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# Sampling distribution of the sample mean

AP.STATS:
UNC‑3 (EU)
,
UNC‑3.H (LO)
,
UNC‑3.H.2 (EK)
,
UNC‑3.H.3 (EK)
,
UNC‑3.H.5 (EK)

## Video transcript

in the last video we learned about what is quite possibly the most profound idea in statistics and that's the central limit theorem and the reason why it's so neat is we can start with any distribution that has a well-defined mean and variance actually I made this I wrote the standard deviation in the last video that should be the mean and let's say it has some variance I could write it like that or I could write the standard deviation there but as long as it has a well-defined mean and standard deviation I don't care what the distribution looks like what I can do is take samples in the last video of save size 4 so in that I take that means I take literally 4 instances of this random variable this is one example I take their mean and I this I consider this the sample mean for my first trial or you could almost say for my first sample I know it's very confusing because you can consider that a sample the set to be a sample or you could consider each of its members of the each member of the set as a sample so that can be a little bit confusing there but I have this first sample mean and then I keep doing that over and over my in my second sample my sample size is 4 I got 4 instances of this random variable I averaged them I have another sample mean and the cool thing about the central limit theorem is as I keep plotting the frequency distribution of my sample means it starts to approach something that approximates the normal distribution and it's going to do a better job of approximating that normal distribution as n gets larger and just so we have a little terminology on our belt this frequency distribution right here that I've plotted out and even or here or up here that I started parting plotting out that is called it's kind of confusing because we use the word sample so much that is called the sampling sampling distribution distribution of the sample mean and let's dissect this a little bit just so that this long description of this distribution starts to make a little bit of sense when we say it's the sampling distribution that's telling us that it's being derived from its the distribution of some statistic which in this case happens to be the sample mean and we're deriving it from samples of an original distribution so each of these so this is my first sample my sample size is four I'm using the statistic the mean actually could have done with other things I could have done the mode or the or the range or other statistics the but the sampling distribution of the sample mean of the sample mean is the most common one it's probably in my mind the best place to start learning about the central limit theorem and and even frankly sampling distribution so that's what it's called and this is a little bit of background I'll prove this to you experimentally not mathematically but I think the experimental is on some levels more satisfying with statistics that this will have the same mean this will have the same mean as your original distribution as your original distribution right here so it has the same mean but we'll see in the next video that this is actually going to be it's going to start approximating a normal distribution even though my original distribution that this is kind of generated from is completely non normal so let's do that with this with this app right here and just to give proper credit where credit is due this is I think was developed at Rice University this is from online stat book stat book.com this is their app and which i think is a really neat app because it really helps you to visualize what a sampling distribution of the sample mean is so I can literally create my own custom distribution here so let me make something kind of crazy so that you could do this in theory with a with a discrete or a continuous probability density function but what they have here is we can take on one of 32 values and I'm just going to set the different probabilities of getting any of those 32 values so clearly this right here is not a normal distribution it looks a little bit bimodal but it doesn't have long tails but what I want to do is first just use the simulation to understand or to better understand what the sampling distribution is all about so what I'm going to do is I'm going to take let's start with five at a time so my sample size is going to be five and so I when I click animated what it's going to do is going to take five samples from this from this probability distribution function it's going to take five samples and you're gonna see them when I click animate it it's going to average them and plot the average down here and then I'm going to click it again it's going to do it again so there you go I've got five samples from there average them and it hit there what did I just do I clicked oh I wanted to clear that let me let me make this bottom one none so let me do that it over again so I'm gonna take five at a time so I took five samples from up here and then it took its mean and plotted the mean there let me do it again five samples from this probability distribution function plotted it right there I could keep doing it it'll take some time but you can see I plotted it right there now I could do this a thousand times it's going to take forever let's say I just wanted to do it a thousand times so it's this program just to be clear it's actually generating the random numbers this isn't like a rigged program it's actually going to generate the random numbers according to this probability distribution function it's going to take five at a time find their means and plot the mean so if I click 10,000 it's going to do that ten thousand times so it's going to take five numbers from here 10,000 times and find their means 10,000 times and then plot the 