Is the sample median an unbiased estimator of the population mean when the distribution is symmetric?
I think so yes, so long as
- The population mean is defined, and
- 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).