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
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.
Now that we established how good it is to square numbers, to get variance, the standard deviation has a very easy explanation: it's the only way to get back from variance to something with the dimension or original set. That is, suppose you numbers are some lengths written in meters. Since the variance is meters squared, you have to take the square root to get something that can be compared with the original set.
(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.)
Now, honestly, this not the only way, since you could also, e.g., multiply it by 2. That's why it's called standard deviation — to show that among different numerical constants we've chosen a specific one.
Standard deviation, defined this way, is very useful. For example, a large class of random distributions are completely defined by their mean and their standard deviation.
