Dictionary:Standard deviation

The standard deviation σ of $n$ measurements of a quantity $X_{i}$ with respect to the mean ${\overline {X}}$ is

$\sigma ={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}\left(X_{i}-{\overline {X}}\right)^{2}}}$

With a normal or Gaussian distribution of data, 68.3% of the data fall within a standard deviation about the mean. The square of the standard deviation is the variance. See statistical measures.

For two degrees of freedom, measurements $(X_{i},Y_{i})$with respect to the means $({\overline {X}},{\overline {Y}})$, $\sigma$ is

$\sigma ={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}\left[\left(X_{i}-{\overline {X}}\right)^{2}+\left(Y_{i}-{\overline {Y}}\right)^{2}\right]}}$

For a Rayleigh distribution of data, 40.5% fall within a circle of radius $\sigma$ (called one sigma).

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