Q:

Which converges faster, mean or median?

If I draw i.i.d. variables from N(0,1), will the mean or the median converge faster?  How much faster?
To be more specific, let $x_1, x_2, \ldots $ be a sequence of i.i.d. variables drawn from N(0,1).  Define $\bar{x}_n = \frac{1}{n}\sum_{i=1}^n x_i$, and $\tilde{x}_n$ to be the median of $\{x_1, x_2, \ldots x_n\}$.  Which converges to 0 faster, $\{\bar{x}_n\}$ or $\{\tilde{x}_n\}$?
For concreteness on what it means to converge faster: does $\lim_{n \to \infty} Var(\bar{X}_n)/Var(\tilde{X}_n)$ exist?  If so, what is it?

A:

The mean and median are the same, in this particular case. It is known that the median is 64% efficient as the mean, so the mean is faster to converge. I can write more details but wikipedia deals with your question exactly.

