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Concept

Variance

Variance measures average squared distance from the mean and is one of the first ways actuaries quantify uncertainty.

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variance

Formula

Variance is the expected squared distance from the mean. In exam problems, the computational form is often faster once you can find the first and second moments.

Definition
Var⁡(X)=E[(X−E[X])2]\operatorname{Var}(X)=E\left[(X-E[X])^2\right]
Computational form
Var⁡(X)=E[X2]−(E[X])2\operatorname{Var}(X)=E[X^2]-(E[X])^2

Worked Example

If X is 0 with probability 0.5 and 10 with probability 0.5, E[X]=5 and E[X^2]=50. So Var(X)=50-25=25.

Common Mistake

For constants, Var(aX+b)=a^2 Var(X), not a Var(X)+b. Shifts change the mean, not the spread.

References And Official Sources