the total sum of squares for these nine data points

right here.

And these nine data points are grouped

into three different groups, or if we want to speak generally,

into m different groups.

What I want to do in this video is

to figure out how much of this total sum of squares

is due to variation within each group versus variation

between the actual groups.

So first, let's figure out the total variation

within the group.

So let's call that the sum of squares within.

So let's calculate the sum of squares within.

I'll do that in yellow.

Actually, I already used yellow, so let me do blue.

So the sum of squares within.

Let me make it clear.

That stands for within.

So we want to see how much of the variation

is due to how far each of these data points

are from their central tendency, from their respective mean.

So this is going to be equal to-- let's

start with these guys.

So instead of taking the distance between each data

point and the mean of means, I'm going

to find the distance between each data

point and that group's mean, because we

want to square the total sum of squares between each data

point and their respective mean.

So let's do that.

So it's 3 minus-- the mean here is 2-- squared, plus 2 minus 2

squared, plus 2 minus 2 squared, plus 1 minus 2 squared.

1 minus 2 squared plus-- I'm going

to do this for all of the groups,

but for each group, the distance between each data point

and its mean.

So plus 5 minus 4, plus 5 minus 4 squared,

plus 4 minus 4 squared-- sorry, the next point was

3-- plus 3 minus 4 squared, plus 4 minus 4 squared.

And then finally, we have the third group.

But we're finding that all of the sum of squares

from each point to its central tendency within that,

but we're going to add them all up.

And then we find the third group.

So we have 5 minus-- oh, its mean is 6-- 5 minus 6 squared,

plus 6 minus 6 squared, plus 7 minus 6 squared.

And what is this going to equal?

So this is going to be equal to-- up here,

it's going to be 1 plus 0 plus 1.

So that's going to be equal to 2 plus.

And then this is going to be equal to 1, 1 plus 1 plus 0--

so another 2-- plus this is going

to be equal to 1 plus 0 plus 1.

7 minus 6 is 1 squared is 1.

So plus.

So that's 2 over here.

So this is going to be equal to our sum of squares

within, I should say, is 6.

So one way to think about it-- our total variation was 30.

And based on this calculation, 6 of that 30

comes from a variation within these samples.

Now, the next thing I want to think about

is how many degrees of freedom do we have in this calculation?

How many independent data points do we actually have?

Well, for each of these-- so over here,

we have n data points in one.

In particular, n is 3 here.

But if you know n minus 1 of them,

you can always figure out the nth one

if you know the actual sample mean.

So in this case, for any of these groups,

if you know two of these data points,

you can always figure out the third.

If you know these two, you can always figure out the third

if you know the sample mean.

So in general, let's figure out the degrees of freedom here.

For each group, when you did this,

you had n minus 1 degrees of freedom.

Remember, n is the number of data points

you had in each group.

So you have n minus 1 degrees of freedom

for each of these groups.

So it's n minus 1, n minus 1, n minus 1.

Or let me put it this way-- you have n minus 1

for each of these groups, and there are m groups.

So there's m times n minus 1 degrees of freedom.

And in this case in particular, each group-- n minus 1 is 2.

Or in each case, you had 2 degrees of freedom,

and there's three groups of that.

So there are 6 degrees of freedom.

And in the future, we might do a more detailed discussion

of what degrees of freedom mean, and how

to mathematically think about it.

But the best-- the simplest way to think about it

is really, truly independent data

points, assuming you knew, in this case,

the central statistic that we used

to calculate the squared distance in each of them.

If you know them already, the third data point

could actually be calculated from the other two.

So we have 6 degrees of freedom over here.

Now, that was how much of the total variation

is due to variation within each sample.

Now let's think about how much of the variation

is due to variation between the samples.

And to do that, we're going to calculate.

Let me get a nice color here.

I think I've run out all the colors.

We'll call this sum of squares between.

The B stands for between.

So another way to think about it--

how much of this total variation is

due to the variation between the means,

between the central tendency-- that's

what we're going to calculate right now--

and how much is due to variation from each data

point to its mean?

So let's figure out how much is due to variation

between these guys over here.

Actually, let's think about just this first group.

For this first group, how much variation

for each of these guys is due to the variation between this mean

and the mean of means?

