot it down here, and average there. Let's do that 10,000 times. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
So when you compare an n of two to an n of 16, you know, let's do it once. So you got one, two trials, you average them, and then it's gonna do it 16. And then it's gonna plot it down here, and average there. Let's do that 10,000 times. So notice, when you took an n of two, even though we did it 10,000 times, this is not approaching a normal distribution. You can actually see it in the skew and kurtosis numbers. It has a rightward positive skew, which means it has a longer tail to the right than to the left. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Let's do that 10,000 times. So notice, when you took an n of two, even though we did it 10,000 times, this is not approaching a normal distribution. You can actually see it in the skew and kurtosis numbers. It has a rightward positive skew, which means it has a longer tail to the right than to the left. And then it has a negative kurtosis, which means that it's a little bit, it has shorter tails and smaller peaks than a standard normal distribution. Now, when n is equal to 16, you do the same, so every time we took 16 samples from this distribution function up here and averaged them, and each of these dots represent an average, we did it 10,001 times. Here, and notice, the mean is the same in both places, but here all of a sudden, our kurtosis is much smaller, and our skew is much smaller, so we are more normal in this situation. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
It has a rightward positive skew, which means it has a longer tail to the right than to the left. And then it has a negative kurtosis, which means that it's a little bit, it has shorter tails and smaller peaks than a standard normal distribution. Now, when n is equal to 16, you do the same, so every time we took 16 samples from this distribution function up here and averaged them, and each of these dots represent an average, we did it 10,001 times. Here, and notice, the mean is the same in both places, but here all of a sudden, our kurtosis is much smaller, and our skew is much smaller, so we are more normal in this situation. But even a more interesting thing is our standard deviation is smaller, right? This is more squeezed in than that is, and it's definitely more squeezed in than our original distribution. Now, let me do it with two, let me clear everything again. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Here, and notice, the mean is the same in both places, but here all of a sudden, our kurtosis is much smaller, and our skew is much smaller, so we are more normal in this situation. But even a more interesting thing is our standard deviation is smaller, right? This is more squeezed in than that is, and it's definitely more squeezed in than our original distribution. Now, let me do it with two, let me clear everything again. I like this distribution, because it's a very non-normal distribution. It looks like a bimodal distribution of some kind. And let's take a scenario where I take an n of, let's take two good n's, let's take an n of 16, that's a nice healthy n, and let's take an n of 25, and let's compare them a little bit. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Now, let me do it with two, let me clear everything again. I like this distribution, because it's a very non-normal distribution. It looks like a bimodal distribution of some kind. And let's take a scenario where I take an n of, let's take two good n's, let's take an n of 16, that's a nice healthy n, and let's take an n of 25, and let's compare them a little bit. Let's compare them a little bit. So if we, let's just do one trial animated, just to, it's always nice to see it. So first it's gonna do 16 of these trials and average them, and there we go. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
And let's take a scenario where I take an n of, let's take two good n's, let's take an n of 16, that's a nice healthy n, and let's take an n of 25, and let's compare them a little bit. Let's compare them a little bit. So if we, let's just do one trial animated, just to, it's always nice to see it. So first it's gonna do 16 of these trials and average them, and there we go. Then it's gonna do 25 of these trials, and then average them, and then there we go. Now let's do that, what I just did, animated, let's do it 10,000 times, miracles of computers. Now notice something, this is 10,000 times, these are both pretty good approximations of normal distributions. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
So first it's gonna do 16 of these trials and average them, and there we go. Then it's gonna do 25 of these trials, and then average them, and then there we go. Now let's do that, what I just did, animated, let's do it 10,000 times, miracles of computers. Now notice something, this is 10,000 times, these are both pretty good approximations of normal distributions. The n is equal to 25 is more normal, it has less skew, slightly less skew than n is equal to 16. It has slightly less kurtosis, which means it's closer to being a normal distribution than n is equal to 16. But even more interesting, it's more squeezed in, it has a lower standard deviation. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
