Standard deviation

Definition

Let $X$ be a real random variable with finite variance. The standard deviation of $X$ is the square root $$\sigma_X = \sqrt{\operatorname{Var}(X)} = \sqrt{E\left[(X - E[X])^2\right]}.$$

Remarks

The standard deviation is in the same units as $X$, but the variance is in squared units. For $a, b \in \mathbb{R}$, $\sigma_{aX + b} = |a| \, \sigma_X$. For the normal distribution $\mathcal{N}(\mu, \sigma^2)$, the standard deviation is $\sigma$. About $68\%$ of the mass lies within one standard deviation of the mean.

Explanations

L1 human

The standard deviation is a typical size of the distance between a value and the mean. A small standard deviation means that the values stay close to the mean. A large one means that they spread out.