Covarianssit

Covarianssit, or covariances, describe the degree to which two random variables vary together. A positive covariance indicates that, on average, when one variable is above its mean the other tends to be above its mean as well; a negative covariance indicates that one tends to be above its mean while the other is below. Covariance provides a directional sense of association but not its strength in a standardized way.

Mathematical definition: For random variables X and Y with finite means μ_X and μ_Y, Cov(X,Y) = E[(X − μ_X)(Y − μ_Y)]. The sample covariance is s_xy = (1/(n−1)) Σ (x_i − x̄)(y_i − ȳ) for a data set of n paired observations. The units of covariance are the product of the units of X and Y.

Properties: Covariance is bilinear and symmetric, satisfying Cov(X,Y) = Cov(Y,X) and Cov(X,X) = Var(X). The value can range from negative to positive infinity. A zero covariance does not, in general, imply independence unless the joint distribution has a specific form (for example, a multivariate normal distribution). The correlation coefficient ρ(X,Y) = Cov(X,Y) / (σ_X σ_Y) normalizes covariance to a dimensionless measure between −1 and 1.

Uses and interpretation: Covariances are fundamental in multivariate analysis, regression, and financial risk assessment, where they are used to build the covariance matrix and to quantify joint variability. Estimation relies on sample moments and can be sensitive to outliers and scale. Nonlinear relationships may produce near-zero covariance even when the variables are clearly related.

Related concepts: The covariance matrix, correlation, and independence.