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So it's going to be a very low standard deviation. It's gonna look something like that. And I'll show you that on the simulation app in the next, or probably later in this video. So two things happen. As you increase your sample size for every time you do the average, two things are happening. You're becoming more norm... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So two things happen. As you increase your sample size for every time you do the average, two things are happening. You're becoming more normal, and your standard deviation is getting smaller. So the question might arise, is there a formula? So if I know the standard deviation, so this is my standard deviation of just ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So the question might arise, is there a formula? So if I know the standard deviation, so this is my standard deviation of just my original probability density function. This is the mean of my original probability density function. So if I know the standard deviation, and I know n, n is gonna change depending on how man... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So if I know the standard deviation, and I know n, n is gonna change depending on how many samples I'm taking every time I do a sample mean. If I know my standard deviation, or maybe if I know my variance, the variance is just the standard deviation squared. If you don't remember that, you might wanna review those vide... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
But if I know the variance of my original distribution, and if I know what my n is, how many samples I'm gonna take every time before I average them in order to plot one thing in my sampling distribution of my sample mean, is there a way to predict what the mean of these distributions are? And so this, sorry, the stand... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
One is just the square root of the other. So this is the variance of our original distribution. Now, to show that this is the variance of our sampling distribution of our sample mean, we'll write it right here. This is the variance of our mean, of our sample mean. Remember, the sample, our true mean is this, that the G... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
This is the variance of our mean, of our sample mean. Remember, the sample, our true mean is this, that the Greek letter mu is your true mean. This is equal to the mean, while an X with a line over it means sample mean. Sample mean. So here, what we're saying is this is the variance of our sample means. Now, this is go... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
Sample mean. So here, what we're saying is this is the variance of our sample means. Now, this is gonna be a true distribution. This isn't an estimate. This is, there's some, you know, if we magically knew this distribution, there's some true variance here. And of course, the mean, so this has a mean. This right here, ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
This isn't an estimate. This is, there's some, you know, if we magically knew this distribution, there's some true variance here. And of course, the mean, so this has a mean. This right here, we can just get our notation right. This is the mean of the sampling distribution of the sampling mean. So this is the mean of o... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
This right here, we can just get our notation right. This is the mean of the sampling distribution of the sampling mean. So this is the mean of our means. It just happens to be the same thing. This is the mean of our sample means. It's gonna be the same thing as that, especially if we do the trial over and over again. ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
It just happens to be the same thing. This is the mean of our sample means. It's gonna be the same thing as that, especially if we do the trial over and over again. But anyway, the point of this video, is there any way to figure out this variance, given the variance of the original distribution and your N? And it turns... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
But anyway, the point of this video, is there any way to figure out this variance, given the variance of the original distribution and your N? And it turns out there is. And I'm not gonna do a proof here. I really wanna give you the intuition of it. And I think you already do have the sense that every trial you take, i... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
I really wanna give you the intuition of it. And I think you already do have the sense that every trial you take, if you take 100, you're much more likely, when you average those out, to get close to the true mean than if you took an N of two or an N of five. You're just very unlikely to be far away, right, if you took... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So I think you know that, in some way, it should be inversely proportional to N. The larger your N, the smaller standard deviation. And actually, it turns out it's about as simple as possible. It's one of those magical things about mathematics. And I'll prove it to you one day. I want to give you a working knowledge fi... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
And I'll prove it to you one day. I want to give you a working knowledge first. In statistics, I'm always struggling whether I should be formal and giving you rigorous proofs, but I've kind of come to the conclusion that it's more important to get the working knowledge first in statistics. And then later, once you've g... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
And then later, once you've gotten all of that down, we can get into the real deep math of it and prove it to you. But I think experimental proofs are kind of all you need for right now, using those simulations to show that they're really true. So it turns out that the variance of your sampling distribution of your sam... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So if this up here has a variance of, let's say this up here has a variance of 20, I'm just making that number up, then, and then let's say your N is 20, then the variance of your sampling distribution of your sample mean for N of 20, well, you're just gonna take that, the variance up here, your variance is 20, divided... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
