https:/.../2020-02-21_psy317l_standard_error.mp4

Hello, in this video, I want to talk about
the standard error and this is really

extending our understanding of sampling
distributions and essential limit theorem.

So, let's talk about what
a standard error is.

First of all, we'll go back to
this penguin example and

you've seen this distribution before
as a uniform distribution of data.

It has, like any distribution, it has--
there's descriptive statistics.

So, it has a population mean.

The average is 5.04.

The average penguin is 5.04 meters
from the edge of the ice sheet.

You can calculate a standard
deviation for this.

So, the deviation is 2.88.

So, that's the, you know,
a measure of the spread.

And there was 5,000 penguins floating
on this ice sheet, that's the n,

the population size.

We then discussed about how if you were
just to sample either just randomly select

five penguins at a time or 50 penguins
at a time, that each of those samples

of, let's pick the n equals five for now,
each of those five penguins,

you could calculate how, what the average
distance from the front of the edge sheet

was for each of those individual penguins,
sample of five penguins and if you were

to do that over and over and over again
and in this histogram, we did it

1,000 times, we would be able to generate
what's called the sampling distribution.

And it's the sampling distribution of
the sample means, that's what it is

and I told you that we could calculate
from that what the average of

those sample means across
the 1,000 samples was and

that's this value and the notation that
we use for that is this mu and then

subscript x bar and that's the mean
of the sample means and I've forgotten

what it was, the exact value, but it's
pretty much going to approximate

very, very close.

So, I just put approximately equal to
5.05, just go back, it's 5.04.

So, it was-- it's going to approximate
the population average and you can

do that for any sample size.

So, that was sample size five,
let's look at the sample size 50.

Again, we have the mean of the sampling
distribution-- sorry, the mean of

the sample means and that is also going
to be very close to 5.04, it might be

a little bit closer because
our sample size is larger.

Two other things to notice about
these distributions, number one

they're normally distributed or approx--
sorry, the approximate to normal

distributions despite the fact for
the original distribution of penguins.

The population distribution was
a uniform distribution.

Second thing to notice, the sample size
doesn't really effect where the value

of the mean, of the sample means, it does
effect the standard deviation of

the sample means.

So, if this is a normal distribution,
or we believe it to approximate,

and then also this approximates
a normal distribution, then, it's clear

that the distance here, let's just assume
that's a standard deviation and I put it

in the right place.

This standard deviation, it's greater than
whatever the corresponding value is

over here, if that's also
the standard deviation.

So, as the sample size gets larger,
the spead of the sample means

gets smaller, so, we can say
the standard deviation gets smaller.

Now, does this standard deviation have any
relationship at all to the original

standard deviation of
the original population.

The original standard deviation was 2.88,
so, I'll just say population of

the original-- standard deviation was 2.88

Is there any relationship at all between
these two standard deviations?

Because it's not like the mean of
the sample means, which is pretty

much the same, regardless of the sample
size, I mean it does get better with

larger samples but it approximates,
it's close, especially if you have

enough of these samples.

What's the relationship of these standard
deviations because it's clear that when

you change n, this value is going to
change, so is there a relationship?

And it turns out that there is
a relationship and we're going to

look into that.

This graph here just shows you that
the normal distribution for becomes

better and better the larger
the sample size, so, it's a little

tricky to see but let me, I just want to
really point out one or two things here.

I'm going to pick a color
that represents that.

So, this value here, actually in red, so,
if I was just to pick one penguin at

a time, a sample size of one, this is
my estimate of the sample-- I'm going

for the red line here.

That's my estimate of the sample--
sorry, let's say that again.

That's the distribution of
the sample means.

It looks like the original population.

So, for a sample size of one, you don't
get a normal distribution of the sample

means, you get whatever
the original population was.

Let's look at two and I've got to find it
on here, so, it's the orange one and

I believe it's this one here.

It is this one here.

This is what it looks like.

This is the n is two.

So, again, not a really
normal distribution.

Now, let's skip to 50.

This is 50 here and you can see it
really, you don't need me to help

you too much.

This is the 50 value, it's very normal.

And then, we got blue at ten--
sorry, 25 here.

This is the 25 one and so on.

This is the ten.

This is the five.

I wanted to just show you this graph
because I wanted to show you that

even with very, very, very small
sample sizes of like five, we already

get very close to a normal distribution.

It's only with sample sizes of ridiculous
sample sizes of like one or two that

we don't do a very good job,

So, even with small sample sizes,
we get to the normal distribution

of the normal distribution of
the sample means.

So, back to the problem
I just posted a moment ago.

This is our original standard deviation
of a population, this is our population

and whenever we get a sample,
and again, this is just the sample

size of five.

This is the distribution of sample means.

The mean is going to approximate the mean
here but what is the relationship of

the standard deviation to
this original population.

What is the relationship?

It must be also related to the sample size
because it changes with its sample size.

And it's just a formula and we're not
going to talk too much about--

we're not going to talk much really at all
about how it's derived but this formula

here, very neatly, just tells us
about their relationship and

so, what we have here is this is
our standard deviation of

the sampling distribution of
the sample means.

So, we call that sigma subscript x bar,

sigma x bar.

The standard deviation, so just to really
reiterate what we're looking at, this is

the distribution of sample means,
this is-- we're looking for this value

what's this standard deviation?

And actually, technically, that's
the notation, what is that standard

deviation?

So, what we do is, we just take
the original population.

This is the population standard deviation
from the original population and we're

going to divide it by the square root of n
and that gives us that this value,

this standard deviation.

Its technical name is the standard
deviation of the sampling distribution

of the sample means, which is an awful
mouthful but we just call

it the standard error of

the mean, which is what we call it
the standard error of the mean.

