Why use the squarred version of the sum of deviations from the mean?

Answer 1

You cannot use deviations from the mean because (by definition) their sum is zero.

Absolute deviations are one way of getting around that problem and they are used. Their main drawback is that they treat deviations linearly. That is to say, one large deviation is only twice as important as two deviations that are half as big. That model may be appropriate in some cases.

But in many cases, big deviations are much more serious than that a squared (not squarred) version is more appropriate.

Conveniently the squared version is also a feature of many parametric statistical distributions and so the distribution of the "sum of squares" is well studied and understood.