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ANOVA 2: Calculating SSW and SSB (total sum of squares within and between) | Khan Academy - YouTube
Channel: Khan Academy
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In the last video, we
were able to calculate
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the total sum of squares
for these nine data points
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right here.
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And these nine data
points are grouped
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into three different groups, or
if we want to speak generally,
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into m different groups.
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What I want to do
in this video is
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to figure out how much of
this total sum of squares
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is due to variation within
each group versus variation
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between the actual groups.
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So first, let's figure
out the total variation
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within the group.
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So let's call that the
sum of squares within.
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So let's calculate the
sum of squares within.
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I'll do that in yellow.
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Actually, I already used
yellow, so let me do blue.
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So the sum of squares within.
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Let me make it clear.
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That stands for within.
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So we want to see how
much of the variation
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is due to how far each
of these data points
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are from their central tendency,
from their respective mean.
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So this is going to
be equal to-- let's
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start with these guys.
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So instead of taking the
distance between each data
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point and the mean
of means, I'm going
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to find the distance
between each data
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point and that group's
mean, because we
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want to square the total sum
of squares between each data
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point and their respective mean.
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So let's do that.
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So it's 3 minus-- the mean here
is 2-- squared, plus 2 minus 2
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squared, plus 2 minus 2
squared, plus 1 minus 2 squared.
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1 minus 2 squared
plus-- I'm going
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to do this for
all of the groups,
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but for each group, the
distance between each data point
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and its mean.
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So plus 5 minus 4,
plus 5 minus 4 squared,
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plus 4 minus 4 squared--
sorry, the next point was
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3-- plus 3 minus 4 squared,
plus 4 minus 4 squared.
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And then finally, we
have the third group.
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But we're finding that
all of the sum of squares
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from each point to its
central tendency within that,
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but we're going to
add them all up.
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And then we find
the third group.
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So we have 5 minus-- oh, its
mean is 6-- 5 minus 6 squared,
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plus 6 minus 6 squared,
plus 7 minus 6 squared.
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And what is this going to equal?
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So this is going to
be equal to-- up here,
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it's going to be
1 plus 0 plus 1.
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So that's going to
be equal to 2 plus.
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And then this is going to be
equal to 1, 1 plus 1 plus 0--
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so another 2--
plus this is going
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to be equal to 1 plus 0 plus 1.
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7 minus 6 is 1 squared is 1.
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So plus.
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So that's 2 over here.
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So this is going to be
equal to our sum of squares
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within, I should say, is 6.
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So one way to think about it--
our total variation was 30.
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And based on this
calculation, 6 of that 30
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comes from a variation
within these samples.
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Now, the next thing
I want to think about
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is how many degrees of freedom
do we have in this calculation?
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How many independent data
points do we actually have?
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Well, for each of
these-- so over here,
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we have n data points in one.
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In particular, n is 3 here.
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But if you know n
minus 1 of them,
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you can always figure
out the nth one
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if you know the
actual sample mean.
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So in this case, for
any of these groups,
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if you know two of
these data points,
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you can always
figure out the third.
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If you know these two, you can
always figure out the third
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if you know the sample mean.
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So in general, let's figure out
the degrees of freedom here.
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For each group,
when you did this,
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you had n minus 1
degrees of freedom.
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Remember, n is the
number of data points
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you had in each group.
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So you have n minus
1 degrees of freedom
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for each of these groups.
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So it's n minus 1, n
minus 1, n minus 1.
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Or let me put it this
way-- you have n minus 1
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for each of these groups,
and there are m groups.
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So there's m times n minus
1 degrees of freedom.
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And in this case in particular,
each group-- n minus 1 is 2.
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Or in each case, you had
2 degrees of freedom,
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and there's three
groups of that.
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So there are 6
degrees of freedom.
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And in the future, we might
do a more detailed discussion
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of what degrees of
freedom mean, and how
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to mathematically
think about it.
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But the best-- the simplest
way to think about it
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is really, truly
independent data
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points, assuming you
knew, in this case,
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the central statistic
that we used
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to calculate the squared
distance in each of them.
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If you know them already,
the third data point
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could actually be calculated
from the other two.
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So we have 6 degrees
of freedom over here.
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Now, that was how much
of the total variation
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is due to variation
within each sample.
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Now let's think about
how much of the variation
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is due to variation
between the samples.
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And to do that, we're
going to calculate.
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Let me get a nice color here.
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I think I've run
out all the colors.
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We'll call this sum
of squares between.
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The B stands for between.
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So another way to
think about it--
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how much of this
total variation is
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due to the variation
between the means,
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between the central
tendency-- that's
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what we're going to
calculate right now--
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and how much is due to
variation from each data
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point to its mean?
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So let's figure out how
much is due to variation
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between these guys over here.
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Actually, let's think about
just this first group.
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For this first group,
how much variation
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for each of these guys is due to
the variation between this mean
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and the mean of means?
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Well, so for this
first guy up here--
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I'll just write it
all out explicitly--
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the variation is going
to be its sample mean.
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So it's going to be 2 minus
the mean of means squared.
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And then for this
guy, it's going
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to be the same thing--
his sample mean,
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2 minus the mean
of mean squared.
