I realized I made a slight error in the last video,
when I talk about the population and the sample mean.
So I will rewrite the equations.
I realize I made a slight notational error,
and it might have confused you a little bit.
So just to review a little bit, it never hurts.
The mean of a population-- once again,
that's mu-- the mean of a population is equal to,
you take the sum of each of the data points.
So you take the sum-- that's that big sigma is
for-- of each of the data points.
So x sub i.
I had written x sub n, before.
And if you review that last video,
you can see why might be a little confusing.
And you start with the first data point.
So i is equal to 1.
You start with the first data point,
and you take the sum all the way to the n-th data point.
Right?
Where we have a big capital N, where
N is the total number of elements in the population.
And then you divide that by N.
So that's another way of writing x sub 1 plus x sub 2, plus--
and you just keep adding, bum, bum, bum,
however many there are-- x sub N.
And then, you divide that by N. And I think
that's what you're familiar with as just the arithmetic mean,
or the average.
You just add up all of the elements,
and you divide by the total number of elements there are.
That's just a fancy way of writing that.
And then, the sample mean is essentially the same thing,
although you use slightly different notation.
The sample mean is written as x with a line over it.
And that's equal to, once again, the sum
of the elements in the sample, right,
and then you have a slight notational difference.
You start at the first element in the sample.
And you go to the number of elements in the sample.
And that's why they use that lowercase n.
There are a big N elements in the whole population.
And if you took some subset of that-- we're
assuming that small n is less than or equal to big N.
And anyway, you divide that by the total number
of elements in the sample.
So once again, this would be x1 plus x2, plus dun, dun, dun,
plus x lower case n, divided by lowercase n.
These are essentially the same thing.
If your sample was the entire population, then
these ends would be equal to each other.
And these numbers would be equal to each other.
But just the notational difference,
if you ever see this, you know you're dealing with a sample.
Here you know you're dealing with the entire population.
And similarly, big N, entire population, small n,
the sample.
Fair enough.
I think we're now ready to learn a little bit about measures
of dispersion.
So the mean, and the mode, and the median,
that we covered in the first video in this playlist,
were all ways of measuring the central tendency of a data set.
Or kind of, picking a number that
is most representative of all of the numbers.
But we lose a lot of information.
We don't know whether all of the numbers in the data set
are close to that number, close to the mean.
Or maybe they're all really far away from the mean.
And that's why we want to come up with measures of dispersion.
And let me tell you, let me show you what I mean.
So let's say I have one set, and let's say, I don't know,
let's say it is a 2, a 2, and a 3, and a 3.
Let's say this is a population.
Let's just deal with population means and population
disbursements for now.
So what's the mean here?
The mean here is going to be 2 plus 2 plus 3 plus 3,
all of that over 4.
And what is that?
That's equal to 2 and 1/2, right?
4 plus 6, divided by 4, right.
That's equal to 2.5.
Fair enough.
Now what if we had this?
What if we had the numbers, I don't know, 0, 0, and 5, and 5.
So these are the numbers in the set.
I'll put commas, just so you know
these are separate numbers.
What's the mean here?
Well, the mean here-- let's say this is the population
that we're sampling, This isn't a sample,
this is the entire population.
And you'll see why I'm making that distinction later--
so it'll be 0 plus 0 plus 5 plus 5.
Well that's 10, divided by 4 is equal to 2.5.
So the arithmetic mean of both of these populations
are the same number.
They're both 2.5.
Right?
But you see that these sets are kind of, they are different.
Here, all of the numbers are pretty close to 2.5, right?
Well, here, sure their mean, their arithmetic mean, is 2.5,
but they're further away from 2.5.
Or the distances of each of these numbers,
each of the numbers in the set, their distance from the mean
is further.
So you can kind of view them that they're more dispersed,
they're further away from the mean.
Or another way you could think of it
is, the mean, although it does measure its central tendency,
it's not quite as indicative of all the numbers.
The numbers are much further away from the mean, on average.
So how do you measure that?
Well, you measure that with a variance.
And this is something I found, and even in my own,
it seems complicated when you first look at it.
And most statistics textbooks use fairly complex notations.
But the idea is almost as straightforward
as the arithmetic mean.
So what they'll do is, they'll write, you know, the variance.
And they'll write it as this letter sigma,
this Greek letter-- I wrote the top part too long.
Let me actually undo that.
I don't want you to spend the rest of your life writing
with a big top part-- they write it as sigma squared.
And we'll talk, in a few seconds,
about why it's written as-- you know, why don't they
just write v for variance?
Why do they write this weird letter squared?
I'll talk about it in a second.
But the variance of a population is defined--
and once again, these are just human-derived constructs
of trying to get our minds around data, being
able to describe a set of data, without having
to list all the numbers.
And then, being able to understand
what that data might represent, or what
can represent that data.
