WEBVTT

00:00:00.060 --> 00:00:01.740
The mathematics of
the correlation

00:00:01.740 --> 00:00:03.810
coefficient are important.

00:00:03.810 --> 00:00:06.410
But it is perhaps more important
to be able to

00:00:06.410 --> 00:00:08.370
interpret it correctly.

00:00:08.370 --> 00:00:12.400
A correlation coefficient of
let's say 0.5, tells us that

00:00:12.400 --> 00:00:15.790
something interesting is going
on as far as the relation of X

00:00:15.790 --> 00:00:17.910
and Y is concerned.

00:00:17.910 --> 00:00:19.300
But what exactly?

00:00:19.300 --> 00:00:21.570
It tells us that the two
random variables are

00:00:21.570 --> 00:00:23.770
associated in some sense.

00:00:23.770 --> 00:00:26.740
But this is often misinterpreted
to mean that

00:00:26.740 --> 00:00:29.950
there is a causal relation
between the two.

00:00:29.950 --> 00:00:32.155
But this is wrong.

00:00:32.155 --> 00:00:35.950
A large correlation coefficient
in general does

00:00:35.950 --> 00:00:39.580
not indicate that there is a
causal relation between the

00:00:39.580 --> 00:00:41.450
random variables.

00:00:41.450 --> 00:00:45.220
As an example, suppose that
X somehow quantifies the

00:00:45.220 --> 00:00:47.310
mathematical aptitude
of a person.

00:00:47.310 --> 00:00:52.290
And Y somehow quantifies the
musical ability of a person.

00:00:52.290 --> 00:00:55.450
In general, it has been found
that mathematical aptitude and

00:00:55.450 --> 00:00:57.780
musical ability are
correlated.

00:00:57.780 --> 00:01:00.800
People who score high on
one will score high on

00:01:00.800 --> 00:01:02.420
the other as well.

00:01:02.420 --> 00:01:04.720
Is there a causal relation?

00:01:04.720 --> 00:01:09.789
If you study math a lot and you
become very good at math,

00:01:09.789 --> 00:01:12.890
does it mean that you would
become a better musician?

00:01:12.890 --> 00:01:14.140
Not necessarily.

00:01:14.140 --> 00:01:18.210
Or if you practice the violin
day in and day out, does it

00:01:18.210 --> 00:01:21.510
mean that you will score better
in the math exam?

00:01:21.510 --> 00:01:23.360
Again, not necessarily.

00:01:23.360 --> 00:01:26.760
Perhaps what is going on is that
there's a certain feature

00:01:26.760 --> 00:01:29.630
of the human brain and when
that feature is well

00:01:29.630 --> 00:01:35.289
developed, then that feature
helps both in math and in

00:01:35.289 --> 00:01:36.750
musical ability.

00:01:36.750 --> 00:01:39.920
And this is a typical situation
of how a correlation

00:01:39.920 --> 00:01:41.940
coefficient may arise.

00:01:41.940 --> 00:01:44.340
That is often a correlation
coefficient that's

00:01:44.340 --> 00:01:48.570
significant, reflects that there
is an underlying common

00:01:48.570 --> 00:01:52.810
but perhaps hidden factor that
affects both of the random

00:01:52.810 --> 00:01:54.710
variables X and Y.

00:01:54.710 --> 00:01:58.640
Let's us go through a simple
numerical example that models

00:01:58.640 --> 00:02:00.980
a situation of this kind.

00:02:00.980 --> 00:02:04.440
Suppose that Z, V, and w are
independent random variables.

00:02:04.440 --> 00:02:07.120
And that we have two more random
variables defined by

00:02:07.120 --> 00:02:08.490
these relations.

00:02:08.490 --> 00:02:13.110
Not that there's no direct
influence from X to Y or from

00:02:13.110 --> 00:02:14.520
Y to X.

00:02:14.520 --> 00:02:16.990
But on the other hand, there's
a common underlying factor,

00:02:16.990 --> 00:02:20.880
this random variable Z that
affects both X and Y. Because

00:02:20.880 --> 00:02:25.270
of this, we expect that X and Y
will somehow have some kind

00:02:25.270 --> 00:02:28.000
of relation or association
between them.

00:02:28.000 --> 00:02:30.210
And we would like to measure
the strength of that

00:02:30.210 --> 00:02:31.460
association.

00:02:31.460 --> 00:02:34.329
The way to measure it will be
in terms of the correlation

00:02:34.329 --> 00:02:37.800
coefficient, which we will
now proceed to compute.

