---
tag: 0D86GJEZ39
kind: statement
role: proposition
name: Variance of a sum
msc: [60A05]
uses: [007QED3E0Y, 0DJ8W3HP2D, 0P7PTC8DDE]
sources:
  - url: https://en.wikipedia.org/wiki/Variance
    locator: Sum of correlated variables
---

## 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|0DJ8W3HP2D]]: $\operatorname{Cov}(S, S) = \sum_{i, j} \operatorname{Cov}(X_i, X_j)$.
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.