- [Tutor] So we are interested in studying
the relationship between the amount
that folks study for a test and their score on a test,
where the score is between zero and six
and so what we're going to do is go look at the people
who took the test, we're going to plot for each person
the amount that they studied and their score,
so for example, this data point is someone
who studied an hour and they got a one on the test
and then we're going to fit a regression line
and this blue regression line is the actual regression line
for these four data points
and here is the equation for that regression line.
Now there's a couple of things to keep in mind,
normally when you're doing this type of analysis,
you would do it with far more than four data points,
the reason why I kept this to four
is because we are actually going to calculate
how good a fit this regression line is by hand
and typically you would not do it by hand,
we have computers for that.
Now the way that we're going to measure
how good a fit this regression line is
to the data has several names,
one name is the standard deviation of the residuals,
another name is the root mean square deviation,
sometimes abbreviated RMSD,
sometimes it's called root mean square error,
so what we're going to do is is for every point,
we're going to calculate the residual
and then we're going to square it
and then we're gonna add up
the sum of those squared residuals,
so we're gonna take the sum
of the residuals,
residuals squared
and then we're going to divide that
by the number of data points we have minus two
and we can talk in future videos
or a more advanced statistics class
of why you divide by two, but it's related to the idea
that what we're calculating here is a statistic
and we're trying to estimate a true parameter
as best as possible and n minus two
actually does the trick for us.
But to calculate the root mean square deviation,
we would then take a square root of this
and some of you might recognize
strong parallels between this and how we calculated
sample standard deviation early in our statistics career
and I encourage you to think about it.
But let's actually calculate it by hand,
as I mentioned earlier in this video,
to see how things actually play out.
So to do that, I'm going to ourselves a little table here,
so let's say that is our x value in that column,
let's make this our y value,
let's make this
y hat, which is going to be equal
to 2.5x minus two
and then let's make this the residual squared,
which is going to be our y value minus our y hat value,
our actual minus our estimate for that given x squared
and then we're going to sum them all up,
divide by n minus two and take the square root.
So first let's do this data point,
so that's the point 1,1,
1,1,
now what is the estimate from our regression line?
Well, for that x value, when x is equal to one,
it's gonna be 2.5 times one minus two,
so it's gonna be 2.5
times one minus two,
which is equal to 0.5
and so our residual squared is going to be one minus 0.5,
one minus 0.5 squared,
which is equal to, that's gonna be .5 squared,
which is gonna be 0.25.
Alright, let's do the next data point,
we have this one right over here,
it is 2,2,
now our estimate from the regression line
when x equals two is going to be equal
to 2.5 times our x value,
times two minus two,
which is going to be equal to three
and so our residual squared is going to be two minus three,
two minus three squared, which is negative one squared,
which is going to be equal to one,
then we can go to this point,
so that's the point 2,3, 2,3,
now our estimate from our regression line
is going to be 2.5 times our x value, times two
minus two, which is going to be equal to three
and so our residual here is going to be zero
and you can see that that point sits on the regression line,
so it's going to be three minus three,
three minus three squared, which is equal to zero
and then last but not least,
we have this point right over here, when x is three,
our y value, this person studied three hours
and they got a six on the test, so y is equal to six,
and so our estimate from the regression line,
you can see what you would have expected to get,
based on the regression line is 2.5
times our x value, times three minus two
is equal to 5.5
and so our residual squared is six minus 5.5 squared,
it is 5.5 squared,
so it's .5 squared, which is 0.25.
So now the next step,
let me take the sum of all of these squared residuals,
so this is, let me just write it this way, do it like this,
so the sum of the residuals,
residuals squared
is equal to,
if I just sum all of this up,
it's going to be 1.5,
1.5
and then if I divide that by n minus two,
so if I divide by n minus two,
that's going to be equal to, I have four data points,
so I'm gonna divide by four minus two,
so I'm gonna divide by two
and then I'm gonna wanna take the square root of that,
then I'm gonna take the square root of that
and so this is going to get us
1.5 over two is the same thing as three-fourths,
so it's the square root of three-fourths
or the square root of three over two
and you could use a calculator to figure what that is,
to figure out what that is as a decimal,
but this gives us a sense
of how good a fit this regression line is,
the closer this is to zero,
the better the fit of the regression line,
the further away from zero, the worst fit
and what would be the units
for the root mean square deviation?
Well, it would be in terms of
whatever your units are for your y axis,
in this case, it would be the score on the test
and that's one of the other values of this calculation,
of taking the square root of the sum
of the squares of the residuals dividing by n minus two.
So big picture, this square root of three over two
can be viewed as the approximate size
of a typical or average prediction error
between these points
and what the regression line would have predicted
or you could view it as the approximate size
of a typical or average residual.