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As I understand it, variance of a random variable is defined as follows: \begin{equation} \text{Var}(X) = \text{E}[(X-\mu)^2] \end{equation} $X-\mu$ is obviously the difference between the value of the random variable and the expected value of the random variable.

Why don't we define variance as \begin{equation} \text{Var}(X) = \text{E}[X-\mu] \end{equation}

To me, this would make more sense because it would express the difference between the expected value and the value it takes...which to me sounds like variation.

Why do we use the $(X-\mu)^2$ instead?


marked as duplicate by Milo Brandt, user147263, Daniel W. Farlow, Winther, Jonas Meyer Feb 9 '15 at 2:17

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  • $\begingroup$ On the right side of this page, under Related, you will find the question why the definition of variance is such. That (and the duplicates pointed to) will answer your question. $\endgroup$ – André Nicolas Feb 9 '15 at 0:06
  • $\begingroup$ $E[X-\mu]$ is always $0$, so it does not express anything about $X$. $\endgroup$ – user147263 Feb 9 '15 at 0:15
  • $\begingroup$ I think I have received enough help I am satisfied. Thanks $\endgroup$ – Stan Shunpike Feb 9 '15 at 1:53

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