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My question comes out of curiosity, and not enough knowledge.

I'm asking about the measurement of dispersion.

We know stdev is the square root of variance, which is in turn the mean of the squared distance of each observation to the mean of observations.

Is there a reason to prefer squaring for variance and rooting for stdev, over something more simplistic, in the spirit of :-

$$mydispersion = \frac {\sum_{i=0}^n abs(N_i - \mu)}{n}$$ $$where$$ $$\mu = \frac {\sum_{i=0}^n N_i}{n}$$

It was even interesting to build the MathJax formulae.

Thank you in advance.

alex
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    see https://math.stackexchange.com/questions/4709270/whats-the-significance-of-mean-squared-error-why-not-something-else/4709289#4709289 or https://math.stackexchange.com/questions/63238/why-do-we-use-a-least-squares-fit/ or https://thestatsguy.rbind.io/post/2020/12/26/the-intuition-behind-averaging/ or https://math.stackexchange.com/questions/3392170/linear-fit-why-do-we-minimize-the-variance-and-not-the-sum-of-all-deviations?noredirect=1&lq=1 or https://math.stackexchange.com/questions/967883/why-get-the-sum-of-squares-instead-of-the-sum-of-absolute-values?noredirect=1&lq=1 – Matthew Towers Jun 13 '23 at 13:46
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    there is no "need" ... measuring Mean Absolute Deviation as you propose is completely standard. The best argument for preferring the usual variance is that it plays a unique roll in the Central Limit Theorem. Other arguments include analytic simplicity (the absolute value function lacks a derivative at $0$ and that complicates computations). – lulu Jun 13 '23 at 13:48
  • thank you very much @MatthewTowers – alex Jun 14 '23 at 14:28
  • thank you very much @lulu – alex Jun 14 '23 at 14:29

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