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In this video and
the next few videos,

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we're just really going to be
doing a bunch of calculations

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about this data set
right over here.

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And hopefully, just going
through those calculations

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will give you an
intuitive sense of what

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the analysis of
variance is all about.

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Now, the first thing I
want to do in this video

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is calculate the
total sum of squares.

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So I'll call that SST.

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SS-- sum of squares total.

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And you could view it
as really the numerator

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when you calculate variance.

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So you're just going to take
the distance between each

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of these data points and the
mean of all of these data

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points, square them,
and just take that sum.

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We're not going to divide by
the degree of freedom, which

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you would normally do
if you were calculating

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sample variance.

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Now, what is this going to be?

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Well, the first
thing we need to do,

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we have to figure out the mean
of all of this stuff over here.

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And I'm actually going to
call that the grand mean.

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And I'm going to
show you in a second

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that it's the same thing as
the mean of the means of each

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of these data sets.

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So let's calculate
the grand mean.

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So it's going to be 3 plus 2
plus 1 plus 5 plus 3 plus 4

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plus 5 plus 6 plus 7.

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And then we have
nine data points here

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so we'll divide by 9.

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And what is this
going to be equal to?

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3 plus 2 plus 1 is 6.

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6 plus-- let me just add.

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So these are 6.

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5 plus 3 plus 4 is 12.

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And then 5 plus 6 plus 7 is 18.

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And then 6 plus 12 is 18 plus
another 18 is 36, divided by 9

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is equal to 4.

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And let me show you that
that's the exact same thing

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as the mean of the means.

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So the mean of this
group 1 over here--

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let me do it in
that same green--

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the mean of group 1 over
here is 3 plus 2 plus 1.

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That's that 6 right over
here, divided by 3 data

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points so that
will be equal to 2.

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The mean of group 2,
the sum here is 12.

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We saw that right over here.

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5 plus 3 plus 4 is
12, divided by 3

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is 4 because we have
three data points.

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And then the mean
of group 3, 5 plus 6

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plus 7 is 18 divided by 3 is 6.

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So if you were to take the
mean of the means, which

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is another way of viewing this
grand mean, you have 2 plus 4

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plus 6, which is 12,
divided by 3 means here.

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And once again, you would get 4.

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So you could view
this as the mean

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of all of the data
in all of the groups

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or the mean of the means
of each of these groups.

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But either way, now that
we've calculated it,

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we can actually figure out
the total sum of squares.

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So let's do that.

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So it's going to be
equal to 3 minus 4--

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the 4 is this 4 right over
here-- squared plus 2 minus 4

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squared plus 1 minus 4 squared.

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Now, I'll do these guys
over here in purple.

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Plus 5 minus 4 squared plus 3
minus 4 squared plus 4 minus 4

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squared.

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Let me scroll over a little bit.

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Now, we only have three
left, plus 5 minus 4 squared

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plus 6 minus 4 squared
plus 7 minus 4 squared.

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And what does this give us?

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So up here, this is going
to be equal to 3 minus 4.

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Difference is 1.

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You square it.

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It's actually negative 1,
but you square it, you get 1,

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plus you get negative 2 squared
is 4, plus negative 3 squared.

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Negative 3 squared is 9.

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And then we have here
in the magenta 5 minus 4

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is 1 squared is still 1.

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3 minus 4 squared is 1.

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You square it again,
you still get 1.

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And then 4 minus 4 is just 0.

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So we could-- well, I'll
just write the 0 there just

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to show you that we
actually calculated that.

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And then we have these
last three data points.

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5 minus 4 squared.

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That's 1.

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6 minus 4 squared.

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That is 4, right?

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That's 2 squared.

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And then plus 7 minus
4 is 3 squared is 9.

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So what's this going
to be equal to?

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So I have 1 plus 4
plus 9 right over here.

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That's 5 plus 9.

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This right over
here is 14, right?

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5 plus-- yup, 14.

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And then we also have
another 14 right over here

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because we have a
1 plus 4 plus 9.

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So that right over
there is also 14.

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And then we have 2 over here.

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So it's going to be
28-- 14 times 2, 14

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plus 14 is 28-- plus 2 is 30.

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Is equal to 30.

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So our total sum of
squares-- and actually,

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if we wanted the
variance here, we

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would divide this by
the degrees of freedom.

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And we've learned multiple
times the degrees of freedom

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here so let's say
that we have-- so we

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know that we have
m groups over here.

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So let me just write
it as m and I'm not

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going to prove things
rigorously here,

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but I want to show
you where some

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of these strange formulas that
show up in statistics books

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actually come from without
proving it rigorously.

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More to give you the intuition.

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So we have m groups here.

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And each group
here has n members.

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So how many total
members do we have here?

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Well, we had m
times n or 9, right?

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3 times 3 total members.

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So our degrees of
freedom-- and remember,

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you have however
many data points

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you had minus 1
degrees of freedom

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because if you know
the mean of means,

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if you assume you knew
that, then only 9 minus 1,

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only eight of these are going
to give you new information

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because if you know that, you
could calculate the last one.

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Or it really doesn't
have to be the last one.

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If you have the other eight,
you could calculate this one.

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If you have eight of
them, you could always

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calculate the ninth one
using the mean of means.

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So one way to think
about it is that there's

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only eight independent
measurements here.

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Or if we want to
talk generally, there

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are m times n-- so that tells
us the total number of samples--

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minus 1 degrees of freedom.

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And if we were actually
calculating the variance here,

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we would just divide
30 by m times n minus 1

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or this is another way of
saying eight degrees of freedom

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for this exact example.

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We would take 30 divided
by 8 and we would actually

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have the variance for
this entire group,

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for the group of nine
when you combine them.

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I'll leave you
here in this video.

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In the next video, we're
going to try to figure out

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how much of this total
variance, how much of this total

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squared sum, total
variation comes

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from the variation within
each of these groups

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versus the variation
between the groups.

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And I think you get
a sense of where

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this whole analysis of
variance is coming from.

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It's the sense
that, look, there's

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a variance of this
entire sample of nine,

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but some of that variance--
if these groups are

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different in some way--
might come from the variation

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from being in different groups
versus the variation from being

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within a group.

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And we're going to
calculate those two things

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and we're going to
see that they're

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going to add up to the
total squared sum variation.