covariation

Covariation is the tendency of two variables to vary together. In statistics, it refers to the degree to which changes in one variable are associated with changes in another. When two variables tend to increase together, they have positive covariation; when one tends to increase while the other decreases, they have negative covariation.

The population covariance between random variables X and Y is defined as Cov(X, Y) = E[(X − μ_X)(Y − μ_Y)], where μ_X and μ_Y are the respective means. The sample covariance, computed from a data set of n paired observations, is s_xy = sum[(x_i − x̄)(y_i − ȳ)] / (n − 1). The sign of the covariance indicates the direction of co-movement, while its magnitude depends on the units of X and Y.

Covariance is related to the correlation coefficient. The Pearson correlation r = Cov(X, Y) / (σ_X σ_Y) standardizes covariance to a dimensionless measure between −1 and 1, reflecting the strength and direction of a linear relationship. A zero covariance implies no linear relationship, but zero does not necessarily imply independence, except in special cases (e.g., jointly normal variables).

In broader contexts, covariation can be extended to time series via the cross-covariance function, which describes how two processes covary at different lags, and to multivariate data via the covariance matrix that summarizes covariation among many variables. Applications span statistics, genetics (genetic and environmental covariation between traits), and finance (portfolio risk from covariances of asset returns). Important caveats include sensitivity to scale, potential confounding factors, and the distinction that covariation signals co-movement rather than causation.