Variance and Standard Deviation — Section 1: Foundations of Probability

Expectation tells you the average outcome, but not how spread out the outcomes are. Two distributions with the same mean can look completely different. Variance measures that spread.

Variance is the expected squared deviation from the mean:

\mathrm{Var}(X) = E[(X - E[X])^2] = E[X^2] - E[X]^2

Squaring penalizes large deviations more than small ones and ensures the result is always non-negative.

Because variance is in squared units, we usually report the standard deviation instead:

\sigma_X = \sqrt{\mathrm{Var}(X)}

Standard deviation has the same units as $X$ , which makes it interpretable. A few identities you'll use constantly:

For any constants $a, b$ : $\mathrm{Var}(aX + b) = a^2\, \mathrm{Var}(X)$ .
For independent $X$ and $Y$ : $\mathrm{Var}(X + Y) = \mathrm{Var}(X) + \mathrm{Var}(Y)$ .
$\mathrm{Var}(X) = 0$ if and only if $X$ is constant.

In finance, the standard deviation of returns is called volatility — the headline measure of an asset's risk.