Is the sample mean a better point estimate of the population median than the sample median?

A beginner's question to check I've understood correctly. A basic stats textbook says:

"The variance of the sampling distribution of the median is greater than that of the sampling distribution of the mean. It follows that sample mean is likely to be closer to the population mean than the sample median. Therefore, the sample mean is a better point estimate of the population mean than the sample median."

Does it follow that for distributions where the median=mean, the sample mean is a better point estimate of the population median than the sample median?

• Which text? Where did it say this? Did you leave out anything important, like some prior statement of the distribution for which this claim is made? (I imagine it's probably intended to be under i.i.d. sampling from a normal distribution). As a general claim, it is of course false, so if they don't mention that it only applies in some cases, you should probably have doubts about what else you read in that book. The Laplace is a common counterexample to your question at the end there. – Glen_b Mar 18 '16 at 2:21
• – Glen_b Mar 18 '16 at 2:27
• I don't think the text of the question matches the title. – David Lane Apr 5 '17 at 0:50