**Intro by Reid Barton**

I think the answer should involve the additivity of variance for independent variables and the central limit theorem.  Maybe someone can flesh this out.

**Answer by ilya**

Indeed, the standard deviation is defined to have the **additivity property**: if `r_1` is a random variable with mean `m_1` and deviation `d_1` and `r_2` is a random variable with mean `m_2` and deviation `d_2` and *these two variables are independent* then the new random variable `r = r_1+r_2` has the mean `m_1+m_2` and deviation `d_1+d_2`.

This will obviously fail for any other function of standard deviation, be it square, cube or something else. Answers that stress convenience are, unfortunately, missing the crucial point.