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As far as I know, the mean is a value that minimizes the variance but I wonder if the median also a value that minimizes the variance or it is a value that means a value in the middle of the data?

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The median is also known to be a better choice when the data set includes outliers while the mean is affected by outliers. Why is it so? Is it because the way the median measures the variance with the absolute values from the mean?

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  • $\begingroup$ In general, the mean and median are not equal. So the median cannot be a minimizer of the variance. $\endgroup$
    – user65203
    May 3, 2021 at 14:36

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The median is a value that minimize another variation index: the Average Absolute deviation

$$\mathbb{E}|X-X_{me}|$$

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In fact $\Bbb E[(X-c)]^2=\Bbb E[(X-\mu)]^2+(c-\mu)^2$ is minimal (thereby minimizing standard deviation) iff $c=\mu$, but the median $m$ does satisfy something else nice: it minimizes absolute deviation, i.e. $\Bbb E|X-c|\ge\Bbb E|X-m|$.

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