Now notice something, this is 10,000 times, these are both pretty good approximations of normal distributions. The n is equal to 25 is more normal, it has less skew, slightly less skew than n is equal to 16. It has slightly less kurtosis, which means it's closer to being a normal distribution than n is equal to 16. But even more interesting, it's more squeezed in, it has a lower standard deviation. The standard deviation here is 2.1, and the standard deviation here is 2.64. So that's another, I mean, you know, and I kind of touched on that in the last video, and it kind of makes sense. For every sample you do for your average, the more you put in that sample, the less standard deviation. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
But even more interesting, it's more squeezed in, it has a lower standard deviation. The standard deviation here is 2.1, and the standard deviation here is 2.64. So that's another, I mean, you know, and I kind of touched on that in the last video, and it kind of makes sense. For every sample you do for your average, the more you put in that sample, the less standard deviation. Think of the extreme case. If instead of taking 16 samples from our distribution every time, or instead of taking 25, if I were to take a million samples from this distribution every time, if I were to take a million samples from this distribution every time, that sample mean is always gonna be pretty darn close to my mean. If I take a million samples of everything, if I try to essentially try to estimate a mean by taking a million samples, I'm gonna get a pretty good estimate of that mean. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
For every sample you do for your average, the more you put in that sample, the less standard deviation. Think of the extreme case. If instead of taking 16 samples from our distribution every time, or instead of taking 25, if I were to take a million samples from this distribution every time, if I were to take a million samples from this distribution every time, that sample mean is always gonna be pretty darn close to my mean. If I take a million samples of everything, if I try to essentially try to estimate a mean by taking a million samples, I'm gonna get a pretty good estimate of that mean. The probability that a bunch of the, a million numbers are all out here is very low. So if n is a million, of course, all of my sample means when I average them are all gonna be really tightly focused around the mean itself. And it actually, and so hopefully that kind of makes sense to you as well. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
If I take a million samples of everything, if I try to essentially try to estimate a mean by taking a million samples, I'm gonna get a pretty good estimate of that mean. The probability that a bunch of the, a million numbers are all out here is very low. So if n is a million, of course, all of my sample means when I average them are all gonna be really tightly focused around the mean itself. And it actually, and so hopefully that kind of makes sense to you as well. If it doesn't, just think about it, or even use this tool and experiment with it, just so you can trust that that is really the case. And it actually turns out that there's a very clean formula that relates the standard deviation of the original probability distribution function to the standard deviation of the sampling distribution of the sample mean. And as you can imagine, it is a function of your sample size of how many samples you take out in every basket before you average them. | Sampling distribution of the sample mean 2 Probability and Statistics Khan Academy.mp3 |
And then in the next few videos, we'll actually use it to really test how well theoretical distributions explain observed ones, or how good a fit observed results are for theoretical distributions. So let's just think about it a little bit. So let's say I have some random variables. And each of them are independent, standard, normally distributed random variables. Let me just remind you what that means. So let's say I have the random variable X. If X is normally distributed, we could write that X is a normal random variable with a mean of 0 and a variance of 1. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
And each of them are independent, standard, normally distributed random variables. Let me just remind you what that means. So let's say I have the random variable X. If X is normally distributed, we could write that X is a normal random variable with a mean of 0 and a variance of 1. Or you could say that the expected value of X is equal to 0, or in that the variance of our random variable X is equal to 1. Or just to visualize it, is that we're sampling, when we take an instantiation of this variable, we're sampling from a normal distribution, a standardized normal distribution that looks like this. Mean of 0, and then a variance of 1, which would also mean, of course, a standard deviation of 1. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
If X is normally distributed, we could write that X is a normal random variable with a mean of 0 and a variance of 1. Or you could say that the expected value of X is equal to 0, or in that the variance of our random variable X is equal to 1. Or just to visualize it, is that we're sampling, when we take an instantiation of this variable, we're sampling from a normal distribution, a standardized normal distribution that looks like this. Mean of 0, and then a variance of 1, which would also mean, of course, a standard deviation of 1. So that could be the standard deviation, or the variance, or the standard deviation, that would be equal to 1. So a chi-squared distribution, if you just take one of these random variables, and let me define it this way. Let me define a new random variable, q, that is equal to you essentially sampling from this standard normal distribution, and then squaring whatever number you got. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
