Let X be a random variable.  Then E(|m-EX|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-EX|^2) is minimized when m is the mean of x.

A couple weeks ago in a technical stretch of a proof involving the <a href="http://en.wikipedia.org/wiki/Lyapunov_condition">Lyapunov condition for the central limit theorem</a> I ended up with the expression E(|m-EX|^3).  Does this statistic have a name, or any nice properties?