Why Standard Deviation is a better (or more frequently used) measure than Average Deviation?

Question

Why Standard Deviation is a better (or more frequently used) measure than Average Deviation? When average deviation can be understood intuitively.

Answer 1

Variance (of which standard deviation is the square root) has nice mathematical properties. Average (absolute) deviation does not allow any mathematical development that would be statistically useful.

Another advantage of variance is in the assessment of error. Measurement errors are unavoidable, and we have to live with them. But small errors are less harmful than big ones. In effect, variance weights errors according to their size (by squaring them) and so standard deviation is a better guide to measurement quality than average deviation from a practical perspective.

Sometimes we are concerned with maximum error. In that case, standard deviation is a poor guide—but then average deviation would be even worse.