Mean of 0, and then a variance of 1, which would also mean, of course, a standard deviation of 1. So that could be the standard deviation, or the variance, or the standard deviation, that would be equal to 1. So a chi-squared distribution, if you just take one of these random variables, and let me define it this way. Let me define a new random variable, q, that is equal to you essentially sampling from this standard normal distribution, and then squaring whatever number you got. So it is equal to this random variable X squared. The distribution for this random variable right here is going to be an example of the chi-squared distribution. Actually, what we're going to see in this video is that the chi-squared distribution is actually a set of distributions, depending on how many sums you have. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
Let me define a new random variable, q, that is equal to you essentially sampling from this standard normal distribution, and then squaring whatever number you got. So it is equal to this random variable X squared. The distribution for this random variable right here is going to be an example of the chi-squared distribution. Actually, what we're going to see in this video is that the chi-squared distribution is actually a set of distributions, depending on how many sums you have. Right now, we only have one random variable that we're squaring. So this is just one of the examples, and we'll talk more about them in a second. So this right here, we could write that q is a chi-squared distributed random variable, or that we could use this notation right here. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
Actually, what we're going to see in this video is that the chi-squared distribution is actually a set of distributions, depending on how many sums you have. Right now, we only have one random variable that we're squaring. So this is just one of the examples, and we'll talk more about them in a second. So this right here, we could write that q is a chi-squared distributed random variable, or that we could use this notation right here. q is, we could write it like this. So this isn't an X anymore, this is the Greek letter chi, although it looks a lot like a curvy X. So it's a member of chi-squared, and since we're only taking one sum over here, we're only taking the sum of one independent, standard normally distributed variable, we say that this only has one degree of freedom. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
So this right here, we could write that q is a chi-squared distributed random variable, or that we could use this notation right here. q is, we could write it like this. So this isn't an X anymore, this is the Greek letter chi, although it looks a lot like a curvy X. So it's a member of chi-squared, and since we're only taking one sum over here, we're only taking the sum of one independent, standard normally distributed variable, we say that this only has one degree of freedom. And we write that over here. So this right here is our degree of freedom. We have one degree of freedom right over there. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
So it's a member of chi-squared, and since we're only taking one sum over here, we're only taking the sum of one independent, standard normally distributed variable, we say that this only has one degree of freedom. And we write that over here. So this right here is our degree of freedom. We have one degree of freedom right over there. Now, if we defined, so let's call this q1. Let's say I have another random variable. Let's call this q, let me do it in a different color. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
We have one degree of freedom right over there. Now, if we defined, so let's call this q1. Let's say I have another random variable. Let's call this q, let me do it in a different color. Let me do q2 in blue. Let's say I have another random variable q2. That is defined as, let's say I have one independent, standard normally distributed variable. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
Let's call this q, let me do it in a different color. Let me do q2 in blue. Let's say I have another random variable q2. That is defined as, let's say I have one independent, standard normally distributed variable. I'll call that x1, and I square it. And then I have another independent, standard normally distributed variable x2, and I square it. So you can imagine, both of these guys have distributions like this, and they're independent. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
That is defined as, let's say I have one independent, standard normally distributed variable. I'll call that x1, and I square it. And then I have another independent, standard normally distributed variable x2, and I square it. So you can imagine, both of these guys have distributions like this, and they're independent. So to get to sample q2, you essentially sample x1 from this distribution, square that value, sample x2 from the same distribution essentially, square that value, and then add the two, and you're going to get q2. This over here, we would write, so this is q1. q2 here is a chi-squared distributed random variable with two degrees of freedom. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
