- [Instructor] What we're going to do in this video
is calculate by hand
the correlation coefficient for a set of bi-variated data.
Now, when I say bi-variate
it's just a fancy way of saying for each X data point,
there's a corresponding Y data point.
Now, before I calculate the correlation coefficient,
let's just make sure we understand
some of these other statistics that they've given us.
So, we assume that these are samples
of the X and the corresponding Y
from our broader population.
And so, we have the sample mean for X
and the sample standard deviation for X.
The sample mean for X is quite straightforward
to calculate, it would just be one plus two
plus two plus three over four
and this is eight over four
which is indeed equal to two.
The sample standard deviation for X,
we've also seen this before,
this should be a little bit review,
it's gonna be the square root
of the distance from each of these points
to the sample mean squared.
So, one minus two squared
plus two minus two squared
plus three minus two squared,
all of that over, since we're talking about
sample standard deviation,
we have four data points,
so one less than four is all of that over three.
Now, this actually simplifies quite nicely
because this is zero, this is zero,
this is one, this is one
and so you essentially get the square root
of 2/3 which is if you approximate 0.816.
So, that's that.
And the same thing is true for Y.
The sample mean for Y,
if you just add up one plus two plus three
plus six over four, four data points,
this is 12 over four which is indeed equal to three
and then the sample standard deviation
for Y you would calculate the exact same way
we did it for X
and you would get 2.160.
Now, with all of that out of the way,
let's think about how we calculate
the correlation coefficient.
Now, right over here is a representation
for the formula for the correlation coefficient
and at first it might seem a little intimating
until you realize a few things.
All this is saying is for each corresponding X and Y,
find the Z score for X,
so we could call this Z sub X
for that particular X,
so Z sub X sub I
and we could say this is the Z score
for that particular Y.
Z sub Y sub I is one way that you could think about it.
Look, this is just saying for each data point,
find the difference between it and its mean
and then divide by the sample standard deviation.
And so, that's how many sample standard deviations
is it away from its mean,
and so that's the Z score for that X data point
and this is the Z score for the corresponding Y data point.
How many sample standard deviations is it away
from the sample mean?
In the real world you won't have only four pairs
and it'll be very hard to do it by hand
and we typically use software computer tools to do it
but it's really valuable to do it by hand
to get an intuitive understanding of what's going on here.
So, in this particular situation,
R is going to be equal to one over N minus one.
We have four pairs, so it's gonna be 1/3
and it's gonna be times a sum of the products
of the Z scores.
So, this first pair right over here,
so the Z score for this one
is going to be one minus how far it is away
from the X sample mean,
divided by the X sample standard deviation, 0.816,
that times one,
now we're looking at the Y variable,
the Y Z score, so it's one minus three,
one minus three over the Y sample standard deviation, 2.160
and we're just going keep doing that.
I'll do it like this.
So, the next one it's going to be two minus two
over 0.816, this is where I got the two from
and I'm subtracting from that the sample mean
right over here, times, now we're looking at this two,
two minus three over 2.160
plus I'm happy there's only four pairs here,
two minus two again, two minus two
over 0.816 times now we're gonna have three minus three,
three minus three over 2.160
and then the last pair you're going to have three minus two,
three minus two
over 0.816 times six minus three,
six minus three over 2.160.
So, before I get a calculator out,
let's see if there's some simplifications I can do.
Two minus two, that's gonna be zero,
zero times anything is zero,
so this whole thing is zero,
two minus two is zero,
three minus three is zero,
this is actually gonna be zero times zero,
so that whole thing is zero.
Let's see this is going to be one minus two
which is negative one,
one minus three is negative two,
so this is going to be R is equal to 1/3
times negative times negative is positive
and so this is going to be two over 0.816
times 2.160 and then plus three minus two is one,
six minus three is three,
so plus three over 0.816 times 2.160.
