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## AP®︎/College Statistics

### Course: AP®︎/College Statistics > Unit 9

Lesson 2: The central limit theorem# Sampling distribution of the sample mean (part 2)

More on the Central Limit Theorem and the Sampling Distribution of the Sample Mean. Created by Sal Khan.

## Want to join the conversation?

- Whats the point of the central limit Theorem if it doesn't provide you with the actual population distribution. For Ex in this video the population distribution in reality was totally different than the normal distribution. So what the importance of this concept?(12 votes)
- Each separate sample we take from the population will be different - they will have different scores and different sample means. So how do we tell which sample gives us the best description of the population? Can we even predict how well a sample describes the population it is drawn from?

By using the distribution of sample means we have the ability to predict the characteristics of the sample. And one of the basic reasons behind taking a sample is to use the sample data to answer questions about the larger population.

The Central Limit Theorem helps us to describe the distribution of sample means by identifying the basic characteristics of the samples - shape, central tendency and variability. So the distribution of sample means helps us to find the probability associated with each specific sample.

And because there's always some discrepancy or error between a sample statistic and the corresponding population statistic, the CLT enables us to calculate exactly how much error to expect.(17 votes)

- Could you please give a practical example of the utility of the Sampling Distribution of the Sample Mean (SDSM) when used with a NON NORMAL distribution? It would seem that a non-normal distribution generated by some process would mean that the process was "out of control", multiple processes going on, or some such. If that's the case, of what utility is the SDSM when it does not describe the scattered output of said process? Thanks much.(9 votes)
- Why is the mean of the sampling distribution of sample means always equal to the population mean?(5 votes)
- In formulas:
`E[X] = µ`

E[ xbar ] = E[ 1/n Σ xi ]

= 1/n E[ Σ xi ]

= 1/n Σ E[X]

= 1/n ( n * µ )

= µ

Logically, it makes sense this should be the case. If some variable has mean µ, that means we*expect*a given value to be µ. There'll be some variation around that, but that's what we expect, on average. So, we're*expecting*the average to be µ. Then, if we get a lot of such sample means (that is: the sampling distribution), we're getting a whole lot of values which we expect to all be µ. The average of a lot of things that are all µ or very close to it, should also be µ.(7 votes)

- Is there a difference between 1000 times taking samples of 10 (as at7:54) and 10 times taking samples of 1000?

What about taking 1 time taking a sample of 10000?

...Or 10000 times taking a sample of 1?(3 votes)- Yes. There is the difference. The bigger sample size you have - the narrow normal distribution you will get.

For example,

Sample size = 25, number of iteration = 5 || Sample size = 5, number of iteration = 25

mean*___________*13.74*__________*||*____________*14.32*____________*

median*__________*13.00*__________*||*____________*15.00*____________*

SD*______________*1.19*___________*||*____________*2.99*____________*(4 votes)

- @4:30actually, if you pulled a 9 and a 6, you would get 7 and a half.(3 votes)
- You're right! They should have one of those little correction pop-ups.(2 votes)

- Technically, Sal says that the bigger the sample data the normal"er" the distribution is, but if N=population then you would get a vertical line with a value of x=mean and y tends to infinity, right?(2 votes)
- Exactly correct! But we almost always assume that
**the size of the sample is much smaller than the population**. If you could really "sample" the ENTIRE POPULATION, then by definition, that's not a "sample", so of course all this theory sort of becomes nonsense. The whole point of this is to try to answer the question, "what can I say about the population if I only know the values for a sample of the population?"(4 votes)

- at around4:25, Sal says we are never going to get 7.5 when n=2. But if 6 and 9 are randomly selected, then 7.5 would be the average, so I'm not quite understanding his reasoning here?(3 votes)
- You're totally correct, Mike G! Your logic is solid; Sal made a mistake.(1 vote)

- What happens when you take the sampling distribution of the sample mean of the sampling distribution of the sample mean of a set of observations? And what happens when you repeat that again and again on the same set? And how would the results relate to one another?(2 votes)
- Let's say that X is my data, xbar1 is the sample mean, and xbar2 is the sample mean of a collection of sample means (basically, a second-level sample mean)

For sake of argument, let's assume that X is Normally distributed with mean μ and SD σ/√n. Then for a sample of size n, xbar1 is Normally distributed with mean μ and SD σ/√n. If we thought of xbar1 as our random variable, and took samples of size m, we'd apply the same logic, and get that xbar2 is Normally distributed with mean μ and SD σ /√(nm). And so on.

