Editing Standard deviation (section)

==Identities and mathematical properties==
The standard deviation is invariant under changes in [[location parameter|location]], and scales directly with the [[scale parameter|scale]] of the random variable. Thus, for a constant {{mvar|c}} and random variables {{mvar|X}} and {{mvar|Y}}:
<math display="block">\begin{align}
      \sigma(c) &= 0 \\
  \sigma(X + c) &= \sigma(X), \\
     \sigma(cX) &= |c| \sigma(X).
\end{align}</math>

The standard deviation of the sum of two random variables can be related to their individual standard deviations and the [[covariance]] between them:

<math display="block"> \sigma(X + Y) = \sqrt{\operatorname{var}(X) + \operatorname{var}(Y) + 2 \,\operatorname{cov}(X,Y)}. \, </math>

where <math>\textstyle\operatorname{var} \,=\, \sigma^2</math> and <math>\textstyle\operatorname{cov}</math> stand for variance and [[covariance]], respectively.

The calculation of the sum of squared deviations can be related to [[moment (mathematics)|moment]]s calculated directly from the data.  In the following formula, the letter {{mvar|E}} is interpreted to mean expected value, i.e., mean.

<math display="block">\sigma(X) = \sqrt{\operatorname E\left[(X - \operatorname E[X])^2\right]} = \sqrt{\operatorname E\left[X^2\right] - (\operatorname E[X])^2}.</math>

The sample standard deviation can be computed as:
<math display="block">s(X) = \sqrt{\frac{N}{N-1}} \sqrt{\operatorname E\left[(X - \operatorname E[X])^2\right]}.</math>

For a finite population with equal probabilities at all points, we have

<math display="block">
  \sqrt{\frac{1}{N}\sum_{i=1}^N\left(x_i - \bar{x}\right)^2} =
  \sqrt{\frac{1}{N}\left(\sum_{i=1}^N x_i^2\right) - {\bar{x}}^2} =
  \sqrt{\left(\frac{1}{N}\sum_{i=1}^N x_i^2\right) - \left(\frac{1}{N} \sum_{i=1}^{N} x_i\right)^2},
</math>

which means that the standard deviation is equal to the square root of the difference between the average of the squares of the values and the square of the average value.

See computational formula for the variance for proof, and for an analogous result for the sample standard deviation.