Variance and Standard Deviation of a Random Variable

The mean alone does not tell us how spread out the values of $X$ are around it. The variance, denoted $Var(X)$ or $\sigma^2$, measures this spread:

where $E(X^2) = \sum_i x_i^2\, p(x_i)$ is the expectation of $X^2$. The second form on the right — "mean of the square minus square of the mean" — is the one almost always used in numerical work because it avoids computing $(x_i - \mu)$ for every value.

The standard deviation is $SD(X) = \sigma = \sqrt{Var(X)}$, expressed in the same unit as $X$ itself.

Property: $Var(aX + b) = a^2\, Var(X)$ — adding a constant $b$ shifts every value equally and does not change the spread, so $b$ drops out; scaling by $a$ scales the variance by $a^2$ (and hence the SD by $|a|$).

Example (continuing the two-coin case). …