Well, these are the same denominator,
so actually I could rewrite if I have two over this thing
plus three over this thing,
that's gonna be five over this thing,
so I could rewrite this whole thing,
five over 0.816 times 2.160
and now I can just get a calculator out
to actually calculate this,
so we have one divided by three
times five divided by 0.816 times 2.16,
the zero won't make a difference
but I'll just write it down,
and then I will close that parentheses
and let's see what we get.
We get an R of, and since everything else
goes to the thousandth place,
I'll just round to the thousandths place,
an R of 0.946.
So, R is approximately 0.946.
So, what does this tell us?
The correlation coefficient is a measure
of how well a line can describe the relationship
between X and Y.
R is always going to be greater than
or equal to negative one
and less than or equal to one.
If R is positive one,
it means that an upwards sloping line
can completely describe the relationship.
If R is negative one,
it means a downwards sloping line
can completely describe the relationship.
R anywhere in between says well,
it won't be as good.
If R is zero that means that a line isn't describing
the relationships well at all.
Now in our situation here,
not to use a pun,
in our situation here,
our R is pretty close to one
which means that a line can get pretty close
to describing the relationship
between our Xs and our Ys.
So, for example, I'm just going to try to hand draw
a line here
and it does turn out that our least squares line
will always go through the mean
of the X and the Y,
so the mean of the X is two,
mean of the Y is three,
we'll study that in more depth in future videos
but let's see, this actually does look like
a pretty good line.
So, let me just draw it right over there.
You see that I actually can draw a line
that gets pretty close to describing it.
It isn't perfect.
If it went through every point
then I would have an R of one
but it gets pretty close to describing
what is going on.
Now, the next thing I wanna do
is focus on the intuition.
What was actually going on here with these Z scores
and how does taking products of corresponding Z scores
get us this property that I just talked about
where an R of one will be strong, positive correlation,
R of negative one
would be strong, negative correlation?
Well, let's draw the sample means here.
So, the X sample mean is two,
this is our X axis here,
this is X equals two
and our Y sample mean is three.
This is the line Y is equal to three.
Now, we can also draw the standard deviations.
This is, let's see, the standard deviation
for X is 0.816 so I'll be approximating it,
so if I go .816 less than our mean
it'll get us at some place around there,
so that's one standard deviation below the mean,
one standard deviation above the mean
would put us some place right over here,
and if I do the same thing in Y,
one standard deviation above the mean, 2.160
so that'll be 5.160 so it would put us
some place around there
and one standard deviation below the mean,
so let's see we're gonna go, if we took away two,
we would go to one
and then we're gonna go take another .160,
so it's gonna be some place right around here.
So, for example, for this first pair,
one comma one.
What were we doing?
Well, we said alright, how many standard deviations
is this below the mean?
And that turned out to be negative one over 0.816,
that's what we have right over here,
that's what this would have calculated,
and then how many standard deviations
for in the Y direction,
and that is our negative two over 2.160
but notice, since both of them were negative
it contributed to the R,
this would become a positive value
and so, one way to think about it,
it might be helping us get closer to the one.
If both of them have a negative Z score
that means that there's a positive correlation
between the variables.
When one is below the mean,
the other is you could say,
similarly below the mean.
Now, if we go to the next data point,
two comma two right over here, what happened?
Well, the X variable was right on the mean
and because of that that entire term became zero.
The X Z score was zero.
And so, that would have taken away
a little bit from our correlation coefficient.
The reason why it would take away
even though it's not negative,
you're not contributing to the sum
but you're going to be dividing by a slightly higher value
by including that extra pair.
If you had a data point where let's say X was below the mean
and Y was above the mean,
something like this,
if this was one of the points,
this term would have been negative
because the Y Z score would have been positive
and the X Z score would have been negative
and so, when you put it in the sum
it would have actually taken away
from the sum and so,
it would have made the R score even lower.
Similarly something like this
would have made the R score even lower
because you would have a positive Z score for X
and a negative Z score for Y
and so a product of a positive and a negative
would be a negative.