Uncorrelated

Uncorrelated is a term used in statistics to describe a pair of random variables whose linear relationship is zero. In population terms, X and Y are uncorrelated when Cov(X,Y) = 0, which is equivalent to Corr(X,Y) = 0 if Var(X) > 0 and Var(Y) > 0. The Pearson correlation coefficient is ρ(X,Y) = Cov(X,Y) / (σ_X σ_Y). In samples, the analogous statistic is the sample correlation r, used to estimate ρ from data.

However, zero correlation does not imply independence. For example, if Y = X^2 and X is symmetrically distributed around zero, Cov(X,Y) = 0 even though Y is a nontrivial function of X, so X and Y are not independent. In the special case of jointly Gaussian variables, zero correlation does imply independence, but this does not extend to non-Gaussian distributions.

In time series and multivariate analysis, uncorrelatedness refers to zero covariance at the relevant lags or between component series. Uncorrelatedness differs from mutual independence and from being uncorrelated across all moments. In practice, uncorrelated assets are useful for diversification, but non-linear relationships can still transfer risk, so correlation captures only part of the dependency structure.