Variance and Standard Deviation of a Random Variable

Mathematical expectation tells us the centre of a distribution, but says nothing about how spread out or unpredictable the values are around that centre. Two ventures could have the identical expected profit, yet one might always give a result close to that average while the other swings wildly between a big gain and a big loss — from a business risk point of view these are very different situations. Variance and its square root, the standard deviation, measure exactly this spread.

Variance

The variance of a random variable $X$ is defined as the expected value of the squared deviation of $X$ from its own mean $E(X)$:

While this definition is the conceptually correct one, it is rarely the easiest way to compute variance. Expanding it algebraically gives the standard working formula:

where $E(X^2) = \sum x_i^2\, p_i$ is found exactly like $E(X)$, but using the squares of the values. This shortcut formula — "mean of the squares minus the square of the mean" — is the one almost always used in practice, because it needs only one pass through the distribution table (to get both $E(X)$ and $E(X^2)$) rather than first finding the mean, then re-computing every deviation from it.

Standard deviation

The standard deviation is simply the (positive) square root of the variance, brought back to the same unit as $X$ itself (variance is in squared units, which is harder to interpret directly):

Property of variance under a linear transformation

$Var(aX + b) = a^2\, Var(X)$ …