For $(x_i)_{1 \leq i \leq N}$, the standard deviation is of course

$$\sigma = \sqrt{\frac{1}{N} \sum_i (x_i - m)^2},$$

where $m$ is the mean value of the $(x_i)$. We can say that the variance $\sigma^2$ is the average of the squared distance between each sample and the mean.

I know why the fact of having this square is helpful (for many formulas), and that $\sigma$ is much more used than :

$$\alpha =\frac{1}{N} \sum_i |x_i - m|,$$

that could seem a natural measure of the average distance between each sample and the mean.


Even if $\alpha$ is rarely used, is there a single-word name for $\alpha$ ?


Although this does not qualify as a "single word name", this measure is called average absolute deviation or mean absolute deviation.

Some of the reasons that the "standard deviation" is so much more used than this measure (although both achieve the same goal, to measure only positive distances from the mean), has to do with the much better statistical (follows well known distribution) and analytical (differentiable) properties of the variance.


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