Given sets of variates denoted , ..., , the covariance of and is defined by

			(1)
			(2)

where and are the means of and , respectively. The matrix of the quantities is called the covariance matrix. In the special case ,

(3)

giving the usual variance .

Note that statistically independent variables are always uncorrelated, but the converse is not necessarily true.

The covariance of two variates and provides a measure of how strongly correlated these variables are, and the derived quantity

(4)

where , are the standard deviations, is called statistical correlation of and . The covariance is symmetric since

(5)

For two variables, the covariance is related to the variance by

(6)

For two independent variates and ,

(7)

so the covariance is zero. However, if the variables are correlated in some way, then their covariance will be nonzero. In fact, if , then tends to increase as increases. If , then tends to decrease as increases.

The covariance obeys the identity

			(8)
			(9)
			(10)
			(11)

By induction, it therefore follows that

			(12)
			(13)
			(14)
			(15)
			(16)

CITE THIS AS:

Eric W. Weisstein. "Covariance." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Covariance.html

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