Statistics: Sample vs. Population Mean

Before, at the end of the last
video, I actually said that

we'd talk about measures of
dispersion or how things are

distributed, but before I go
into that, I realize that I

have more to talk about,
especially the mean.

And before I do that, I want
to differentiate between a

sample and a population.

I touched on this a little
bit in the last video.

Let's say I wanted to
know-- I don't know.

Let's say I wanted to know
the average height of all

men in America, right?

So let me make the set
of all men in America.

So that's all men in America.

I know there's 300 million
people in the U.S., and half of

them maybe roughly are men,
so this would be 150

million men, right?

And it would be nearly
impossible, even if I was

intent on doing it, to actually
measure the average height

of every man in America.

Frankly, you know, every few
seconds, one of these men is

being born and one of
these men is dying.

So you know by the time I'm
done measuring everything,

someone would have died, and
some new men would have

been born, so it would
almost be impossible.

And if not impossible, it would
be very tiresome to measure the

average, or the mean, or the
median, or the mode of this

entire population, right?

So the best way I can get a
sense of this, because I'm

interested in what the average
of the population is, maybe I

can take the average
of a sample.

So what I could do is I can go
up to, you know-- and I'd try

to be pretty random about it.

I don't want to like-- you
know, hopefully, my sample

wouldn't be my college's
basketball team because that

would be a skewed sample, but
I'd try to find random people

and random situations where it
wouldn't be skewed

based on height.

And I'd maybe collect 10
heights, and I'd get, well,

maybe-- you know, the more
people I get the more

indicative it is, but if I got
10 heights, and those 10

heights were-- I don't know.

I'll do it in, you know, 5
feet, 6 feet, 5 and a half

feet, 5.75 feet, and, well,
let's say I only do 6,

or let's say in 6 and
a half feet, right?

Those are the five people that
I'd sample, and we could talk

more about what's a good way to
generate a random sample from a

population so it's not skewed
one way or the other.

But anyway, if I wanted to get
a sense of it and if I was kind

of lazy, so I only took
five measurements, this

is the way I would do it.

This would be a sample.

This would be a sample
of the population.

So instead of taking the mean--
let's say how I wanted to

calculate the average by
taking the arithmetic mean.

Instead of taking the
arithmetic mean of this entire

group of 150 million people, I
might just be happy taking the

mean of this sample, and
that'll be called

the sample mean.

And I want to introduce you to
some notation, even though it's

kind of-- so in statistics
speak, the mean, this mu, it's

a Greek letter, essentially the
Greek letter that later turns

into m, but mu is the
population mean, and this is

just a convention
population mean.

And x with a line over it, that
is equal to a sample mean.

And these are just notations
that people might see, and you

might have been confused
because sometimes you see

something-- people talk about
means, and you see this mu, and

sometimes you see this x with a
line over it, and it's nice

to know the distinction.

Here they're talking about
the mean of a sample of the

population, and here they're
talking about the mean of

the population as a whole.

Now, the way you calculate
them is essentially the same.

If you want to figure out the
population mean, you'd go to

all 150 million people at one
moment and add up all their

heights, and divide by 150
million to get the

population mean.

The sample mean, you just add
up the numbers in your sample

and divide by the number
of data points you have.

And the formulas I
want to show you.

I think you know how to
calculate averages.

It's a fairly straightforward
operation, and I want to show

you how it's often written in
statistics books, so that

you're not intimidated
when you see it.

The population mean, they'll
write it as-- so just to do,

you know, the convention.

Each member of a-- well, let
me do the sample first.

Each member of a sample, say
this is the first sample.

They'll call that x sub 1.

They'll call this x sub 2.

They'll call this one x
sub 3, x sub 4, and this

one x sub 5, right?

And this is just a way
of referring to each

of the samples.

So in a sample mean, they'll
say, do you know what you do?

You take the sum
of these numbers.

And you know how to do that,
but the fancy way of writing

it is to say, let's
write a capital Sigma.

That means the sum.

Sum of every x sub n, right?

Take the sum of each of
these numbers, right?

This is x sub 1, x sub 2, where
n goes from 1 to-- I mean, you

could say to the size
of the population.

You know, sometimes-- you know,
in this case it would be 5, or

sometimes they'd write a big--
they'd write an n like that.

And you'd divide it by the
number of members there are

in that population,
so divided by n.

You know, when you see this in
a book, you're like, wow, this

is advanced mathematics.

But essentially, they're saying
take the sum of all the data

points, just sum up these
numbers, and divide by the

number of numbers there are.

So this would just be 5
plus 5 plus 5.5 plus 5.75

plus 6.5 divided by 5.

That's all this is telling you.

For the population mean,
it's the same thing.

They just use a slightly
different notation.

They'll say that's equal to the
sum from n is equal to 1 to a

big N-- and I'll explain why
they write a big N-- of each

data point in the population,
not just the sample, all

that divided by big N.

And this is just a way of,
when they're at big N,

they mean 150 million.

They mean, you know, we want
you to get every data point

in the entire population.

So that's what they mean by--
and then divide by the number

of the entire population.

While the small n, they're kind
of-- it's just the convention,

the notation, that they say,
hey, we just want you to get

some smaller number, not
the entire population.

But the way you calculate
them is, you know, they're

essentially equivalent.

Anyway, I wanted to leave you
with that just because this is

something that if you don't get
it clarified early on-- it's a

fairly simple concept-- later
on, it becomes very confusing

when people want to
differentiate between the

population and the sample mean.

And you see these formulas
written slightly different.

Sometimes you'll see a mu, and
sometimes you'll see an x with

a line over it for
the sample mean.

Anyway, I'll see in
the next video.