What's gonna be the square root of that? Standard deviation is gonna be the square root of one, well, that's also going to be one. So we could also write this. We could take the square root of both sides of this and say the standard deviation of the sampling distribution of the, of the standard deviation of the samplin... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
We could take the square root of both sides of this and say the standard deviation of the sampling distribution of the, of the standard deviation of the sampling distribution of the sample mean, it's often called the standard deviation of the mean, and it's also called, I'm gonna write this down, the standard error of ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
But if we just take the square root of both sides, the standard error of the mean, or the standard deviation of the sampling distribution of the sample mean is equal to the standard deviation of your original, of your original function, of your original probability density function, which could be very non-normal, divi... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So here, the standard deviation, when n is 20, the standard deviation of the sampling distribution of the sample mean is gonna be one. Here, when n is 100, well, our variance, our variance here, when n is equal to 100, so our variance of the sampling mean of the sample distribution, or our variance of the mean, of the ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
Now, this guy's standard deviation, or the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean, is gonna be the square root of that. So one over the square root of five. And so, this guy's a little bit under 1 1 2 standard deviation while this guy had a standard deviati... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So you see, it's definitely thinner. Now, I know what you're saying. Well, Sal, you just gave a formula. I don't necessarily believe you. Well, let's see if we can prove it to ourselves using the simulation. So I'll, just for fun, let me make a, I'll just mess with this distribution a little bit. So that's my new distr... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
I don't necessarily believe you. Well, let's see if we can prove it to ourselves using the simulation. So I'll, just for fun, let me make a, I'll just mess with this distribution a little bit. So that's my new distribution. And let me take an n of, let me take two things that's easy to take the square root of, because ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So that's my new distribution. And let me take an n of, let me take two things that's easy to take the square root of, because if we're looking at standard deviation. So let's take, we'll take an n of 16, and an n of 25. And let's, well, I'll do a, let's do 10,000 trials. So in this case, every one of the trials, we're... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
And let's, well, I'll do a, let's do 10,000 trials. So in this case, every one of the trials, we're gonna take 16 samples from here, average them, plot it here, and then do a frequency plot. Here, we're gonna do 25 at a time, and then average them. I'll do it once animated, just to remember. So I'm taking 16 samples, p... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
I'll do it once animated, just to remember. So I'm taking 16 samples, plot it there. I take 16 samples, as described by this probability density function, or 25 now, plot it down here. Now, if I do that 10,000 times, what do I get? What do I get? All right, so here, you know, just visually, you can tell just when n was... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
Now, if I do that 10,000 times, what do I get? What do I get? All right, so here, you know, just visually, you can tell just when n was larger, the standard deviation here is smaller. This is more squeezed together. But actually, let's write, let's write this stuff down. Let's see if I can remember it here. Here, n is,... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
This is more squeezed together. But actually, let's write, let's write this stuff down. Let's see if I can remember it here. Here, n is, so in this random distribution I made, my standard deviation was 9.3. I'm gonna remember these. Our standard deviation for the original thing was 9.3. And so, standard deviation here ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
Here, n is, so in this random distribution I made, my standard deviation was 9.3. I'm gonna remember these. Our standard deviation for the original thing was 9.3. And so, standard deviation here was 2.3, and the standard deviation here is 1.87. Let's see if it, if it conforms to our formula. So I'm gonna take this offs... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
And so, standard deviation here was 2.3, and the standard deviation here is 1.87. Let's see if it, if it conforms to our formula. So I'm gonna take this offscreen for a second, and I'm gonna go back and do some mathematics. So I have this on my other screen, so I can remember those numbers. So in the trial we just did,... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So I have this on my other screen, so I can remember those numbers. So in the trial we just did, my wacky distribution had a standard deviation of 9.3. When n is equal to, let me do this in another color. When n was equal to 16, just doing the experiment, doing a bunch of trials, and averaging, and doing all the things... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
When n was equal to 16, just doing the experiment, doing a bunch of trials, and averaging, and doing all the things, we got the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean. We experimentally determined it to be 2.33. And then when n is equal to 25, when n is equ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
