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# Standard error of the mean

AP.STATS:
UNC‑3 (EU)
,
UNC‑3.Q (LO)
,
UNC‑3.Q.1 (EK)

## Video transcript

we've seen in the last several videos you start off with any crazy distribution and doesn't have to be crazy it could be a nice normal distribution but that to really make the point that you don't have to have a normal distribution I like to use crazy one so let's say you have some kind of crazy distribution that looks something like that it could look like anything so we've seen multiple times you take samples from this crazy distribution so let's say you were to take samples of let's say n is equal to n is equal to 10 so we take 10 instances of this random variable average them out and then plot our average we plot our average we get one instance there we keep doing that we do that again we take ten samples from this random variable average them plot them again you plot again and eventually you do this a gazillion times in theory infinite number of times and you're going to approve approach the sampling distribution of the sample mean and RN equal ten it's not going to be a perfect normal distribution but it's going to be close it would be perfect only if n was infinity but let's say we eventually all of our samples you know we keep we keep if we get a lot of averages that are there that stacks up there that stacks up there and eventually we'll approach something that looks something like that and we've seen from the last video that one if let's say we were to do it again and this time let's say that n is equal to 21 the distribution that we get is going to be more normal and maybe in future videos we'll delve even deeper into things like kurtosis and skewness even more obviously to us and we saw that in the experiment it's going to have a lower standard deviation so they're all gonna have the same mean let's say the mean here is you know I don't know let's say this the mean here is five then the mean here is also going to be five the mean of our sampling distribution of the sample mean is going to be five and it doesn't matter what our end is if I ran this twenty it still going to be five but our standard deviation is going to be less than either of these scenarios and we saw that just by experimenting it might look like this it's going to be normal but is going to have a tighter standard deviation so maybe it'll look like that and if we did it with a even larger sample size let me do that in a different color if we did that with an even larger sample size n is equal to 100 what we're going to get is something that fits the normal distribution even better we do you know we take 100 instances of this random variable average them plot it 100 instances of this random variable average of them plot and we just keep doing that if we keep doing that what we're going to have is something that's even more normal than either of these so it's going to be you know a much closer fit to a true normal distribution but even more obvious to the human eye it's going to be even tighter or so it's going to be a very low standard deviation it's going to look something like that I'll show you that in the on the on the simulation app in the neck 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 normal and your standard deviation is getting smaller so that the question might arise will is there a formula so if I know the standard deviation so this is my standard deviation of 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 and it's going to 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 right the variance is just a standard deviation squared if you don't remember that you might want to review those videos but if I know the variance of of my original distribution and if I know how many what my N is how many samples I'm going to 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 is there a way to predict what the mean of these distributions are and so this the sorry the standard deviation of these distributions and to make so you don't get confused between that and that and let me say the variance if you know the variance you can figure out the standard deviation that one is just the square root of the other so this is the variance of our original of our original distribution now to show that this is the variance of our sampling distribution of our sample mean will 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 Greek letter mu is your true mean this is a 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 going to be a true distribution this isn't an estimate this is there some you know if we if we magically knew this distribution there is some true variance here and of course the mean so this has a mean this right here we can just to 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 going to 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 very this variance given the variance of the original distribution and your N and it turns out there is and I'm not going to do a proof here I really want to give you the intuition of it 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 then if you took an N of 2 or an N of 5 you're you're just very unlikely to be far away right if you took 100 trials as opposed to taking 5 so I think you know that in some way it should be inversely proportional to n the larger your n the smaller standard deviation it actually turns out it's about as simple as possible it's one of those magical things about mathematics and I haven't I'll prove it to you one day I want to give you a working knowledge first I statistics I'm always struggling whether I should be formal and giving you rigorous proofs but I 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 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 are kind of all you need for right now using those simulations so that they're really true so it turns out that the variance of your sampling distribution of your sample mean is equal to the variance of your original distribution that guy right there divided by n that's all it is so if if this if this up here has a variance of let's say this has up here has a variance of 20 I'm just making that number up then the set and then let's say your n is 20 then the variance of your sampling distribution of your sample mean for an N of 20 well you're going to take that the variance up here your variance is twenty divided by your n 20 so here your variance is going to be 20 divided by 20 which is equal to one this is the variance of your original probability