Well, so for this first guy up here--

I'll just write it all out explicitly--

the variation is going to be its sample mean.

So it's going to be 2 minus the mean of means squared.

And then for this guy, it's going

to be the same thing-- his sample mean,

2 minus the mean of mean squared.

Plus same thing for this guy, 2 minus the mean of mean squared.

Or another way to think about it--

this is equal to-- I'll write it over here--

this is equal to 3 times 2 minus 4 squared,

which is the same thing as 3.

This is equal to 3 times 4.

Three times 4 is equal to 12.

And then we could do it for each of them.

And actually, I want to find the total sum.

So let me just write it all out, actually.

I think that might be an easier thing

to do, because I want to find, for all of these guys combined,

the sum of squares due to the differences

between the samples.

So that's from the contribution from the first sample.

And then from the second sample, you have this guy over here.

Oh, sorry.

You don't want to calculate him.

For this data point, the amount of variation

due to the difference between the means

is going to be 4 minus 4 squared.

Same thing for this guy.

It's going to be 4 minus 4 squared.

And we're not taking it into consideration.

We're only taking its sample mean into consideration.

And then finally, plus 4 minus 4 squared.

We're taking this minus this squared

for each of these data points.

And then finally, we'll do that with the last group.

With the last group, sample mean is 6.

So it's going to be 6 minus 4 squared,

plus 6 minus 4 squared, plus 6 minus 4, plus 6 minus 4

squared.

Now, let's think about how many degrees of freedom

we had in this calculation right over here.

Well, in general, I guess the easiest way

to think about is, how much information did we

have, assuming that we knew the mean of means?

If we know the mean of means, how much

here is new information?

Well, if you know the mean of the mean,

and you know two of these sample means,

you can always figure out the third.

If you know this one and this one,

you can figure out that one.

And if you know that one and that one,

you can figure out that one.

And that's because this is the mean of these means over here.

So in general, if you have m groups, or if you have m means,

there are m minus 1 degrees of freedom here.

Let me write that.

But with that said, well, and in this case, m is 3.

So we could say there's two degrees of freedom

for this exact example.

Let's actually, let's calculate the sum of squares between.

So what is this going to be?

I'll just scroll down.

Running out of space.

This is going to be equal to-- this right here is 2 minus 4

is negative 2 squared is 4.

And then we have three 4's over here.

So it's 3 times 4, plus 3 times-- what is this?

3 times 0 plus-- what is this?

The difference between each of these-- 6 minus 4

is 2 squared is 4-- so that means we have 3 times

4, plus 3 times 4.

And we get 3 times 4 is 12, plus 0, plus 12 is equal to 24.

So the sum of squares, or we could

say, the variation due to what's the difference

between the groups, between the means, is 24.

Now, let's put it all together.

We said that the total variation,

that if you looked at all 9 data points, is 30.

Let me write that over here.

So the total sum of squares is equal to 30.

We figured out the sum of squares

between each data point and its central tendency,

its sample mean-- we figured out,

and when you totaled it all up, we got 6.

So the sum of squares within was equal to 6.

And in this case, it was 6 degrees of freedom.

Or if we wanted to write it generally,

there were m times n minus 1 degrees of freedom.

And actually, for the total, we figured out

we have m times n minus 1 degrees of freedom.

Actually, let me just write degrees of freedom

in this column right over here.

In this case, the number turned out to be 8.

And then just now we calculated the sum

of squares between the samples.

The sum of squares between the samples is equal to 24.

And we figured out that it had m minus 1 degrees of freedom,

which ended up being 2.

Now, the interesting thing here-- and this

is why this analysis of variance all fits nicely together,

and in future videos we'll think about how we can actually

test hypotheses using some of the tools

that we're thinking about right now--

is that the sum of squares within,

plus the sum of squares between, is

equal to the total sum of squares.

So a way to think about is that the total variation

in this data right here can be described

as the sum of the variation within each

of these groups, when you take that total,

plus the sum of the variation between the groups.

And even the degrees of freedom work out.

The sum of squares between had 2 degrees of freedom.

The sum of squares within each of the groups

had 6 degrees of freedom.

2 plus 6 is 8.

That's the total degrees of freedom

we had for all of the data combined.