00:02:37.800 --> 00:02:41.890
To have a complete example in
our hands and in order to also

00:02:41.890 --> 00:02:44.630
keep things simple, let's
us assume that the basic

00:02:44.630 --> 00:02:48.540
underlying random variable Z, V,
and W all have 0 means and

00:02:48.540 --> 00:02:50.240
unit variances.

00:02:50.240 --> 00:02:52.880
And now let us take the
definition of the correlation

00:02:52.880 --> 00:02:56.090
coefficient and start
calculating.

00:02:56.090 --> 00:03:01.600
Let us look at the variance of
X. Because X is the sum of two

00:03:01.600 --> 00:03:05.300
independent random variables,
its variance is going to be

00:03:05.300 --> 00:03:08.290
the sum of those variances.

00:03:08.290 --> 00:03:11.370
And we have assumed that each
one of those variances is

00:03:11.370 --> 00:03:12.100
equal to 1.

00:03:12.100 --> 00:03:15.050
So the variance of
X is equal to 2.

00:03:15.050 --> 00:03:18.000
And that implies that the
standard deviation of X is

00:03:18.000 --> 00:03:20.270
equal to the square root of 2.

00:03:20.270 --> 00:03:23.650
By a similar argument, the
standard deviation of Y is

00:03:23.650 --> 00:03:26.440
also equal to the square
root of 2.

00:03:26.440 --> 00:03:31.350
Now, let us look at the
covariance between X and Y.

00:03:31.350 --> 00:03:36.660
Because X and Y have 0 means,
the covariance is just the

00:03:36.660 --> 00:03:40.160
expected value of the product
of the two random variables.

00:03:40.160 --> 00:03:42.990
And using the definition of
what these two random

00:03:42.990 --> 00:03:47.450
variables are, it's this
particular product here.

00:03:47.450 --> 00:03:51.040
We expand the product into
a sum of four terms.

00:03:51.040 --> 00:03:54.590
And take the expected value of
each one of the four terms.

00:04:04.170 --> 00:04:11.580
Which leaves us with this
particular expression here.

00:04:11.580 --> 00:04:15.240
Now, Z has 0 mean and
unit variance.

00:04:15.240 --> 00:04:18.940
Therefore, the expected value
of Z squared is equal to 1.

00:04:18.940 --> 00:04:20.680
How about the next term?

00:04:20.680 --> 00:04:22.240
V and Z are independent.

00:04:22.240 --> 00:04:25.140
So the expected value of the
product is the product of the

00:04:25.140 --> 00:04:26.210
expected values.

00:04:26.210 --> 00:04:30.410
But the expected values are
zero, so this term is zero.

00:04:30.410 --> 00:04:32.500
And with a similar argument,
the other

00:04:32.500 --> 00:04:35.150
terms are zero as well.

00:04:35.150 --> 00:04:37.970
So the co-variance
is equal to 1.

00:04:37.970 --> 00:04:43.350
And from this, we can conclude
our calculation and write that

00:04:43.350 --> 00:04:48.340
the correlation coefficient
between X and Y is equal to 1

00:04:48.340 --> 00:04:53.270
divided by the square root of
2 times square root of 2,

00:04:53.270 --> 00:04:54.520
which is 1/2.

00:04:56.940 --> 00:05:00.610
This example also serves to give
you a rough idea of what

00:05:00.610 --> 00:05:02.790
it may mean to have
a correlation

00:05:02.790 --> 00:05:05.320
coefficient of 1/2.

00:05:05.320 --> 00:05:07.510
It means that the two
random variables

00:05:07.510 --> 00:05:10.310
have some common elements.

00:05:10.310 --> 00:05:13.690
And they also have some
idiosyncratic elements.

00:05:13.690 --> 00:05:18.390
And these two elements are
roughly equal in weight.

00:05:18.390 --> 00:05:22.220
If V and W were completely
absent, the correlation

00:05:22.220 --> 00:05:24.405
coefficient would have been 1.

00:05:24.405 --> 00:05:29.900
If on the other hand V and W had
a huge variance, so as to

00:05:29.900 --> 00:05:33.810
completely hide the effect of
Z, then the value of the

00:05:33.810 --> 00:05:37.490
correlation coefficient would
have been much, much smaller

00:05:37.490 --> 00:05:39.480
perhaps closer to 0.

00:05:39.480 --> 00:05:42.260
And in the extreme case of
course where Z is completely

00:05:42.260 --> 00:05:45.610
absent, then X and Y are
independent, and we get a

00:05:45.610 --> 00:05:47.300
correlation coefficient of 0.