Variance of a sum

Statement

Let $X_1, \dots, X_n$ be real random variables on one probability space with $E[X_i^2] < \infty$ for each $i$. Then $$\operatorname{Var}\left(\sum_{i=1}^n X_i\right) = \sum_{i=1}^n \operatorname{Var}(X_i) + 2 \sum_{1 \le i < j \le n} \operatorname{Cov}(X_i, X_j).$$ In particular, $\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2 \operatorname{Cov}(X, Y)$.

Remarks

Proof: $\operatorname{Var}(S) = \operatorname{Cov}(S, S)$ for $S = \sum_i X_i$. Expand by the bilinearity of the covariance: $\operatorname{Cov}(S, S) = \sum_{i, j} \operatorname{Cov}(X_i, X_j)$. Also $\operatorname{Var}(X - Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) - 2 \operatorname{Cov}(X, Y)$. More generally, $\operatorname{Var}\left(\sum_i a_i X_i\right) = \sum_{i, j} a_i a_j \operatorname{Cov}(X_i, X_j)$ for real $a_i$.

Explanations

L1 human

The spread of a sum is the sum of the spreads plus a correction for how the terms move together. If the terms tend to move in the same direction, the sum spreads more. If they tend to compensate each other, it spreads less.