We generally define standard deviation to be the deviation of the data points about the mean of the whole set, but why about mean? Couldn't we do it about the RMS value? Or any other value? So I was doing some calculation, on the basis of which it seems to me that the standard deviation is a minimum about the mean. Definition of standard deviation is ( about $x_0$)
\begin{align*} \langle（x-x_0)^2\rangle &= \langle x^2 - 2 x\,x_0+ x_0 ^2 \rangle \\ &= \langle x^2 \rangle - \langle x \rangle ^2 + [ \langle x\rangle ^2 - 2 \langle x \rangle x_0+ x_0^2] \\ &=(\Delta x)^2 + [ \langle x \rangle - x_0 ]^2 . \end{align*} So deviation is minimum about the average value! Is this the reason we always calculate average values in statistical mechanics ?