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I just got done reading Gorard's "Revisiting a 90-year-old debate: the advantages of the mean deviation." (https://www.leeds.ac.uk/educol/documents/00003759.htm) I'm no expert at statistics; in fact, I'm quite aware of how much there is that I don't know. So maybe this question is actually silly, but I'm gonna give it a go because information regarding this seems to be sparse.

Gorard talks a lot about how the MAD doesn't lend itself as well as the SD to algebraic manipulation. My question is this: are there pieces of statistics which could not be essentially rephrased using the MAD as the concept of spread rather than the SD? Has anybody even tried? If so I'd love to check out their work! I expected this to be an endless rabbit hole of alternative statistical theory but I haven't been able to find much on it.

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Are there pieces of statistics which could not be essential rephrased using MAD as the concept of spread rather than the SD?

Yes. SD/variance has a unique role in the central limit theorem (CLT), both in what converges to the normal distribution and the requirement of finite variance. Finite MAD is not enough for the CLT to kick in.

That said, I find it much easier to think about what MAD means in a non-normal distribution than SD.

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The use of SD is only justified if the error is a sum of many enough independent noise sources. That's because according to the central limit theorem, the resulting error distribution approaches normal, for which variance/SD is a stable measure of dispersion, and has the useful properties mentioned in Dave's answer. However, SD can break if

  • the total error is a sum of only few error sources, or
  • error sources are dependent.

Gorard essentially argues that this is often the case, and suggests using MAD by default (so does Taleb in this short essay against SD). I found work by Falie and David most insightful.

Skimming this list of publications referencing Gorard's work, I found El Amir's On uses of mean absolute deviation: decomposition, skewness and correlation coefficients. Follow-ups by El Amir (1, 2) and Gorard (3, 4) might be of interest, too.

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