If data is drawn from a Laplace distribution of the form p(x) = exp(-|x|), the mean absolute deviation is more informative than the standard deviation, but if i

If data is drawn from a Laplace distribution of the form p(x) = exp(-|x|), the mean absolute deviation is more informative than the standard deviation, but if its form is close to the normal, p(x) = exp(-x^2), the standard deviation is more important. So whether to use the mean absolute or standard deviation depends on the distribution of the data. There is a field called robust statistics that looks at this question.