# Dictionary:Standard deviation

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

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

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 ${\displaystyle (X_{i},Y_{i})}$with respect to the means ${\displaystyle ({\overline {X}},{\overline {Y}})}$, ${\displaystyle \sigma }$ is

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

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