Variance and standard deviation

Measuring spread

The mean tells us the centre of a distribution; the variance tells us how widely the values are spread around it.

Var(X) = σ² = Σ(xᵢ − μ)²pᵢ, which simplifies to the more convenient form Var(X) = E(X²) − [E(X)]², where E(X²) = Σxᵢ²pᵢ.

The standard deviation is σ = √Var(X). It is in the same units as X, which makes it easier to interpret.

Example

X takes 0, 1, 2 with probabilities 1/4, 1/2, 1/4. E(X) = 1 and E(X²) = 0 + 1/2 + 4/4 = 3/2. So Var(X) = 3/2 − 1 = 1/2 and σ = 1/√2.

Fair die

For X = score on a fair die, E(X) = 7/2 and E(X²) = (1 + 4 + 9 + 16 + 25 + 36)/6 = 91/6. So Var(X) = 91/6 − 49/4 = 35/12.

Properties

• Var(X) ≥ 0; it is 0 only when X is constant.
• Var(aX + b) = a²Var(X): adding a constant shifts values but does not change spread; multiplying by a scales the spread by a².
• SD(aX + b) = |a| SD(X).
• For independent X and Y, Var(X + Y) = Var(X) + Var(Y).

Key points

• Var(X) = E(X²) − μ².
• σ = √Var(X).
• Var(aX + b) = a²Var(X).
• Variance of a constant is 0.

Common mistakes

• Writing Var(X) = E(X²) − E(X) (forgetting to square the mean).
• Writing Var(3X) = 3Var(X) instead of 9Var(X).
• Giving a negative variance, which signals an arithmetic error.