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I think we'd be better off if we recognized that there are statistical distributions in the world besides the plain old Gaussian. For example, wealth does not follow a Gaussian, so why the heck do we throw around ideas like "above average wealth"?

Is MAD any better? Definitely. But I'd like to see a visual demonstration of how well it models exponential-based distributions. How well does it describe their "shape", the skew of the tail?

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In that specific case, I'd submit that the word "average" doesn't belong in a conversation attempting any level of rigor. It only encourages the confusion between median and mean.

Medians and means are interrelated. Averages are best thought of as being on a scale from meanlike to medianlike: a median requires a mean of the middle 2 values when there's an even number of them, and a mean can have outliers on each side to be eliminated before calculation. The word average abstracts away this detail. (I guess there'd be another scale from meanlike to medianlike absolute deviation, but that's another story.)

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