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.
• If it is 0 then there is no linear correlation between the two variables.
  In particular, this is the case for stochastically independent variables. But the inverse is not true because there may still be other – nonlinear – 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 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).

In a more general context of probability theory the covariance is a second-order central moment of the two-dimensional random variable (X,Y), often denoted as μ₁₁.

Finite data

For a finite set of data

$(x_{i},y_{i})\in \mathbb {R} ^{2}\ {\text{with}}\ i=1,\dots ,n$

the covariance is given by

${1 \over n}\sum _{i=1}^{n}(x_{i}-{\overline {x}})(y_{i}-{\overline {y}})\qquad {\text{where}}\ {\overline {x}}:={1 \over n}\sum _{i=1}^{n}x_{i}\ {\text{and}}\ {\overline {y}}:={1 \over n}\sum _{i=1}^{n}y_{i}$

or, using a convenient notation

$[x_{i}]:=\sum _{i=1}^{n}x_{i}$

introduced by Gauss, by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1\over n}( [ x_i y_ ]-[x_i][y_] ) }

This corresponds to taking the uniform distribution where each item (x_i,y_i) has probability 1/n.

Unbiased estimate

The expectation of the covariance of a random sample — taken from a probability distribution — depends on the size n of the sample and is slightly smaller than the covariance of the distribution.

An unbiased estimate of the covariance is

$\mathrm {Cov} (X,Y)={n \over n-1}\mathrm {Cov} (x_{i},y_{i})={1 \over n-1}\sum _{i=1}^{n}(x_{i}-{\overline {x}})(y_{i}-{\overline {y}})$

Remark:
The distinction between the covariance of a sample and the estimated covariance of the distribution is not always clearly made. This explains why one finds both formulae for the covariance — that taking the mean with " 1 / n " and that with " 1 / (n-1) " instead.

Properties

The following properties hold for the covariance

$\operatorname {Cov} (X,Y)=\operatorname {Cov} (Y,X)\qquad {\text{symmetry}}$

$\operatorname {Cov} (aX_{1}+bX_{2},Y)=a\cdot \operatorname {Cov} (X_{1},Y)+b\cdot \operatorname {Cov} (X_{2},Y)\qquad {\text{linearity}}$

and, because of the symmetry, also

$\operatorname {Cov} (X,aY_{1}+bY_{2})=a\cdot \operatorname {Cov} (X,Y_{1})+b\cdot \operatorname {Cov} (X,Y_{2})\qquad {\text{(bi)linearity}}$

$\operatorname {Cov} (X,X)\geq 0$

$\operatorname {Cov} (X,X)=0\Leftrightarrow X=\mu _{X}\ {\textrm {almostsurely}}$