Standard Error of the Mean

Standard Error of the Mean (a.k.a. the standard deviation of the sampling distribution of the sample mean!)

We've seen in the last several
videos you start off with

any crazy distribution.

It doesn't have to be
crazy, it could be a nice

normal distribution.

But to really make the point
that you don't have to have

a normal distribution I
like to use crazy ones.

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 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 1 instance there.

We keep doing that.

We do that again.

We take 10 samples from this
random variable, average

them, plot them again.

You plot again and eventually
you do this a gazillion times--

in theory an infinite number of
times-- and you're going to

approach the sampling
distribution of the sample

mean. n equal 10 is not going
to be a perfect normal

distribution but it's
going to be close.

It'd be perfect only
if n was infinity.

But let's say we eventually--
all of our samples we get a lot

of averages that are there that
stacks up, that stacks up

there, and eventually will
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 20-- one, 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 skew.

But it's going to
be more normal.

But even more important here or
I guess even more obviously

to us, we saw that in the
experiment it's going to have

a lower standard deviation.

So they're all going to
have the same mean.

Let's say the mean here is,
I don't know, let's say

the mean here is 5.

Then the mean here is
also going to be 5.

The mean of our sampling
distribution of the sample

mean is going to be 5.

It doesn't matter
what our n is.

If our n is 20 it's
still going to be 5.

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 more normal
but it's going to have a

tighter standard deviation.

So maybe it'll look like that.

And if we did it with an 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 take a hundred instances
of this random variable,

average them, plot it.

A hundred instances of
this random variable,

average them, plot it.

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 a much
closer fit to a true

normal distribution.

But even more obvious to
the human, it's going

to be even tighter.

So it's going to be a very
low standard deviation.

It's going to look
something like that.

And I'll show you 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 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 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

going to change depending on
how many samples I'm taking

every time I do a sample mean--
if I know that my standard

deviation, or maybe if I
know my variance, right?

The variance to just the
standard deviation squared.

If you don't remember
that you might want to

review those videos.

But if I know the variance of
my original distribution and if

I know 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 to

predict what the mean of
these distributions are?

And so-- I'm sorry, the
standard deviation of

these distributions.

And so you don't get confused
between that and that,

let me say the variance.

If you know the variance
you can figure out the

standard deviation.

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.

The Greek letter Mu
is our true mean.

This is equal to the mean,
while an x a line over

it means sample mean.

So here what we're saying is
this is the variance of our

sample mean, that this is going
to be true distribution.

This isn't an estimate.

There's some-- you know, if we
magically knew 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 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 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 a hundred,
you're much more likely when

you average those out, to get
close to the true mean than if

you took an n of
2 or an n of 5.

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
a 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
working knowledge first.

In statistics, I'm always
struggling whether I should be

formal in 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

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 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 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 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 just going to take

that, the variance up here--
your variance is 20--

divided by your n, 20.

So here your variance is
going to be 20 divided by

20 which is equal to 1.

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 1.

Well that's also going to be 1.

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

standard-- the standard
deviation of the sampling

distribution of the sample mean
is 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.

All of these things that I just
mentioned, they 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 over and over again.

And if it confuses
you let me know.

I'll do another video or pause
and repeat or whatever.

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 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 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, divide it 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, 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 it equals-- n is
100-- so it equals 1/5.

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 going to be the

square root of that.

So 1 over the square root of 5.

And so this guy's will be a
little bit under 1/2 the

standard deviation while
this guy had a standard

deviation of 1.

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 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 we're looking at

standard deviations.

So we take an n of
16 and an n of 25.

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.

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?

All right, so here, 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 this stuff down.

Let's see if I can
remember it here.

So in this random distribution
I made my standard

deviation was 9.3.

I'm going to 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 conforms
to our formula.

So I'm going to 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 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, 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
we got the standard error of

the mean being equal to 1.87.

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-- 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 the standard

error of the mean is equal to
the 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 9.3--
so let me do this case.

So 9.3 divided by the
square root of 16, right?

N is 16.

So divided by the square
root of 16, which is

4, what do I get?

So 9.3 divided by 4.

Let me get a little
calculator out here.

Let's see.

We have-- let me clear it
out-- we want to divide

9.3 divided by 4.

9.3 three divided by our
square root of n. n was 16.

So divided by 4 is
equal to 2.32.

So 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.

Let me scroll over,
that might be better.

So we take our standard
deviation of our

original distribution.

So just that formula that we've
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 square root

of 25, right?

4 was just the
square root of 16.

So this is equal to
9.3 divided by 5.

And let's see if it's 1.87.

So let me get my
calculator back.

So if I take 9.3 divided
by 5, what do I get?

1.86 which is very
close to 1.87.

So we got in this case 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've got another
10,000 trials.

Well we're still
in the ballpark.

We're not going to-- maybe I
can't hope to get the exact

number 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 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
every basket that you average I

guess is the best way
to think about it.

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 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 the one

trial, and then 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.