Consider probability distribution which is unimodal, symmetric and mean value exists. (Of course due to symmetry mean will coincide with mode (position of the maximum)).

**Question** Is sample mean always the "best" estimate of the mean or may be median, whatever can be better ?

Situation: assume the "tails" are heavy, (but still mean exists), so "sample-mean" will not be very stable, because big deviations will occur with big probability and hence mean will deviate of the "ideal" value. Median estimate seems to be more stable - if big deviations happens rarely median will not be affected by this.

So may be one can "cook up" something in the middle between mean and median - just filter out values which are at the ends and take average of the others or something like that...