Home page for accesible maths 2.6 Expectation and Related Summaries 2.6.1 Expectation 2.6.3 Properties of Summary Measures

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2.6.2 Variance

Variance is a measure of the variability between outcomes of the random variable (in squared units). The variance of the random variable $R$ , ${\operatorname{\mathsf{Var}}}\left[{R}\right]$ , is defined as

\displaystyle{\operatorname{\mathsf{Var}}}\left[{R}\right]=\operatorname{% \mathsf{E}}\left[{(R-\operatorname{\mathsf{E}}\left[{R}\right])^{2}}\right],

which is most easily evaluated (see first year probability module for the proof) using:

\displaystyle{\operatorname{\mathsf{Var}}}\left[{R}\right]=\operatorname{% \mathsf{E}}\left[{R^{2}}\right]-[\operatorname{\mathsf{E}}\left[{R}\right]]^{2}.

Standard deviation: a measure of the variability between outcomes of the random variable (in units of the random variable). The standard deviation of the random variable $R$ , is defined as ${\operatorname{\mathsf{StdDev}}}\left[{R}\right]=\sqrt{{\operatorname{\mathsf{% Var}}}\left[{R}\right]}$ , where the positive square root is taken.

Coefficient of variation: a measure of the typical variability of a random variable compared to its typical size: ${\operatorname{\mathsf{StdDev}}}\left[{X}\right]/\operatorname{\mathsf{E}}% \left[{X}\right]$ .