So you can imagine, both of these guys have distributions like this, and they're independent. So to get to sample q2, you essentially sample x1 from this distribution, square that value, sample x2 from the same distribution essentially, square that value, and then add the two, and you're going to get q2. This over here, we would write, so this is q1. q2 here is a chi-squared distributed random variable with two degrees of freedom. And just to visualize the set of chi-squared distributions, let's look at this over here. So I got this off of Wikipedia. This shows us some of the probability density functions for some of the chi-squared distributions. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
q2 here is a chi-squared distributed random variable with two degrees of freedom. And just to visualize the set of chi-squared distributions, let's look at this over here. So I got this off of Wikipedia. This shows us some of the probability density functions for some of the chi-squared distributions. This first one over here, for k equal to 1, that's the degrees of freedom. So this is essentially our q1. This is our probability density function for q1. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
This shows us some of the probability density functions for some of the chi-squared distributions. This first one over here, for k equal to 1, that's the degrees of freedom. So this is essentially our q1. This is our probability density function for q1. And notice, it really spikes close to 0, and that makes sense, because if you're sampling just once from this standard normal distribution, there's a very high likelihood that you're going to get something pretty close to 0. And then if you square something close to 0, remember, these are decimals, they're going to be less than 1, pretty close to 0, it's going to become even smaller. So you have a high probability of getting a very small value. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
This is our probability density function for q1. And notice, it really spikes close to 0, and that makes sense, because if you're sampling just once from this standard normal distribution, there's a very high likelihood that you're going to get something pretty close to 0. And then if you square something close to 0, remember, these are decimals, they're going to be less than 1, pretty close to 0, it's going to become even smaller. So you have a high probability of getting a very small value. You have high probabilities of getting values less than some threshold, this right here, less than 1 right here, so less than 1 half. And you have a very low probability of getting a large number. I mean, to get a 4, you would have to sample a 2 from this distribution, and we know that 2 is, actually, it's 2 variances, or 2 standard deviations from the mean. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
So you have a high probability of getting a very small value. You have high probabilities of getting values less than some threshold, this right here, less than 1 right here, so less than 1 half. And you have a very low probability of getting a large number. I mean, to get a 4, you would have to sample a 2 from this distribution, and we know that 2 is, actually, it's 2 variances, or 2 standard deviations from the mean. So it's less likely. And actually, that's to get a 4. So to get even larger numbers are going to be even less likely, so that's why you see this shape over here. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
I mean, to get a 4, you would have to sample a 2 from this distribution, and we know that 2 is, actually, it's 2 variances, or 2 standard deviations from the mean. So it's less likely. And actually, that's to get a 4. So to get even larger numbers are going to be even less likely, so that's why you see this shape over here. Now, when you have 2 degrees of freedom, it moderates a little bit. This is the shape, this blue line right here, is the shape of q2. And notice, you're a little bit less likely to get values close to 0, and a little bit more likely to get numbers further out, but it still is kind of shifted, or heavily weighted, towards small numbers. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
So to get even larger numbers are going to be even less likely, so that's why you see this shape over here. Now, when you have 2 degrees of freedom, it moderates a little bit. This is the shape, this blue line right here, is the shape of q2. And notice, you're a little bit less likely to get values close to 0, and a little bit more likely to get numbers further out, but it still is kind of shifted, or heavily weighted, towards small numbers. And then if we had another random variable, another chi squared distributed random variable, so then we have, let's say, q3, and let's define it as the sum of 3 of these independent variables, each of them that have a standard normal distribution. So x1, x2 squared, plus x3 squared. Then all of a sudden, our q3, this is q2 right here, has a chi squared distribution with 3 degrees of freedom. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
