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Calculating R-squared | Regression | Probability and Statistics | Khan Academy - YouTube
Channel: Khan Academy
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In the last video, we were able
to find the equation for
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the regression line for these
four data points.
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What I want to do in this video
is figure out the r
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squared for these data points.
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Figure out how good this
line fits the data.
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Or even better, figure out the
percentage-- which is really
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the same thing-- of the
variation of these data
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points, especially the variation
in y, that is due
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to, or that can be explained
by variation in x.
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And to do that, I'm actually
going to get
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a spreadsheet out.
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I've actually tried to do this
with a calculator and it's
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much harder.
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So hopefully this doesn't
confuse you too much to use a
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spreadsheet.
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And I'm a make a couple
of columns here.
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And spreadsheets actually have
functions that'll do all of
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this automatically, but I really
want to do it so that
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you could do it by hand
if you had to.
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So I'm going to make a couple
of columns here.
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This is going to
be my x column.
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This is going to
be my y column.
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This is going to be the column--
I'll call this y
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star-- this'll be the y value
that our line predicts based
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on our x value.
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This is going to be the
error with the line.
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Let me caught it the squared
error with the line.
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I don't want us to take
up too much space.
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And then the next one, I'm
going to have the squared
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variation for that y value
from the mean y.
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And I think these columns by
themselves will be enough for
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us to do everything.
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So let's first put all
the data points in.
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So we had negative 2
comma negative 3.
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That was one data point.
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Negative 1 comma negative 1.
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And we had 1 comma 2.
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Then we have 4 comma 3.
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Now, what does our
line predict?
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Well our line says, you give
me an x value, I'm going to
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tell you what y value
I'll predict.
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So when x is equal to negative
2, the y value on the line is
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going to be the slope.
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So this is going to be equal
to 41 divided by
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42 times our x value.
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And I just selected that cell.
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And just a little bit of a
primer on spreadsheets, I'm
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selecting the cell D2.
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I was able to just move my
cursor over and select that.
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But that tells me the x value.
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Minus 5/21.
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Minus 5 divided by 21.
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Just like that.
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So just to be clear of what
we're even doing.
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This y star here, I
got negative 2.19.
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That tells us at this
point right over
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here is negative 2.19.
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So when we figure out the error,
we're going to figure
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out the distance between
negative 3, that's our y
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value, and negative 2.19.
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So let's do that.
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So the error is just going to
be equal to our y value.
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That's cell E2.
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Minus the value that our
line would predict.
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So just that value is
the actual error.
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But we want to square it.
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And then, the next thing
we want to do
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is the squared distance.
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so this is equal to the squared
distance of our y
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value from the y's mean.
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So what's the mean of the y's?
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Mean of the y's is 1/4.
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So minus 0.25, is the
same thing is 1/4.
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And we also want
to square that.
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Now, this is what's fun
about spreadsheets.
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I can apply those formulas
to every row now.
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And notice, what it did
when I did that.
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Now all of a sudden, this is the
y value that my line would
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predict, it's now using
this x value and
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sticking it over here.
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It's now figuring out the square
distance from the line
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using what the line would
predict and using the
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y value, this one.
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And then does the same
thing over here.
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It's figures out the squared
distance of this y
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value from the mean.
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So what is the total squared
error with the line?
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So let me just sum this up.
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The total squared error
with the line is 2.73.
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And then the total variation
from the mean, squared
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distances from the mean
of the y, are 22.75.
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So let me be very clear
what this is.
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So let me write these
numbers down.
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I'll write it up here so we
can keep looking at this
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actual graph.
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So are squared error versus our
line, our total squared
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error, we just computed
to be 2.74.
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I rounded a little bit.
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And what that is, is you take
each of these data points'
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vertical distance to the line.
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So this distance squared, plus
this distance squared, plus
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this distance squared, plus
this distance squared.
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That's all we just calculated
on Excel.
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And that total squared variation
to the line is 2.74.
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Or total squared error
with the line.
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And then the other number we
figured out was the total
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distance from the mean.
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So the mean here is
y is equal to 1/4.
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So that's going to be
right over here.
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This is 1/2.
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So right over here.
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So this is our mean y value.
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Or the central tendency
for our y values.
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And so what we calculated next
was the total error, the
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squared error, from the
means of our y values.
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That's what we calculated over
here in the spreadsheet.
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You see in the formula.
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It is this number, E2, minus
0.25, which is the mean of our
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y's squared.
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That's exactly what
we calculated.
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We calculated for each
of the y values.
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And then we summed
them all up.
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It's 22.75.
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It is equal to 22.75.
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So this is essentially
the error that the
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line does not explain.
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This is the total error,
this is the total
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variation of the numbers.
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So if you wanted to know the
percentage of the total
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variation that is not explained
by the line, you
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could take this number divided
by this number.
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So 2.74 over 22.75.
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This tells us the percentage
of total variation not
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explained by the line or
by the variation in x.
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And so what is this number
going to be?
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I can just use Excel for this.
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So I'm just going to divide this
number divided by this
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number right over there.
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I get 0.12.
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So this is equal to 0.12.
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Or another way to think about
it is 12% of the total
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variation is not explained
by the variation in x.
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The total squared distance
between each of the points or
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their kind of spread, their
variation, is not explain by
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the variation in x.
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So if you want the amount that
is explained by the variance
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in x, you just subtract
that from 1.
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So let me write it
right over here.
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So we have our r squared, which
is the percent of the
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total variation that is
explained by x, is going to be
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1 the minus that 0.12 that
we just calculated.
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Which is going to be 0.88.
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So our r squared here is 0.88.
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It's very, very close to 1.
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The highest number
it can be is 1.
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So what this tells us, or a way
to interpret this, is that
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88% of the total variation of
these y values is explained by
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the line or by the
variation in x.
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And you can see that it looks
like a pretty good fit.
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Each of these aren't too far.
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Each of these points are
definitely much closer to the
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line than they are
to the mean line.
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In fact, all of them are closer
to our actual line than
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to the mean.
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