10,000 means here so let's do that so there you go notice it's already looking a lot like a normal distribution and like I said the original mean of my crazy distribution here was fourteen point four five and the mean of after doing 10,000 samples or 10,000 trials my mean here's fourteen point four two so I'm already getting pretty close to the mean they're my standard deviation you might notice it's less than that we'll talk about that in a future video and the skew and kurtosis these are idea these these are are things that help us measure how normal a distribution is and I've talked a little bit about it in the past and let me let me actually just diverge a little bit and and just so it's interesting and they're fairly straightforward concepts skew literally tells so if this is let me do it in a different color if this is a Norfolk normal distribution and clearly my drawing is very far from perfect if that's a perfect distribution if this would have a skew of zero if you have a positive skew that means you have a larger right tail than you would have otherwise expect so something with a positive skew might look like this it would have a large tail to the right so this would be a positive skew positive skew which makes it a little less than ideal for a normal distribution and a negative skew would look like this has a long tail to the left so negative skew might look like that so that is a negative skew if you have trouble remembering it just remember which direction the tail is going this tail is going through is the negative direction this tail is going to the positive direction so if something has no skew that means that it's nice and symmetrical around its mean now kurtosis which sounds like a very fancy word is similarly not that not that fancy of an idea kurtosis so once again if I were to draw a perfect normal distribution remember you know there's no one normal distribution you could have different means and different standard deviations let's say that's the perfect normal distribution if I have positive kurtosis if I have positive rotation kurtosis what's going to happen is I'm going to have fatter tails let me draw it a little nicer than that I'm going to have fatter tails but I'm going to have a more pointy peak I didn't you have to draw at that point to let me draw it like this I'm going to have fatter tails and I'm going to have a more pointy peak than a normal distribution so this right here is positive kurtosis so something that has positive kurtosis isn't it's you know it depending on how positive it is it tells you it's a little bit more pointy than a real normal distribution positive kurtosis and negative kurtosis has smaller tails but it's smoother near the middle so it's like this so something like this would have negative kurtosis negative kurtosis and maybe in future videos we'll explore that in more detail but in the context of this simulation it's just telling us how normal this distribution is so when our sample size was n equal five and we did 10,000 trials we got it pretty close to a normal distribution let's do another 10,000 trials just to see what happens it looks even more like a normal distribution our mean is now the exact same number but we still have a little bit of skew and a little bit of kurtosis now let's see what happens if we were to do the same thing with a larger sample size and we can actually do them simultaneously so here is n equal 5 let's do here N equals 25 let's go let me clear them I'm going to do the sample sampling distribution of the sample mean and so I'm going to run 10,000 trial so I'll do 1 animated trial just so you remember what's going on so I'm literally taking first 5 samples from over here find their mean now I'm taking 25 samples from up here find its mean and then plotting it down here so here the sample size is 25 here it's 5 I'll do it one more time I take 5 get the mean plot it take 25 get the mean and then plot it down there this is a larger sample size now that that thing that I just did I'm going to do 10,000 times 10,000 times and that's interesting remember our first distribution was just this really crazy very non normal distribution but once we did it whoops I didn't want to make it that big but once we scroll up a little bit so here what's interesting I mean they both look a little normal but if you look at the skew and the kurtosis when our sample size is larger it's more normal this has a lower skew then when our sample size was only 5 and it has a less negative kurtosis then when our sample size was 5 so this is this is a more normal distribution and one thing that we're going to explore further in a future video is not only is it more normal in its shape but it's also tighter fit around the mean and you can even think about it why that kind of makes sense when your sample size is larger your odds of getting really far away from the mean is lower because it's very low likelihood if you're taking 25 samples or 100 samples that you're going to get a bunch of stuff way out here a bunch of stuff way out here you're very likely to get a reasonable spread of things so it makes sense that your mean your sample mean is less likely to be far away from the mean we're going to talk a little bit more about that in the future but hopefully this kind of satisfies you that at least experimentally I haven't proven it to you with mathematical rigor which hopefully we'll do in the future but hopefully this satisfies you at least experimentally that the central limit theorem really does apply to any distribution I mean this is a crazy distribution I encourage you to to use this applet that online stat book comm and experiment with other crazy distributions to believe it for yourself but the interesting things there are is that we're approaching a normal distribution but as my sample size got larger it's a better fit for a normal distribution
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