And notice, you're a little bit less likely to get values close to 0, and a little bit more likely to get numbers further out, but it still is kind of shifted, or heavily weighted, towards small numbers. And then if we had another random variable, another chi squared distributed random variable, so then we have, let's say, q3, and let's define it as the sum of 3 of these independent variables, each of them that have a standard normal distribution. So x1, x2 squared, plus x3 squared. Then all of a sudden, our q3, this is q2 right here, has a chi squared distribution with 3 degrees of freedom. And so this guy right over here, that will be this green line, maybe I should have done this in green. This will be this green line over here. And then notice, now it's starting to become a little bit more likely that you get values in this range over here, because you're taking the sum. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
Then all of a sudden, our q3, this is q2 right here, has a chi squared distribution with 3 degrees of freedom. And so this guy right over here, that will be this green line, maybe I should have done this in green. This will be this green line over here. And then notice, now it's starting to become a little bit more likely that you get values in this range over here, because you're taking the sum. Each of these are going to be pretty small values, but you're taking the sum, so it starts to shift it a little over to the right. And so the more degrees of freedom you have, the further this lump starts to move to the right, and to some degree, the more symmetric it gets. And what's interesting about this, I guess it's different than almost every other distribution we've looked at, although we've looked at others that have this property as well, is that you can't have a value below 0, because we're always just squaring these values. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
And then notice, now it's starting to become a little bit more likely that you get values in this range over here, because you're taking the sum. Each of these are going to be pretty small values, but you're taking the sum, so it starts to shift it a little over to the right. And so the more degrees of freedom you have, the further this lump starts to move to the right, and to some degree, the more symmetric it gets. And what's interesting about this, I guess it's different than almost every other distribution we've looked at, although we've looked at others that have this property as well, is that you can't have a value below 0, because we're always just squaring these values. Each of these guys can have values below 0. They're normally distributed. They could have negative values. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
And what's interesting about this, I guess it's different than almost every other distribution we've looked at, although we've looked at others that have this property as well, is that you can't have a value below 0, because we're always just squaring these values. Each of these guys can have values below 0. They're normally distributed. They could have negative values. But since we're squaring and taking the sum of squares, this is always going to be positive. And the place that this is going to be useful, and we're going to see in the next few videos, is in measuring essentially error from an expected value. And if you take this total error, you can figure out the probability of getting that total error if you hold some parameters, or if you assume some parameters. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
They could have negative values. But since we're squaring and taking the sum of squares, this is always going to be positive. And the place that this is going to be useful, and we're going to see in the next few videos, is in measuring essentially error from an expected value. And if you take this total error, you can figure out the probability of getting that total error if you hold some parameters, or if you assume some parameters. And we'll talk more about it in the next video. Now with that said, I just want to show you how to read a chi squared distribution table. So if I were to ask you if this is our distribution, let me pick this blue one right here. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
And if you take this total error, you can figure out the probability of getting that total error if you hold some parameters, or if you assume some parameters. And we'll talk more about it in the next video. Now with that said, I just want to show you how to read a chi squared distribution table. So if I were to ask you if this is our distribution, let me pick this blue one right here. So over here we have 2 degrees of freedom, because we're adding 2 of these guys right here. If I were to ask you what is the probability of Q2 being greater than 2.41? And I'm picking that value for a reason. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
So if I were to ask you if this is our distribution, let me pick this blue one right here. So over here we have 2 degrees of freedom, because we're adding 2 of these guys right here. If I were to ask you what is the probability of Q2 being greater than 2.41? And I'm picking that value for a reason. So I want the probability of Q2 being greater than 2.41. What I want to do is I'll look at a chi squared table like this. Q2 is a version of chi squared with 2 degrees of freedom. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
