Is the sample median a good estimator of the mean?

Question

Someone proposes using the sample median as an estimator of the population mean \mu. Is the sample median unbiased and consistent for \mu? Answer for a symmetric distribution, then for a skewed one.

Accepted Answer

The key fact: as n 	o \infty the sample median converges to the population median m, not automatically to the mean \mu. So its behavior depends on the shape of the distribution. Symmetric distribution (e.g. normal). Here the population median equals the mean, m = \mu. The sample median is then consistent for \mu, and for a symmetric law it is also unbiased at every n (the sampling distribution of the median is symmetric about \mu). So it is a valid, if less efficient, estimator of the mean. Skewed distribution (e.g. lognormal income). Now m 
e \mu: the mean is pulled toward the long tail wh…

Is the sample median a good estimator of the mean?

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