This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

Standard deviation

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The standard deviation of a distribution is a measurement of variability or diversity of its values from the "average".

The standard deviation show much the values differ -- on average -- from the mean (or expected value). A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

The standard deviation of a sample is $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2},$$ where $\scriptstyle\{x_1,\,x_2,\,\ldots,\,x_N\}$ are the observed values of the sample items and $\scriptstyle\overline{x}$ is the mean value of these observations.