Let's see if it conforms to our formulas. So we know that the variance, or we could almost say the variance of the mean, or the standard error, well, you know, the variance of the sampling distribution of the sample mean is equal to the variance of our original distribution divided by n. Take the square roots of both s... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So 9.3 divided by the square root of 16, right? N is 16. So divided by the square root of 16, which is four, what do I get? So 9.3 divided by four. Let me get a little calculator out here. Let's see, we have, let me clear it out. We wanted to divide 9.3 divided by four. | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So 9.3 divided by four. Let me get a little calculator out here. Let's see, we have, let me clear it out. We wanted to divide 9.3 divided by four. 9.3 divided by our square root of n, n was 16, so divided by four is equal to 2.32. 2.32. So this is equal to, this is equal to 2.32, which is pretty darn close to 2.33. | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
We wanted to divide 9.3 divided by four. 9.3 divided by our square root of n, n was 16, so divided by four is equal to 2.32. 2.32. So this is equal to, this is equal to 2.32, which is pretty darn close to 2.33. This was after 10,000 trials. Maybe right after this, I'll see what happens if we did 20,000 or 30,000 trials... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So this is equal to, this is equal to 2.32, which is pretty darn close to 2.33. This was after 10,000 trials. Maybe right after this, I'll see what happens if we did 20,000 or 30,000 trials where we take samples of 16 and average them. Now let's look at this. Here, we would take 9.3. So let me draw a little line here. ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
Now let's look at this. Here, we would take 9.3. So let me draw a little line here. Maybe scroll over, that might be better. So we take our standard deviation of our original distribution. So just that formula that we derived right here would tell us that our standard error should be equal to the standard deviation of ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
Maybe scroll over, that might be better. So we take our standard deviation of our original distribution. So just that formula that we derived right here would tell us that our standard error should be equal to the standard deviation of our original distribution, 9.3, divided by the square root of n, divided by the squa... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
Four was just the square root of 16. So this is equal to 9.3 divided by five, and let's see if it's 1.87. So let me get my calculator back. So if I take 9.3 divided by five, what do I get? 1.86, which is very close to 1.87. So we got, we got in this case, 1.86. 1.86. | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So if I take 9.3 divided by five, what do I get? 1.86, which is very close to 1.87. So we got, we got in this case, 1.86. 1.86. So as you can see, what we got experimentally was almost exactly, and this was after 10,000 trials, of what you would expect. Let's do another 10,000. So you got another 10,000 trials. | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
1.86. So as you can see, what we got experimentally was almost exactly, and this was after 10,000 trials, of what you would expect. Let's do another 10,000. So you got another 10,000 trials. Well, we're still in the ballpark. We're not gonna, maybe I can't hope to get the exact number, you know, rounded or whatever. Bu... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So you got another 10,000 trials. Well, we're still in the ballpark. We're not gonna, maybe I can't hope to get the exact number, you know, rounded or whatever. But as you can see, hopefully that'll be pretty satisfying to you, that the variance of the sampling distribution of the sample mean, the variance of the sampl... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
But as you can see, hopefully that'll be pretty satisfying to you, that the variance of the sampling distribution of the sample mean, the variance of the sampling distribution sampling mean, is just going to be equal to the variance of your original distribution, no matter how wacky that distribution might be, divided ... | Standard error of the mean Inferential statistics Probability and Statistics Khan Academy.mp3 |
So let's say I have a population and I care about some proportion. Let's say I care about the proportion of folks that are left-handed. I don't know what that is, and so I take a sample of size n, and then from that sample, I can calculate a sample proportion. That's why I put that little hat on top of it. It's a sampl... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
That's why I put that little hat on top of it. It's a sample proportion that's estimating our true proportion. Now I wanna construct a confidence interval, but before I go down the path, I need to actually set up my conditions for inference, make sure that I meet them, and we've done this many times. So the first condi... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
So the first condition for inference is the random condition. I need to feel good that this is truly a random sample from the population. The second one is often known as the normal condition, and that's the condition that, hey, in order to feel like the sampling distribution for the sample proportions is roughly norma... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
We've seen that multiple times before. And then the third one is the independence condition, and there's two ways to meet this. Either the individual observations in our sample should be done with replacement, or if it's not done with replacement, we can feel pretty confident about this if our sample size is no more th... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
But let's say that we meet these conditions for inference. What do we do? Well, we set up a confidence level, confidence level, for our confidence interval that we're about to construct. And let's say we said it was a 95% confidence level. That would mean that 95% of the time that we went through this exercise, the con... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
