The reason we don’t use absolute values for calculating variance

Question

I know the reason for using squared values is so the values don’t get cancelled out. Why not absolute values? I get it when calculating error metrics that squaring can punish the model more for bigger errors, but I don’t get the reason here.

In: Other

Answer 1

Variances have a lot of useful properties, re: distributions and random variables that are easy to calculate. For some of these measurements, the mean absolute difference might have similar properties, but usually variance is cleaner and have more fundamental properties.

For example, for independent random variables X and Y:

Mean(X+Y)=Mean(X)+Mean(Y)

Mean(X-Y)=Mean(X)-Mean(Y)

Var(X+Y)=Var(X)+Var(Y)

Var(X-Y)=Var(X)+Var(Y)