Variance Metric: Difference between revisions

** QUOTE: If a random variable ''X'' has the [[expected value]] (mean) {{nowrap|1 = ''μ'' = E[''X'']}}, then the variance of ''X'' is given by: :<math>\operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 \right]. \,</math>  <P> That is, the variance is the expected value of the squared difference between the variable's realization and the variable's mean.  This definition encompasses random variables that are [[discrete random variable|discrete]], [[continuous random variable|continuous]], or neither (or mixed). It can be expanded as follows:  :<math>\begin{align}  \operatorname{Var}(X)  &= \operatorname{E}\left[(X - \mu)^2 \right] \\      &= \operatorname{E}\left[X^2 - 2\mu X + \mu^2 \right] \\      &= \operatorname{E}\left[X^2 \right] - 2\mu\,\operatorname{E}[X] + \mu^2 \\      &= \operatorname{E}\left[X^2 \right] - 2\mu^2 + \mu^2 \\      &= \operatorname{E}\left[X^2 \right] - \mu^2 \\      &= \operatorname{E}\left[X^2 \right] - (\operatorname{E}[X])^2.  \end{align}</math>  <P> A mnemonic for the above expression is "mean of square minus square of mean". <P> The variance of random variable ''X'' is typically designated as Var(''X''), <math>\scriptstyle\sigma_X^2</math>, or simply σ² (pronounced "[[sigma]] squared").