So, this graph illustrates how
the standard error of the mean

changes by sample size.

So, if I just go back to-- maybe,
I'll just go back to this slide here

and we were asking the question of,
you know, what's this value over

sample size 50 compared to this
value of a sample size of five?

So, that was the question and I'm going to
plot-- maybe here I'll plot it or write it

sorry.

So, this is the formula, the standard
error of the mean or the standard

deviation of the sampling distribution
of the sample means is equal to

the original population standard deviation
divided by the square root of n.

So, when we had that sample size of five,
which is this one up here, what we're

really looking at is this, the original
standard deviation was 2.88 and

we're going to divide by the square root
of the sample size which is five, so that

equals 1.3.

So, the standard deviation here is 1.3 and
that standard error we call that is 1.3.

So, what this is saying is this value here
is 1.3 higher that was it, I forget.

I think it was 5.04 was the mean of
the sample means and so this value here

is going to be a 6.5-- nope, nope, not five.

It's going to be at 6.34.

This is one standard deviation above
the sample mean but if we have

a sample size of fifty, then
the calculation becomes this.

Becomes the original standard deviation
of the population divided by the square

root of 50, which is equal to and I've

written this down so I can check, 0.4.

So, back to this graph,
this value is 0.4,

and this value is 1.3.

And so, it gets smaller the bigger the
sample size.

This graph here that I got to previously
is actually showing us

how the standard error changes by
the sample size.

So we just had a sample size of 50,
which is approximately here.

If we go across to this value on this
axis, it tells us that's about 0.4,

sample size of 50, and if we had
a sample size of 5,

which is approximately here --
I'm doing a line, not very well,

but it goes to about there.
This was about 1.3.

And I just want you to -- there's nothing
really too much for you to take home

from this graph other than showing you
that as the sample size increases,

that the -- any population
standard deviation that we have,

the standard error is going to get
much smaller very rapidly.

A sample size of 5 is still quite high up
on this curve,

but once you come down to sample sizes
of 20 or 30 or more,

then we get a very, very small
standard error.

This is just to reiterate that point so
you can see what these are on this graph.

So let's put together what we've
just learned about the standard error

with what we have learned previously about
the Central Limit Theorem.

So what we have just been discussing is
that we just know that we have

an original population,
it could be any distribution,

here's our uniform distribution.

If we take many samples from it,
we get our sampling distribution.

In this case, of the sample means,
is normally distributed

or approximately normally distributed.

And we know that the sampling distribution
has a mean that is approximately equal to

the population mean and we've just learned
that we just know now that

the standard deviation of this
approximately normal distribution,

this is the standard error.

I'll write here, "standard error."

So we can actually write this in
notation form,

and we say that this sampling distribution
is approximately normal,

this is what this tilde squiggle means,
is approximately normal,

approximately normal and it has a mean
of the population mean,

so I'll just write here,
the mean is the population mean.

And the standard deviation of that
distribution,

and we're talking about this distribution
down here,

the standard deviation of that
distribution is the standard error,

that's what we call it.

And it's approximately equal to the
standard deviation of the

original population divided by the
square root of the sample size n.

So, this is a key thing that we know.
If we have at a population of any --

I'll just write "uniform" in here,
of any type, it could bimodal,

it could be uniform, it could be skewed,
we know that if we were to take

thousands and thousands of samples
or just one thousand -- or just a few,

hundred samples, the sample means that
we get from all those samples

are going to approximate
a normal distribution

if our sample size is larger,
it's going to approximate

a normal distribution even more.
And we can already determine what the

shape of that distribution is going to be
because we know that the population mean

is approximately equal to the mean
of the sample means,

and we know that the standard deviation,
this is the standard error,

we know that that, the standard error,
is the standard deviation of the

sampling distribution.

Okay, so we can work that out.

But the thing is, what you're probably
already thinking is,

"why do you care?" And you may not care,
and that's fine.

There's no reason to particularly.

But, it can be very, very helpful.
I'm just going to just float this idea

and we'll return to it in future videos.

Hopefully it's gone through your head
that why is this strange person

taking thousands of samples all the time?

You know, you're not going to go to this
penguin ice sheet and just keep

randomly picking 5 penguins at random
1,000 times.

Science and other types of time --
when we collect data,

it doesn't work like that.
We pretty much usually only just collect

one sample of data.

And so, when we collect one sample of data
and this here -- I've got

sampling distribution of n = 5 penguins.

This is when we did do it 1,000 times.

But let's just say that we did it one time
and we got a value around about here,

around about 7 meters,
that was our sample.

We just got one sample.

If we just got one sample,
we don't know anything really about that

in terms of how certain or how uncertain
are we that this truly is the sample mean.

We knew if we did this many, many times
the average of al the sample means

would converge on the true
population mean.

And that's our ultimate goal,
we're trying to est--

normally we don't know the population mean
we're trying to estimate it.

So in our one sample, we just got this
value of 7, say.

How confident are we that that is
the population mean?

And so, what we're able to do by having
this belief that we're able to know

that this value of 7 does come from
in theory,

a sampling distribution that exists.

And in theory, this sampling distribution
exists with a standard deviation

that we call the standard error.
We're able to understand how far

this value of 7, or any value that
we collected, it could be some other value

but our one sample was 7 meters,
we get a sense of how far away

from the mean that is in the units
of standard deviations

or technically,
with a sampling distribution,

standard errors.

So we're going to come back to this topic,
but really the value of the standard error

is that enables us to determine
when we collect one sample,

we're able to work out how far away
or how confident we are in our value,

is how far away is it from the
population mean,

how confident we are that this is a true
representation of the population mean.

We're going to come back to this
in future videos.