Varianssummen

Varianssummen is a statistical term that denotes the sum of variances of a collection of random variables. For random variables X1, X2, …, Xn, varianssummen is defined as S = Var(X1) + Var(X2) + … + Var(Xn). It provides a single scalar measure of total dispersion contributed by the individual components when they are treated separately.

Relation to the variance of a sum: If Y = Σ_i X_i, then Var(Y) = Σ_i Var(X_i) + 2 Σ_{i<j} Cov(X_i, X_j). Therefore, varianssummen equals Var(Y) only when all pairwise covariances are zero (for example, when the X_i are independent or uncorrelated). In general, S and Var(Y) differ because the covariances can add or subtract from the total dispersion.

When the variables are independent, the covariances vanish and Var(Σ_i X_i) = S. For the mean, if X̄ = (1/n) Σ_i X_i, then Var(X̄) = (1/n^2) Var(Σ_i X_i) = (1/n^2)[S + 2 Σ_{i<j} Cov(X_i, X_j)]. If the X_i are independent, this reduces to Var(X̄) = S / n^2.

Applications and limitations: Varianssummen is useful as a compact measure of aggregate dispersion across multiple components, for example in risk assessment or when combining independent estimates. However, it does not capture the structure of dependencies among variables; understanding the full dispersion of a sum requires considering covariances.