12.8 Variance and standard deviation

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The variance (\(\sigma^2\)) is a measure of spread of a data set, and indicates how much each data point deviates from the mean.

\(\sigma^2 = \dfrac{\sum{(x-\bar{x})^2}}{n} = \dfrac{\sum{x^2}}{n} - \Big(\dfrac{\sum{x}}{n}\Big)^2\)

Alternatively, using the summary statistic \(S_{xx}\):

\(S_{xx} = \sum{(x-\bar{x})^2} = \sum{x^2} - \dfrac{(\sum{x})^2}{n}\)

\(\sigma^2 = \dfrac{S_{xx}}{n}\)

The standard deviation (\(\sigma\)) is the square root of the variance.

\(\sigma = \sqrt{\dfrac{\sum{(x-\bar{x})^2}}{n}} = \sqrt{\dfrac{\sum{x^2}}{n} - \Big(\dfrac{\sum{x}}{n}\Big)^2}\)

Alternatively:

\(\sigma = \sqrt{\dfrac{S_{xx}}{n}}\)

Grouped frequency

\(\sigma^2 = \dfrac{\sum{f(x-\bar{x})^2}}{\sum{f}} = \dfrac{\sum{fx^2}}{\sum{f}} - \Big(\dfrac{\sum{fx}}{\sum{f}}\Big)^2\)

\(\sigma = \sqrt{\dfrac{\sum{f(x-\bar{x})^2}}{\sum{f}}} = \sqrt{\dfrac{\sum{fx^2}}{\sum{f}} - \Big(\dfrac{\sum{fx}}{\sum{f}}\Big)^2}\)
Important
Variance

\(\sigma^2 = \dfrac{\sum{(x-\bar{x})^2}}{n} = \dfrac{\sum{x^2}}{n} - \Big(\dfrac{\sum{x}}{n}\Big)^2\)

For grouped frequency:

\(\sigma^2 = \dfrac{\sum{f(x-\bar{x})^2}}{\sum{f}} = \dfrac{\sum{fx^2}}{\sum{f}} - \Big(\dfrac{\sum{fx}}{\sum{f}}\Big)^2\)

Standard deviation

\(\sigma = \sqrt{\dfrac{\sum{(x-\bar{x})^2}}{n}} = \sqrt{\dfrac{\sum{x^2}}{n} - \Big(\dfrac{\sum{x}}{n}\Big)^2}\)

For grouped frequency:

\(\sigma = \sqrt{\dfrac{\sum{f(x-\bar{x})^2}}{\sum{f}}} = \sqrt{\dfrac{\sum{fx^2}}{\sum{f}} - \Big(\dfrac{\sum{fx}}{\sum{f}}\Big)^2}\)
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