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Let $X_1,\cdots,X_n\overset{iid}{\sim} F_X(x)$ be a random sample from a symmetric distribution with a defined mean. If need be, assume that $n$ is odd and that $F_X(x)$ is continuous.

Is it always the case that the empirical median is an unbiased estimator for the expected value?

Simulations suggest this is the case, and it certainly fits with my intuition, but then I go to prove it, remember the equation for the PDF of an order statistic, and do something else with my life. Perhaps I am missing a simple argument based on symmetry.

This seems to show that what I posit is true in the limit. What about in a finite sample?

Dave
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  • Maybe [this](https://math.stackexchange.com/questions/119414/prove-that-the-sample-median-is-an-unbiased-estimator) thread is helpful. – COOLSerdash Jan 12 '22 at 12:10
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    The distribution of the sample median must also be symmetric, whence the conclusion follows immediately. – whuber Jan 12 '22 at 16:33

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