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.