What is wrong with calculating the average of absolute difference? what is the motivation to square it?

Question

I have a beautiful function
$$f(x)= \frac{\sum_0^n|x-E(x)|}{n} $$ which calculates the average differences between all values and the mean value
The results are very simple to understand
I m trying to understand the motivation behind creating the variance
$$f(x)= \frac{\sum_0^n(x-E(x))^2}{n} $$ What is this squaring good for?
Tnx

Answer 1

The main motivation for using the square root of the mean squared deviation instead of using the mean absolute deviation is that it makes possible this identity: $$ \operatorname{var}(X_1+\cdots+X_n) = \operatorname{var}(X_1) + \cdots + \operatorname{var}(X_n) \text{ if } X_1,\ldots,X_n \text{ are independent.} $$

Abraham de Moivre used this in the 18th century when he showed that the probability mass function of the number of "heads" that appear when a coin is tossed 1800 times is well approximated by the integral of a Gaussian function over a bounded interval.