And let's say we said it was a 95% confidence level. That would mean that 95% of the time that we went through this exercise, the confidence interval that we get would actually overlap with the true population proportion. And 95% is actually a fairly typical one. But from that confidence level, you can calculate a crit... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
But from that confidence level, you can calculate a critical value. And the way that you do that, you just look up in a z-table, and once again, all of this is review. You would say, hey, how many standard deviations above and below the mean of a normal distribution would you need to go in order to get, say, 95%, that ... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
And now we're ready to calculate the confidence interval. Confidence interval, it is going to be equal to our sample proportion plus or minus our critical value, our critical value, times the standard deviation of the sampling distribution of the sample proportion. Now, there is a way to calculate this exactly if we kn... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
If we knew what p is, this would be the square root of p times one minus p over n. But if we knew what p is, then we wouldn't even have to do this business of constructing confidence intervals. So instead, we estimate this. We say, look, an estimate of the standard deviation of the sampling distribution, often known as... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
So p hat times one minus p hat, all of that over n. Now, the whole reason why I did this, this is covered in much more detail and much slower in other videos, is to see the parallels between this and a situation when we're constructing a two-sample confidence interval, or z-interval, for a difference between proportion... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
So this is the first population, and it has some true proportion of the folks that, let's say, are left-handed. And then there's another population. So let's call that p two. Maybe this is a freshman in your high school or college, and maybe this is sophomores. So two different populations. And you wanna see if there's... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
Maybe this is a freshman in your high school or college, and maybe this is sophomores. So two different populations. And you wanna see if there's a difference between the proportion that are left-handed, say. And so what you could do, just like we've done here, is for each of these populations, you will take a sample h... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
And so what you could do, just like we've done here, is for each of these populations, you will take a sample here, we'll call that n one, and then from that sample, you calculate a sample proportion, let's call that p one. And then from this second population, we do the same thing. This is n two. Notice, n one and n t... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
Notice, n one and n two do not have to be the same sample size. That's a common misconception when doing these things. These could be different sample sizes. And then from that sample, you calculate the sample proportion. Now, after you do that, you would wanna check your conditions for inference. And it turns out that... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
And then from that sample, you calculate the sample proportion. Now, after you do that, you would wanna check your conditions for inference. And it turns out that the conditions for inference would be exactly the same. Do both of these samples meet the random condition? Do both of these samples meet the normal conditio... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
Do both of these samples meet the random condition? Do both of these samples meet the normal condition? And do both of these samples meet the independence condition? And if both samples meet these conditions for inference, then we would have to calculate our critical value. And you would do it the exact same way. I'll ... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
And if both samples meet these conditions for inference, then we would have to calculate our critical value. And you would do it the exact same way. I'll just write it down again. So first, you need to check all of these. Then you would take your confidence level, confidence level, and from that, get a critical z. And ... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
So first, you need to check all of these. Then you would take your confidence level, confidence level, and from that, get a critical z. And then you're ready to say what your confidence interval's going to be. So your confidence, confidence interval, interval for p one minus p two. So it's the confidence interval for t... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
So your confidence, confidence interval, interval for p one minus p two. So it's the confidence interval for the difference between these true population proportions. That is going to be equal to the difference between your sample proportions. So p hat one minus p hat two, p hat two, plus or minus, plus or minus your c... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
So p hat one minus p hat two, p hat two, plus or minus, plus or minus your critical value right over here, times the standard deviation of the sampling distribution of the difference between the sample proportions. So it would be p hat one minus p hat two. And so we already know how to calculate this, how to calculate ... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
How do we calculate that? Well, I will just give you the formula first. But then we just have to appreciate that this just comes out of the properties of standard deviations and variances that we have studied in the past. So the standard deviation of the sampling distribution of the difference between the sample propor... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
So the standard deviation of the sampling distribution of the difference between the sample proportions, it is a mouthful. This is going to be approximately equal to, approximately equal to the square root of p hat one times one minus p hat one over n one, over n one, plus, plus p hat two times one minus p hat two over... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
