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.