I'll describe and compare various point estimators for quantiles, including the not commonly used Harrell-Davis estimator, which is superior to more commonly used estimators in a wide variety of situations. Attempts to improve the performance of the Harrell-Davis estimator using bias correction by bootstrapping will be discussed.

Particular attention will be given to estimating distribution medians, for both symmetric and skewed distributions. The use of the Harrell-Davis estimator for tests about the median will also be discussed.