Covariance

The covariance — usually denoted as Cov — is a statistical parameter used to compare two real random variables on the same sample space.
It is defined as the expectation (or mean value) of the product of the deviations (from their respective mean values) of the two variables.

The sign of the covariance indicates a linear trend between the two variables.

• If one variable increases (in the mean) with the other, then the covariance is positive.
• It is negative if one variable tends to decrease when the other increases.
• And it is 0 if the two variables are not linearly correlated. In particular, this is the case for stochastically independent variables. The inverse is not true, however, because there may still be other dependencies.

The value of the covariance is scale-dependent and therefore does not show how strong the correlation is. For this purpose a normed version of the covariance is used — the correlation coefficient which is independent of scale.

Formal definition

The covariance of two real random variables X and Y

$X\quad {\text{and}}\quad Y$

with expectation (mean value)

$\mathrm {E} (X)=\mu _{X}\quad {\text{and}}\quad \mathrm {E} (Y)=\mu _{Y}$

is defined by

$\operatorname {Cov} (X,Y):=\mathrm {E} ((X-\mu _{X})(Y-\mu _{Y}))=\mathrm {E} (XY)-\mathrm {E} (X)\mathrm {E} (Y)$

Remark:
If the two random variables are the same then their covariance is equal to the variance of the single variable: Cov(X,X) = Var(X).