Kovariancie

Kovariancie, commonly referred to as covariance in English, is a statistical measure of how two random variables vary together. It captures the directional relationship between variables: positive covariance indicates that they tend to increase together, while negative covariance indicates that one tends to increase as the other decreases.

Mathematically, the population covariance of two variables X and Y is defined as Cov(X,Y) = E[(X − μ_X)(Y − μ_Y)], where μ_X and μ_Y are their means and E denotes expectation. For a sample of n paired observations (x_i, y_i), the sample covariance is cov(X,Y) = (1/(n−1)) Σ (x_i − x̄)(y_i − ȳ), with x̄ and ȳ the sample means. Covariance is measured in the product of the units of X and Y, making it scale-dependent and not directly interpretable in isolation.

Covariance and correlation are related by the correlation coefficient ρ = Cov(X,Y) / (σ_X σ_Y); this standardizes covariance to lie between −1 and 1, providing a dimensionless measure of linear association. If X and Y are independent, Cov(X,Y) = 0; however, zero covariance does not necessarily imply independence except in special cases (for example, when the variables are jointly normal).

In multivariate statistics, covariances are organized into a covariance matrix Σ, where diagonal entries are variances and off-diagonal entries are covariances. This matrix underpins techniques such as principal component analysis and many linear models. Covariance is also used in time series (autocovariance and cross-covariance) and in portfolio theory to assess joint risk of asset returns.