However, I'm not sure if you're quite thinking along the right lines. The purpose of a sampling distribution is to understand how a statistic varies from sample to sample. Generally speaking, we will have a sample of data, and so only the "first level" sampling distribution would be relevant. While it's possible to think of a "second level" sampling distribution (as you have: the sampling distribution of the sample mean of the sampling distribution), by and large we just won't have any use for it.(2 votes)

- why SD_p = SD_x / n(1/2)? As Sal said before in scaling random variable video, if x_new = X/n, then SD(x_new) = SD(X)/n. But here, why take square root?(2 votes)
- Is there any video or topic explicitly on skewness and Kurtosis?(2 votes)

## Video transcript

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 can just start off with any crazy
distribution, maybe it looks something like this. I'll do a discrete
distribution. Really to model anything
at some point you have to make it discreet. It could be a very granular
discrete distribution. But let's say it's 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 4. So if you took 4 numbers from
this distribution, 4 random numbers where let's say this
is the probably of a 1, 2, 3, 4, 5, 6, 7, 8, 9. If you took 4 numbers at a time
and averaged them-- let me do that here-- if you took 4
numbers at a time, let's say we used this distribution to
generate 4 random numbers, right? We're very likely to get a 9. We're definitely not going
to get any 7's or 8's. We're definitely not
going to get a 4. We might get a 1 or 2. 3 is also very likely. Five is very likely. So we use this function
to essentially generate random numbers for us. And we take samples of 4 and
then we average them up. So let's say our first average
is, I don't know, let's say 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 4, is going to be 6, right? 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'd do it again,
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. 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 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 are
some average values. It won't be a perfect normal
distribution because you can never get anything less than
a 0, or anything less than a 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. 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 from this population, you could say, or from this
random variable, average them, and then plotted them here. And in the last video
we ran the simulation. I'm going to go back to
that simulation a second. We saw a couple of things. And I'll show it to you
at 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 1 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 the 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-- you're
going to have a couple of 1's, you're going to
have a couple of 2's. You're going to have
more 3's like that. You're going to have no 4's. You're going to have
a bunch of 5's. You're going to have some
6's that'll look like that. And you're going to
have a bunch of 9's. 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 said you do a bunch of trials, it'll look like a
normal distribution, it definitely doesn't
work for n equal 1. As n gets larger though
it starts to make sense. That let's see if we've 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's still would be difficult for it to become an
exact normal distribution. But then you can get more
instance-- that you could get more-- you know, you might get
things from all of the above. But you can only get two
in each of your baskets that your averaging. You're only going to
get 2 numbers, right? So? You're never going to for
example, you're never going to get 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 it, maybe it looks like this. But there will be a gap at 7.5
because that's impossible and maybe it looks
something like that. So it's 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 because I really wanted you to
get the gut sense 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. But in kind of every day
practice, you don't have to get that much beyond n equals two. 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. Now the other thing is
you obviously wants 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 size of 4. And 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 one. So that's our sample size. So as that approaches infinity
your actual sampling distribution of the sample of
the sample mean will approach a normal distribution. Now in order to actually see
that normal distribution and actually to prove it to
yourself, you would have to do this many, many-- remember the
normal distribution happens, this is essentially the
population or this is the random variable. That tells you all of
the possibilities. In real life, we seldom know
all the possibilities. In fact in real life, we seldom
know the pure probability generating function. Only if we're writing
it or if we're writing a computer program. Normally we're doing
samples and we're trying estimate things. So normally there's some random
variable and then maybe we'll do a bunch of-- we'd take it a
bunch of samples, we'd take their means and we'd plot them
and 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 it's
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. It has a pure mean. So the mean of the sampling
distribution of the sample mean, we'll write it like that. Notice I didn't write it is
just the x with-- what this is, this is actually saying that
this is a real population mean, this is a real random
variable mean. If you look 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 see sample size. Let's see were dealing
with the world where a 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 1,000--
if you did the trial 1,000 times-- so 1,000 times you took
10 samples from some distribution and took 1,000
averages and then plotted them, you're going
to get pretty close. Now the next thing I want to
touch on is what happens as n-- 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 to 5, our graph after we try-- we take samples
of 5, average them and we do it 10,000 times-- our graph
look 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 going to 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-- or 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 want to show-- or this
distribution is as good as any-- the first thing I want to
show you is that n of 2 is really not that good. So let's compare an n of 2
to let's say an n of 16. So when you compare an
n of 2 to an n of 16, let's do it once. So you get 1, 2 trials,
you average them. And then it's going to do 16
and then it's going to plot it down here and average there. Let's do that 10,000 times. So notice, when you took an n
of 2, 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
tales 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 and we did
it 10,001 times-- 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 into 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 then our original distribution. Now let me do it with 2-- 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. So if we-- I'll do one trial
animated just because it's always nice to see. So first it's going to do 16
of these trials and average them and there we go. And then it's going to 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 16. It has slightly 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 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 into 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 1,000,000 samples from this distribution
every time that sample mean is always going to be pretty
darn close to my mean. If I take 1,000,000 samples of
everything, if I essentially try to estimate a mean by
taking 1,000,000 samples, I'm going to get a pretty good
estimate of that mean. The probability that a
million numbers are all out here is very low. So if n is 1,000,000 of course
all of my sample means when I average them are all going to
be really tightly focused around the mean itself. 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 is really the case. And it actually turns out that
there's a very clean formula that relates to 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. And I'll go over that
in the next video.