Covariance

Covariance provides a measure of the strength of the correlation between two random variables.

Suppose X and Y are random variables with means E(X) and E(Y), and standard deviations sd(X) and sd(Y), respectively. The covariance of X and Y is defined by

cov(X, Y) = E([X - E(X)][Y - E(Y)])

cov(X, Y) = E(XY) - E(X)E(Y)

Example

What is the covariance of the data set X = (1, 2, 4, 6, 7) and Y = (2, 3, 4, 7, 9)?

E: expected value or mean
	X	Y	X-E(X)	Y-E(Y)	[X-E(X)][Y-E(Y)]
	1	2	-3	-3	9
	2	3	-2	-2	4
	4	4	0	-1	0
	6	7	2	2	4
	7	9	3	4	12
sum	20	25			29
E	4	5			5.8

E: expected value or mean
	X	Y	XY
	1	2	2
	2	3	6
	4	4	16
	6	7	42
	7	9	63
sum	20	25	129
E	4	5	25.8

From the table above,

cov(X, Y)	= E(XY) - E(X)E(Y)
	= 25.8 - (4)(5)
	= 5.8