Weight of mammal (normal, µ=1000 σ=200)
The sample median and mean are both estimates of µ in a normal population — it is both the population mean and median. The diagrams on this page make the point that the SE of the mean is lower so it is a better estimator.
(N.B. Although this is true for a normal popn, it is not always true. The general point is that the SE is a good way to compare estimators.)
Simulation
Take about 150 samples (hold down Take sample.)
Theory (approx)
This diagram shows the theoretical distns for samples of different sizes. The SE of the sample median is always about 25% higher than that of the sample mean.