---
tag: 0VTN5RKNC5
kind: definition
name: Standard deviation
msc: [60A05]
defines: 0P4FTHHMYS
uses: [007QED3E0Y, 0VB8BFRABH]
sources:
  - url: https://en.wikipedia.org/wiki/Standard_deviation
    locator: Definition of population values
---

## Definition

Let $X$ be a real random variable with finite [[variance|007QED3E0Y]].
The standard deviation of $X$ is the [[square root|0VB8BFRABH]]
$$\sigma_X = \sqrt{\operatorname{Var}(X)} = \sqrt{E\left[(X - E[X])^2\right]}.$$

## 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|07RXBVTVZB]] $\mathcal{N}(\mu, \sigma^2)$, 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.