Why is the square root of the average squared distance from the mean more important than…

[Anonymous]

the average absolute distance from the mean?

The latter seems much more straightforward.

[Anonymous]

There is a right triangle and Pythagoras somewhere in the reason

[Anonymous]

They both have utility, but the former is preferred because it’s actually simpler when you are doing a line of best fit. (Doing the calculus with absolute values is messy). It’s also more sensitive to outliers, which is often what people want.
(The average of 2,3, and 10 is 5, the root squared average is 5.3. The average of 2,3,100 is 35. The root squared average is 58.)

[Anonymous]

The first definition is variance, or standard deviation squared, and a useful statistic when dealing with a large number of samples from a uniform distribution. It allows you to make predictions about the samples.