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Standard deviation is the square root of the variance of a random variable, an estimator thereof, or a similar measure of the spread of a batch of data.

Overview

The standard deviation (usually denoted by $\sigma$) is a measure of spread of a random variable or of data. It is expressed in the same units as the original data, unlike the variance. The standard deviation is defined as the square root of the second central moment of a random variable. For a random variable $X$ with mean (expected value) $\mathrm{E}[X]$, the standard deviation is expressed as:

$$\sigma = \sqrt{\mathrm{E}\left[ (X- \mathrm{E}(X) )^2 \right]} = \sqrt{\mathrm{E}[X^2] - \left(\mathrm{E}[X] \right)^2}$$

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