And I'm picking that value for a reason. So I want the probability of Q2 being greater than 2.41. What I want to do is I'll look at a chi squared table like this. Q2 is a version of chi squared with 2 degrees of freedom. So I look at this row right here under 2 degrees of freedom, and I want the probability of getting a value above 2.41. And I picked 2.41 because it's actually at this table. And so most of these chi squared, the reason why we have these weird numbers like this instead of whole numbers or easy to read fractions, is it's actually driven by the p-value. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
Q2 is a version of chi squared with 2 degrees of freedom. So I look at this row right here under 2 degrees of freedom, and I want the probability of getting a value above 2.41. And I picked 2.41 because it's actually at this table. And so most of these chi squared, the reason why we have these weird numbers like this instead of whole numbers or easy to read fractions, is it's actually driven by the p-value. It's driven by the probability of getting something larger than that value. So normally you would look at it the other way. You'd say, OK, if I want to see what chi squared value for 2 degrees of freedom, there's a 30% chance of getting something larger than that, then I would look up 2.41. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
And so most of these chi squared, the reason why we have these weird numbers like this instead of whole numbers or easy to read fractions, is it's actually driven by the p-value. It's driven by the probability of getting something larger than that value. So normally you would look at it the other way. You'd say, OK, if I want to see what chi squared value for 2 degrees of freedom, there's a 30% chance of getting something larger than that, then I would look up 2.41. But I'm doing it the other way just for the sake of this video. So if I want the probability of this random variable right here being greater than 2.41, or its p-value, we read it right here, it is 30%. And just to visualize it on this chart, this chi squared distribution, this was q2, the blue one over here, 2.41 is going to sit, let's see, this is 3, this is 2.5, so 2.41 is going to be someplace right around here. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
You'd say, OK, if I want to see what chi squared value for 2 degrees of freedom, there's a 30% chance of getting something larger than that, then I would look up 2.41. But I'm doing it the other way just for the sake of this video. So if I want the probability of this random variable right here being greater than 2.41, or its p-value, we read it right here, it is 30%. And just to visualize it on this chart, this chi squared distribution, this was q2, the blue one over here, 2.41 is going to sit, let's see, this is 3, this is 2.5, so 2.41 is going to be someplace right around here. So essentially what that table is telling us is this entire area under this blue line right here, what is that? And that right there is going to be 30% of, well, it's going to be 0.3. Or you could view it as 30% of the entire area under this curve, because obviously all the probabilities have to add up to 1. | Chi-square distribution introduction Probability and Statistics Khan Academy.mp3 |
At the Olympic Games, many events have several rounds of competition. One of these is the men's 100-meter backstroke. The upper dot plot shows the times in seconds of the top eight finishers in the semi-final round at the 2012 Olympics. The lower dot plot shows the times of the same eight swimmers, but in the final round. Which pieces of information can be gathered from these dot plots? In the semi-final round, we see that these are the eight times of the eight swimmers. 53 swimmers finished in exactly 53.5 seconds. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
The lower dot plot shows the times of the same eight swimmers, but in the final round. Which pieces of information can be gathered from these dot plots? In the semi-final round, we see that these are the eight times of the eight swimmers. 53 swimmers finished in exactly 53.5 seconds. One swimmer finished in 53.7 seconds right here. And one swimmer right over here finished in 52.7 seconds. We can think about similar things for each of these dots. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
53 swimmers finished in exactly 53.5 seconds. One swimmer finished in 53.7 seconds right here. And one swimmer right over here finished in 52.7 seconds. We can think about similar things for each of these dots. Now in the final round, one swimmer here went much, much, much faster. So this is in 52.2 seconds. While this swimmer right over here went slower. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
We can think about similar things for each of these dots. Now in the final round, one swimmer here went much, much, much faster. So this is in 52.2 seconds. While this swimmer right over here went slower. We don't know which dot he was up here, but regardless of which dot he was up here, this dot took more time than all of these dots. So his time definitely got worse. This is at 53.8 seconds. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
While this swimmer right over here went slower. We don't know which dot he was up here, but regardless of which dot he was up here, this dot took more time than all of these dots. So his time definitely got worse. This is at 53.8 seconds. Let's look at the statements and see which of these apply. The swimmers had faster times on average in the finals. Is this true? | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
