or equally, the mean is the central point which minimises the sum of squares of deviation.
Use least mean square regression?
Do you mean this:
Show that the mean $\displaystyle \bar{x}$ is the value of $\displaystyle k $ which minimises the second moment of a RV $\displaystyle X$ about $\displaystyle k$?
That is $\displaystyle k=\bar{x}$ minimises:
$\displaystyle
\int (x-k)^2 p(x) ~dx
$
Differentiate wrt $\displaystyle k$ and solve:
$\displaystyle
\frac{d}{dk}f(k)=0
$
RonL