MAD and Stdev?

Here’s the short answer, it’s not logical to just ignore +/- signs just because.
 
Without your calculator, tell me what (200 - -102.19135)^2 + (200 - 103.29)^2 is?
Now tell me what |200 - -102.19135| + |200 - 103.29| is?
It’s a good lazy measure of dispersion.
 
Why would you use std. dev.? (Ans: Because it’s the natural measure of dispersion for normally distributed things and there’s lots of math that says that random diffusion is characterized by normality. Also there’s lots of mathematical tractability because of the Pythagorean theorem in using variance).
What if you have non-normal data, should you use std. dev. and if so why? Why should distances from the mean become increasing significant beyond just their distance? Who says that the non-normal world ought to use the squared-error loss metric?
I think you need to determine what the purpose is of the measurement and find the one most appropriate.
 
MAD is one of the most intuitive measures of dispersion (certainly more intuitive than SD). It answers the question “on average, how far is a data point from the mean.” It’s a pain to deal with mathematically, because it’s difficult to do algebraic operations on absolute values without getting into “do this if > 0; do that if < 0”.
SD is easier to deal with algebraically, and there are lots of reasons to think that lots of stuff is normal or approximately normally distributed, which, as Joey says, is where SD is most useful. However, the farther away something is from the mean, the more it contributes to increasing the SD, so the interpretation as an average distance from the mean is conceptually meaningful, but slightly distorted.
 
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