**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 variance is defined to have the **additivity property**: if `r_1` is a random variable with mean `m_1` and variance `d_1` and `r_2` is a random variable with mean `m_2` and variance `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 variance `d_1+d_2`.

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

To get back to something in the same units as the original variable, we take the square root of the variance and call it the standard deviation.