$$ Mean\;Absolute\;Deviation = \frac{1}{n}\sum_{i=1}^n |x_i-mean(X)| $$ Almost all textbooks and papers are using Standard Deviation as a measurement of dispersion. And of course, almost all the models are built based on Standard Deviation.

But I don't understand how Standard Deviation has gained such popularity. I mean, Mean Absolute Deviation is a very intuitive measurement of dispersion. It tells you exact average distant that each value deviates from their mean. Standard Deviation, on the other hand, makes the result more sensitive to outliers. Why we need this sensitivity. Is there any historical reason leads us to use SD way more often than MAD?


marked as duplicate by Martijn Weterings, gung Oct 10 '18 at 13:22

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    $\begingroup$ Something something absolute function not differentiable. $\endgroup$ – user2974951 Oct 10 '18 at 8:11
  • $\begingroup$ Both are used in empirical research. Standard deviation is more convenient mathematically, probably among other reasons. $\endgroup$ – Xiaomi Oct 10 '18 at 8:23
  • $\begingroup$ A few potential reasons: it's differentiable, the Normal Distribution naturally parammetrises in terms of its standard deviation, the true mean minimises the expected variance (whereas the true median minimises the expected MAD) $\endgroup$ – gazza89 Oct 10 '18 at 8:43
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    $\begingroup$ I am voting to close this question because it has been asked several times before. However, it is useful and it could serve some purpose as helping in searches. For instance I seem not to find the post that I had in mind when marking this as duplicate (although several other posts came up that are obvious duplicates as well). $\endgroup$ – Martijn Weterings Oct 10 '18 at 11:03
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    $\begingroup$ stats.stackexchange.com/search?q=gorard it will relate to the 118th question on this website $\endgroup$ – Martijn Weterings Oct 10 '18 at 11:13

Historically, Laplace started with the expected absolute deviation from the expectation and got mired into computational issues, beyond the Laplace (or double exponential) distribution, while Legendre and Gauss advocated the expected square difference from the expectation, which is more naturally connected with the Normal or Gaussian distribution. Portnoy and Koenker wrote a nice paper called the Gaussian Hare and the Laplacian Tortoise (!) on that issue, including a parody of the Hare and the Tortoise with Laplace's and Gauss' heads:

enter image description here

The issue is covered in depth in this earlier (2015) X Validated question. (Which makes the current one a potential duplicate.)


One possible answer, among many others, is that the mean is precisely defined from the standard deviation as

$mean\left( X \right) = \mathop {\arg \min }\limits_x \frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{x_i} - x} \right)}^2}} $

Hence, mean and standard deviation come together. This is not the case for the MAD.

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    $\begingroup$ Can you explain how this formula for the mean comes about? $\endgroup$ – JacKeown Nov 29 '18 at 4:00
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    $\begingroup$ @JacKeown Sorry I missed your comment. 1) Compute the derivate of the sum of the squares with respect to x. 2) Cancel this derivate: you'll find the sample mean. This is the most basic application of the Least Squares Method. $\endgroup$ – Fabrice Pautot Sep 4 at 11:42

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