TITLE: Prove that the sample median is an unbiased estimator
QUESTION [13 upvotes]: My book says that sample median of a normal distribution is an unbiased estimator of its mean, by virtue of the symmetry of normal distribution. Please advice how can this be proved.

REPLY [14 votes]: Let $\mu$ be the population mean (so that's assumed to exist), and assume the distribution is symmetric and there's a density.  (So those are weaker assumptions than normality, and maybe the density assumption can be dropped too.)  Let $X_1,\ldots,X_n$ be the sample; let $Y_i=X_i-\mu$ for $i=1,\ldots,n$.  Let $m=\operatorname{E}(\operatorname{median})=\operatorname{E}(\operatorname{median}(Y_1,\ldots,Y_n))$.  By symmetry of the distribution of the $Y$s about $0$, $-m=\operatorname{E}(-{\operatorname{median}})=\operatorname{E}(\operatorname{median})$.  Since $m=-m$, we must have $m=0$.  Since $\operatorname{E}(\operatorname{median})=0$, we conclude $\operatorname{E}(\operatorname{median}(X_1,\ldots,X_n))$ $=\operatorname{E}(\operatorname{median}(Y_1+\mu,\ldots,Y_n+\mu))$ $=\operatorname{E}(\mu + \operatorname{median}(Y_1,\ldots,Y_n))=\mu$.