And once again, how would you interpret that? Well, let's say your confidence level is 90%. And from that, you're able to construct this confidence interval. That would mean that 90% of the time that you go through this exercise, your confidence interval would overlap with the true difference between these population p... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
That would mean that 90% of the time that you go through this exercise, your confidence interval would overlap with the true difference between these population parameters, the true difference between these population proportions. Now, where did this thing come from? Well, you might notice some similarities here. This ... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
This part over here is an estimate, or it's approximately equal to the variance of the sampling distribution of the sample proportion for our first population. And then this right over here, once again, is approximately going to be equal to the variance of the sampling distribution for the sample proportions for this p... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
Well, look, if this is approximately the standard deviation, you square that, you approximately get the variance. And so the big takeaway is is that the variance for the sampling distribution of the difference is just the sum of the variances of each of those sampling distributions. That's a lot of big mouthful, I know... | Confidence intervals for the difference between two proportions AP Statistics Khan Academy.mp3 |
So here we have a scenario. I keep picking cards from a standard deck until I get a king. So this is a classic geometric random variable here, and it's important that in this parentheses, it says I replace the cards if they are not a king. And this is important, as we talk about in other videos, because the probability... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
And this is important, as we talk about in other videos, because the probability of success each time can't change. And so we could define some random variable, X, this is a geometric random variable, as being equal to the number of picks until we get a king, until we get a king, when we replace the cards if they are n... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
And remember, one of the conditions for a geometric random variable is that probability of success does not change on each trial. Well, the probability of success is going to be equal to, there's four kings in a standard deck of 52, and this is the same thing as one over 13. So this first question is, what is the proba... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
Well, this would be the probability that our geometric random variable, X, is equal to five. And you could actually figure this out by hand, but the whole point here is to think about how to use a calculator. And there is a function called geometpdf, which stands for geometric probability distribution function where wh... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
So five right over there. Now, just to be clear, if you're doing this on an AP exam, and this is one of the reasons why a calculator is useful, you actually can use this on an AP exam, AP statistics exam, it's important to tell the graders, if you're doing it on the free response, that this right over here is your P, a... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
All right, so I have my calculator now, and I just need to type in geometpdf and then those parameters. And so the place where I find that function, I press second, distribution right over here, it's this little blue above the VARS button, and then I can click up, I can scroll down, or I could just go to the bottom of ... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
And so then click Enter, click Enter again, and there you have it. It's about 0.056. So this is approximately, approximately 0.056. Now let's answer another question. So here they say, what is the probability that I need to pick less than 10 cards? So this is the probability that X is less than 10, or I could say this ... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
Now let's answer another question. So here they say, what is the probability that I need to pick less than 10 cards? So this is the probability that X is less than 10, or I could say this is equal to the probability that X is less than or equal to nine. And I could say, well, this is the probability that X is equal to ... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
And I could say, well, this is the probability that X is equal to one, plus the probability that X is equal to two, all the way to the probability that X is equal to nine, but that would take a while, even if I used this function right over here. But lucky for us, there's a cumulative distribution function. Take some s... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
This is going to be equal to geometpdf, cumulative distribution function. And once again, I pass the probability of success on any trial, and then up to and including nine. And so let's get the calculator out again. So we go to second, distribution. I click up, and there we have geomet cumulative distribution function.... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
So we go to second, distribution. I click up, and there we have geomet cumulative distribution function. Press Enter. One out of 13 chance of success on any trial, up to and including nine, and then Enter. And there you have it. It's approximately 51.3%, or 0.513. So this is approximately 0.513. | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
One out of 13 chance of success on any trial, up to and including nine, and then Enter. And there you have it. It's approximately 51.3%, or 0.513. So this is approximately 0.513. Now let's do one more. What is the probability that I need to pick more than 12 cards? And like, I'll pause the video and see if you can figu... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
So this is approximately 0.513. Now let's do one more. What is the probability that I need to pick more than 12 cards? And like, I'll pause the video and see if you can figure this one out. What function would I use on my calculator? How would I set it up? Well, the probability, this is the probability that X is going ... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