So what you do is, you take the sum,
and you start with all of the points in the population.
But instead of taking the sum of the points,
you take each point, x sub i, and you subtract from that--
and actually, it doesn't matter if you subtract from that,
or you subtract that from the mean,
the population mean, and then you square it.
So what is this?
This is the distance between each number and the mean.
And then, when you square it, it becomes a positive number.
So you can kind of view it as the squared absolute distance
between each number and the mean of that set.
And then you take the average of all of those.
You divide that by n.
Now that might seem like a very complicated notion,
but let's calculate it for these two data sets.
So here-- let me rewrite that first data set-- it's 2, 2, 3,
and 3.
So what is-- let me, actually, let me write it this way.
I think this will help explain it for you a little bit better.
So if I wrote i.
i1, i2, i3, i4.
That's i.
Then x sub i.
And you know, it's kind of arbitrary.
It's just, you know, this is saying
the first term, the second term, the third term.
I could have had this in any order.
It doesn't matter.
Maybe this was the first term, and this was the second,
and this was the third.
It doesn't matter.
Because we're just going to add them all up, and then
divide them.
So it doesn't matter what order we do it.
But anyway, x sub 1 is equal to 2, x sub 2 is equal to 2,
x sub 3 is equal to 3-- I'll stop
writing this equal thing-- x sub 4 is equal to 4.
Right?
What's the mean?
Well, we figured out the mean up here.
We just took these numbers and added them and divided by 4.
The mean is 2.5.
So what is x sub i minus the mean?
Right?
We're slowly building up to this equation.
What is x sub i minus the mean?
Well, 2 minus 2.5, that's minus 0.5.
2 minus 2.5, that's once again, minus 0.5.
3 minus 2.5, that's 0.5.
Fair enough.
Now this equation, they want us to square this.
So x sub i minus the mean squared.
There's several other properties we'll talk about later.
But the most important thing that the squaring does,
and the absolute value could have done it as well.
But the squaring makes all of these positive.
So minus 0.5 squared is positive 0.25.
This is positive 0.25.
Plus 0.5 squared is also positive 0.25.
And this is positive 0.25.
And so what is-- so if we wanted to know the sum from i
is equal to 1 to 4 of x sub i minus the mean, which
is 2.5 squared.
This is equal to, what, the sum of all of these numbers.
This is just saying sum all of these.
So sum all of these.
0.25, so that's equal to 1.
But this isn't the variance yet.
The variance is this thing-- let's look
at the original formula-- the variance is this thing, divided
by the total number of numbers you have.
So you take this, and then you-- so the variance
is equal to this thing, divided by the total number of numbers,
which is 4.
Which is equal to 0.25.
And you see, I mean here, the distance
from every number to the mean squared was 0.25.
So the average of all of these, which is essentially
what the variance is, the average was also 0.25.
And I'll do another example where these are different.
The other example in this video, actually, they're
not different.
But you see here, the average squared distance from the mean,
in that first data set, is 0.25.
And here, what's the average squared distance from the mean?
So let's see.
This is how far from the mean?
It is-- So let's say x-- let me write-- x sub i.
And then, x sub i minus the mean for this population.
So x sub i, there's a 0, there's a 0, there's a 5,
and a 5, right?
There's the first term x sub 1, this is x sub 2, and so forth.
And then, each of these numbers 0 minus 2, that's minus 2.5.
0 minus 2.5.
Right?
This could be 2.5, right?
That's the mean.
It's minus 2.5.
5 minus 2.5 is 2.5, 5 minus 2.5 is 2.5.
Now, if you took x sub i minus the mean squared, 2.5 squared
is what?
6.25 and it becomes positive.
So it's 6.25.
That's the same thing, 6.25.
That's already positive, so 6.25, 6.25.
And so, the variance is the sum of all
of these, divided by the total number of numbers there are,
right?
So we take the sum of all-- so this is the average of these,
and that's pretty easy to calculate.
If you added all of these up and divided by 4,
you're just going to get 6.25.
So the variance of this population is 6.25.
So there you have it.
You have two data sets where their means are the same.
But the variance of this data set is equal to,
we figured out, was 0.25.
While the variance of this data set is equal to 6.25.
And it's hard right now to have an intuition of what this--
you know, how does this 6 relate to the 0.25--
but you know that this is a larger number.
This is a much larger number than this is.
Which tells you, just kind of at a intuitive feel,
that the numbers in this that are, on average, much further
away from the mean than the numbers in this data set.
Anyway, I'm out of time.
I'll see you in the next video, and we'll
talk a little bit about this.
And we'll talk-- and we'll move into the standard deviation.
And then, what happens if you take
these of a sample instead of a population.
Everything we're doing here, we're taking the mean
and the variance of every number in the data set.
Later we'll do it for the sample.
See you soon.