This is at 53.8 seconds. Let's look at the statements and see which of these apply. The swimmers had faster times on average in the finals. Is this true? Faster times on average in the finals. If we look at the finals right over here, we could take each of these times, add them up and then divide by eight the number of times we have. Let's see if we can get an intuition for where this is. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
Is this true? Faster times on average in the finals. If we look at the finals right over here, we could take each of these times, add them up and then divide by eight the number of times we have. Let's see if we can get an intuition for where this is. We're really just comparing these two plots or these two distributions. Let's see. If all the data was these three points and these three points, we could intuit that the mean would be right around there. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
Let's see if we can get an intuition for where this is. We're really just comparing these two plots or these two distributions. Let's see. If all the data was these three points and these three points, we could intuit that the mean would be right around there. It would be around 53.2 or 53.3 seconds, right around there. Then we have this point and this point. If you just found the mean of that point and that point, so halfway between that point and that point would get you right around there. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
If all the data was these three points and these three points, we could intuit that the mean would be right around there. It would be around 53.2 or 53.3 seconds, right around there. Then we have this point and this point. If you just found the mean of that point and that point, so halfway between that point and that point would get you right around there. The mean of those two points would bring down the mean a little bit. Once again, I'm not figuring out the exact number, but maybe it would be around 53.1 or 53.2 seconds. That's my intuition for the mean of the final round. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
If you just found the mean of that point and that point, so halfway between that point and that point would get you right around there. The mean of those two points would bring down the mean a little bit. Once again, I'm not figuring out the exact number, but maybe it would be around 53.1 or 53.2 seconds. That's my intuition for the mean of the final round. Now let's think about the mean of the semifinal round. Let's just look at these bottom five dots. If you find the mean, you could intuit it would be maybe right around, someplace around here, pretty close to around 53.3 seconds. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
That's my intuition for the mean of the final round. Now let's think about the mean of the semifinal round. Let's just look at these bottom five dots. If you find the mean, you could intuit it would be maybe right around, someplace around here, pretty close to around 53.3 seconds. Then you have all these other ones that are at 53.5 and 53.3, which will bring the mean even higher. I think it's fair to say that the mean in the final round, the time, is less than the mean up here. You could calculate it yourself, but I'm just trying to look at the distributions and get an intuition here. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
If you find the mean, you could intuit it would be maybe right around, someplace around here, pretty close to around 53.3 seconds. Then you have all these other ones that are at 53.5 and 53.3, which will bring the mean even higher. I think it's fair to say that the mean in the final round, the time, is less than the mean up here. You could calculate it yourself, but I'm just trying to look at the distributions and get an intuition here. At least in this case, it looks pretty clear that the swimmers had faster times on average in the finals. It took them less time. One of the swimmers was disqualified from the finals. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
You could calculate it yourself, but I'm just trying to look at the distributions and get an intuition here. At least in this case, it looks pretty clear that the swimmers had faster times on average in the finals. It took them less time. One of the swimmers was disqualified from the finals. That's not true. We have eight swimmers in the semifinal round and we have eight swimmers in the final round, so that one's not true. The times in the finals vary noticeably more than the times in the semifinals. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
One of the swimmers was disqualified from the finals. That's not true. We have eight swimmers in the semifinal round and we have eight swimmers in the final round, so that one's not true. The times in the finals vary noticeably more than the times in the semifinals. That does look to be true. We see in the semifinals a lot of the times were clumped up right around here, at 53.3 seconds and 53.5 seconds. The high time isn't as high as this time. | Comparing distributions with dot plots (example problem) 7th grade Khan Academy.mp3 |
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