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.