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Plus same thing for this guy, 2
minus the mean of mean squared.
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Or another way to
think about it--
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this is equal to-- I'll
write it over here--
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this is equal to 3
times 2 minus 4 squared,
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which is the same thing as 3.
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This is equal to 3 times 4.
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Three times 4 is equal to 12.
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And then we could do
it for each of them.
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And actually, I want
to find the total sum.
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So let me just write
it all out, actually.
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I think that might
be an easier thing
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to do, because I want to find,
for all of these guys combined,
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the sum of squares
due to the differences
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between the samples.
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So that's from the contribution
from the first sample.
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And then from the second sample,
you have this guy over here.
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Oh, sorry.
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You don't want to calculate him.
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For this data point,
the amount of variation
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due to the difference
between the means
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is going to be 4
minus 4 squared.
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Same thing for this guy.
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It's going to be
4 minus 4 squared.
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And we're not taking
it into consideration.
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We're only taking its sample
mean into consideration.
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And then finally, plus
4 minus 4 squared.
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We're taking this
minus this squared
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for each of these data points.
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And then finally, we'll do
that with the last group.
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With the last group,
sample mean is 6.
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So it's going to be
6 minus 4 squared,
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plus 6 minus 4 squared, plus
6 minus 4, plus 6 minus 4
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squared.
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Now, let's think about how
many degrees of freedom
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we had in this calculation
right over here.
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Well, in general, I
guess the easiest way
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to think about is, how
much information did we
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have, assuming that we
knew the mean of means?
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If we know the mean
of means, how much
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here is new information?
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Well, if you know
the mean of the mean,
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and you know two of
these sample means,
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you can always
figure out the third.
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If you know this
one and this one,
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you can figure out that one.
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And if you know that
one and that one,
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you can figure out that one.
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And that's because this is the
mean of these means over here.
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So in general, if you have m
groups, or if you have m means,
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there are m minus 1
degrees of freedom here.
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Let me write that.
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But with that said, well,
and in this case, m is 3.
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So we could say there's
two degrees of freedom
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for this exact example.
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Let's actually, let's calculate
the sum of squares between.
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So what is this going to be?
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I'll just scroll down.
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Running out of space.
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This is going to be equal to--
this right here is 2 minus 4
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is negative 2 squared is 4.
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And then we have
three 4's over here.
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So it's 3 times 4, plus
3 times-- what is this?
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3 times 0 plus-- what is this?
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The difference between
each of these-- 6 minus 4
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is 2 squared is 4-- so
that means we have 3 times
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4, plus 3 times 4.
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And we get 3 times 4 is 12,
plus 0, plus 12 is equal to 24.
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So the sum of
squares, or we could
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say, the variation due
to what's the difference
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between the groups,
between the means, is 24.
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Now, let's put it all together.
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We said that the
total variation,
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that if you looked at
all 9 data points, is 30.
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Let me write that over here.
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So the total sum of
squares is equal to 30.
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We figured out
the sum of squares
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between each data point
and its central tendency,
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its sample mean--
we figured out,
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and when you totaled
it all up, we got 6.
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So the sum of squares
within was equal to 6.
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And in this case, it was
6 degrees of freedom.
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Or if we wanted to
write it generally,
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there were m times n minus
1 degrees of freedom.
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And actually, for the
total, we figured out
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we have m times n minus
1 degrees of freedom.
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Actually, let me just
write degrees of freedom
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in this column right over here.
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In this case, the number
turned out to be 8.
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And then just now we
calculated the sum
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of squares between the samples.
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The sum of squares between
the samples is equal to 24.
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And we figured out that it had
m minus 1 degrees of freedom,
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which ended up being 2.
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Now, the interesting
thing here-- and this
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is why this analysis of variance
all fits nicely together,
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and in future videos we'll
think about how we can actually
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test hypotheses using
some of the tools
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that we're thinking
about right now--
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is that the sum
of squares within,
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plus the sum of
squares between, is
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equal to the total
sum of squares.
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So a way to think about is
that the total variation
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in this data right
here can be described
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as the sum of the
variation within each
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of these groups, when
you take that total,
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plus the sum of the
variation between the groups.
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And even the degrees
of freedom work out.
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The sum of squares between
had 2 degrees of freedom.
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The sum of squares
within each of the groups
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had 6 degrees of freedom.
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2 plus 6 is 8.
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That's the total
degrees of freedom
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we had for all of
the data combined.
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It even works if you
look at the more general.
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So our sum of
squares between had
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m minus 1 degrees of freedom.
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Our sum of squares
within had m times n
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minus 1 degrees of freedom.
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So this is equal to m
minus 1, plus mn minus m.
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These guys cancel out.
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This is equal to mn minus
1 degrees of freedom, which
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is exactly the total
degrees of freedom we
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had for the total
sum of squares.
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So the whole point
of the calculations
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that we did in the last
video and in this video
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is just to appreciate that
this total variation over here,
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this total variation
that we first calculated,
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can be viewed as the sum
of these two component
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variations-- how much
variation is there
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within each of the samples plus
how much variation is there
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between the means
of the samples?
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Hopefully that's
not too confusing.
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