# Expectation of the sample median for symmetric distributions

Is the sample median an unbiased estimator of the population mean when the distribution is symmetric?

I think so yes, so long as

1. The population mean is defined, and
2. The sample is composed of iid draws.

For a symmetric distribution, the median is an unbiased estimate of the population median and the mean equals the median.

However, what it won't be, in general, is the most efficient or minimum variance unbiased estimator (the exception being the Laplace distribution for which median is the best estimator of the mean).

• (+1) I've reworded things slightly to try to avoid a possible misinterpretation. We can get away with somewhat less in terms of the sampling scheme: exchangeability is enough. – cardinal Feb 5 '13 at 18:59
• The additional caveat left unstated is that there is an implicit assumption here that the population median is unique. – cardinal Feb 5 '13 at 19:07
• Is that a caveat though? The sample median is always an unbiased estimate of one of the population medians, and if that one happens to be the mean, then this still holds. It seems to me that if the sample median for even samples is the mean of the two either side, then this should work? – Corone Feb 5 '13 at 19:40
• The sample median is not always unbiased for the population median. Just take $n = 1$ and your favorite asymmetric distribution to get a counterexample, though, of course, the inequality is true much more generally. – cardinal Feb 5 '13 at 19:47
• But for a symmetric distribution, it is surely unbiased? – Corone Feb 5 '13 at 21:11