distribution and this is your n what's your standard deviation going to be what's going to be the square root of that right standard deviation is going to be 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 step of the step 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 going to write this down the standard error of the mean standard error of the mean all of these things that I just mentioned these all just mean the standard deviation of the sampling distribution of the sample mean that's why this is confusing because you use the word mean and sample it over and over again and if it confuses you let me know I'll do another video or pause and repeat it whatever but if we just take the square root of both sides the standard error 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 divided by the square root of n I just took the square root of both sides of this equation I personally I like to remember this that the variance is just inversely proportional to n and then I like to go back to this because this is very simple in my head you just take the variance divided by n oh and if I want the standard deviation I just take the square roots of both sides and I get this formula so here the standard deviation when n is 20 the standard deviation of the sampling distribution of the sample mean is going to be 1 here when n is 100 what 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 sample mean we could say is going to be equal to 20 this guy's variance divided by n so the equals and is 100 so it equals 1/5 now this guy's standard deviation of or the standard deviation of the of the sampling distribution of the sample mean or the standard error the mean is going to be the square root of that so one over the square root of five and you so you know this guy's a lot have you you know a little bit under one-half standard deviation about this guy had a standard deviation of one 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 let's see if we can prove it to ourselves using the simulation so I'll just for fun let me make an you know 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 it's easy to take the square root of because if you're looking at standard deviation so let's take if you take an N of 16 and an N of 25 and let's well I'll do it let's do 10,000 trials so in this case every one of the trials we're going to take 16 samples from here average them plot it here and then do a frequency plot here we're going to do 25 at a time and then average them I'll do it once animated just to remember so I'm taking 16 samples plot it there and 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 alright 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 say so in this random distribution I made my standard deviation was nine point three I'm going to remember these our standard deviation for the original thing was nine point three and so standard deviation here was two point three and the standard deviation here is one point eight seven let's see if it confer if it if it conforms to our formula so I'm gonna take this off screen for a second and I'm going to 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 my wacky distribution had a standard deviation of nine point three when n is equal to let me just in another color when n was equal to 16 just doing the experiment doing a bunch of trials and averaging and duelled thing we got the standard deviation of the sampling distribution of the sample mean the standard of the mean we experimental II determined it to be two point three three and then when n is equal to twenty five when n is equal to twenty-five we got the standard error of the mean being equal to one point eight seven 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 I will 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 sides and then you get standard error of the mean is equal to standard deviation of your original distribution divided by the square root of n so let's see if this works out for these two things so if I were to take nine point three so let me do this case so nine point three divided by the square root of sixteen right n is 16 so divided by the square root of 16 which is 4 what do I get so nine point three divided by four let me a little calculator out here let's see we have let me clear it out we wanted to divide nine point three divided by four nine point three divided by R square root of N and with sixteen so divided by four is equal to 2 point 3 - 2 point 3 - so this is equal to this is equal to two point three - which is pretty darn close to two point three three 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 of them now let's look at this here R we would take nine point three so let me draw a little line here maybe scroll over that's might be better so we take the dish 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 nine point three divided by the square root of n divided by square root of 25 right four was just the square root of 16 so this is equal to nine point three divided by five and see if it's 1.87 so let me get my calculator back so if I take nine point three divided by five what do I get one point eight six which is very close to one point eight seven so we got we got in this case one point eight six one point eight six so as you can see what we do got experimentally was almost exactly and this is after 10000 trials of what you would expect let's do another 10,000 so you got another 10,000 trials while we're still in the ballpark we're not going to maybe I can't hope to get the exact the exact number you know you know round it 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 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 by your sample size by the number of samples you take for when you for every basket that you averaged I guess is the best way to think about it and you know sometimes this can get confusing because you are taking samples of averages based on samples so when someone says sample size you're like is sample size the number of times I took averages or the number of things I'm taking averages of each time and you know it doesn't hurt it to clarify that normally when they talk about sample size they're talking about N and at least in my head when I think of the trials as you take a sample size of 16 you average it that's one trial and you plot it then you do it again and you do another trial and you do it over and over again but anyway hopefully this makes everything clear and then you now also understand how to get to the standard error of the mean