I am still confused, despite similar questions being asked, about the difference between the variance and standard deviation in statistics. Why is the variance squared?
It is not easy to get an intuition behind standard deviation $\sigma$, as one gets easily about mean as soon as one sees them.
Now why variance is squared of standard deviation is perhaps an outcome of the definition of the variance ($Var[X] = \Sigma_i (x-u)^2 f(x) dx)$ which requires that the difference term $(x - u)$ should be squared. Now one can protest that one need not use $(x - u)^2$ in this definition and live with some other metric such as $|(x-u)|$.
If you define variance in this way than standard deviation ceases to be squared for the variance. Now one can wonder why use square function in the definition in variance at all and not mod function. IMHO, I suspect since mod changes its direction rapidly at 0, it is not as useful as square function.