Example of a Covariance Matrix?

Question

Can someone provide an example of a covariance matrix for any set of data? For example, if given:

2 3 4

5 1 8

9 7 6

how would I take this 3x3 matrix and convert it to the covariance matrix? I see the formula involves taking the means, but I'm not quite sure how that works in this case...

Answer 1

Here is a session from R statistical software with means and variances of three variables similar to yours, and then a variance-covariance matrix of all three.

x1 = c(2,3,4)
x2 = c(5,8,1)
x3 = c(9,7,5)
mean(x1); mean(x2); mean(x3)
## 3                         # sample mean of x1
## 4.666667
## 7
var(x1); var(x2); var(x3)
## 1                         # sample variance of x1
## 12.33333
## 4
cbind(x1, x2, x3)            # puts 3 column vectors into matrix
       x1 x2 x3
##[1,]  2  5  9
##[2,]  3  8  7
##[3,]  4  1  5
cov(cbind(x1,x2,x3))         # makes covariance matrix from data matrix
##    x1       x2 x3
## x1  1 -2.00000 -2
## x2 -2 12.33333  4
## x3 -2  4.00000  4

Notice that the variance $1$ of data vector $x_1$ is in the upper-left corner of the variance-covariance matrix. And that the other two variances 12.3333 and 4 are also on the principal diagonal of the matrix.

The covariance of $x_1$ and $x_3$ is computed as $$S_{13} = \frac{(2-3)(9-7)+(3-3)(7-7)+(4-3)(5-7)}{2} = \frac{-2+0-2}{2} -4/2 = -2,$$ which is shown at the top of right-hand column of the variance-covariance matrix. (Also again, at the left of the bottom row.)

You do not show the formula for the covariance as given in your book so I have shown the arithmetic without trying to guess the notation. I will leave it to you to match my arithmetic with the formula in your book.