And like, I'll pause the video and see if you can figure this one out. What function would I use on my calculator? How would I set it up? Well, the probability, this is the probability that X is going to be greater than 12, which is equal to one minus the probability that X is less than or equal to 12. And now this, we... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
Well, the probability, this is the probability that X is going to be greater than 12, which is equal to one minus the probability that X is less than or equal to 12. And now this, we can just use the cumulative distribution function again. So this is one minus geomet CDF, cumulative distribution function, CDF of one ov... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
And so what is this going to be equal to? So second distribution, I click up, I get to the function, click Enter. And so I already have that first, the probability of success on every trial is one over 13, and then cumulative up to 12. And so I click Enter. And then, well, I could click Enter there, but I really wanna ... | TI-84 geometpdf and geometcdf functions Random variables AP Statistics Khan Academy.mp3 |
What we're going to do in this video is think about how to visualize distributions of data, then to analyze those visualizations, and we will eventually get to something known as a density curve. But let's start with a simple example, just to review some concepts. Let's say I go to 16 students, and I ask them to measur... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
And so this data point right over here tells us one student drank an average of 0.5 glasses of water per day. That person is probably very dehydrated. This person drank 8.1 glasses of water per day on average for the last 30 days. They are better hydrated. If we want to visualize that, we can set up a frequency histogr... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
They are better hydrated. If we want to visualize that, we can set up a frequency histogram, where we can create some categories. So this first category would be for data points that are greater than or equal to zero and less than one. And we can see that two data points fall into that category, and that's why the bar ... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
And we can see that two data points fall into that category, and that's why the bar right over here for that category is up to two. This category right over here is greater than or equal to three and less than four. Notice there are four data points in that category, and on this frequency histogram, the height of the b... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
So this is a nice way of looking at a distribution, but you might be more concerned with what percentage of my data falls into each of these categories. And that becomes especially interesting if we have many, many, many, many data points. And if we had 1,600,432,507 data points, well, just knowing the absolute number ... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
The percent that fits into each category is a lot more useful. And so for that, we could set up a relative frequency histogram. So notice, this is representing the same data. But in that first category, instead of the bar height being two, the bar height is now 12.5%. Why is that? Because two of the 16 data points fall... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
But in that first category, instead of the bar height being two, the bar height is now 12.5%. Why is that? Because two of the 16 data points fall into this category. 2 16ths is 1 8th, which is 12.5%. And this one right over here, notice, instead of the height being four, for four data points, it's now 25%. But these ar... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
2 16ths is 1 8th, which is 12.5%. And this one right over here, notice, instead of the height being four, for four data points, it's now 25%. But these are saying the same thing. Four out of the 16 data points fall into this category. 4 16ths is 1 4th, which is 25%. So both of these types of histograms are really usefu... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
Four out of the 16 data points fall into this category. 4 16ths is 1 4th, which is 25%. So both of these types of histograms are really useful, and you will see them used all of the time. But there are also cases where you have many, many, many more data points, and you want more granular categories. So what you could ... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
But there are also cases where you have many, many, many more data points, and you want more granular categories. So what you could do is, well, let's just make our categories a little more granular. So for example, instead of them being one glass of water wide, maybe you make them half a glass of water wide. So this f... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
So this first category could be greater than or equal to zero, and less than 0.5. And that will give you a clearer picture. And I'm now assuming a world where we have more than 16 data points. Maybe we have 16 million data points. This would be percentages on the left-hand side. But maybe that isn't good enough for you... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
Maybe we have 16 million data points. This would be percentages on the left-hand side. But maybe that isn't good enough for you. Maybe you wanna get even more granular. So you make everything, each category, a quarter of a glass. But maybe that doesn't satisfy you. You wanna get more and more and more granular. | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
Maybe you wanna get even more granular. So you make everything, each category, a quarter of a glass. But maybe that doesn't satisfy you. You wanna get more and more and more granular. Well, you could imagine where this is going. You could get to a point where you're approaching an infinite number of categories, and eac... | Density Curves Modeling data distributions AP Statistics Khan Academy.mp3 |
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