When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do


$$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$. 


Why do we need to square and then square-root the numbers? 



Why is it so cool to square numbers (in terms of finding the standard deviation)?
One answer is mathematical convenience. The theory is much simpler using powers of two rather than other powers.


There are justifications. Squaring makes small numbers smaller and makes big numbers bigger. You could argue that a useful measure of dispersion should be forgiving of small errors but weigh larger errors more heavily, so it makes sense to square the deviations from the mean.


