  For $J \ge 2$ independent groups, the paper deals with testing the global
hypothesis that all $J$ groups have a common population median or identical
quantiles, with an emphasis on the quartiles. Classic rank-based methods are
sometimes suggested for comparing medians, but it is well known that under
general conditions they do not adequately address this goal. Extant methods
based on the usual sample median are unsatisfactory when there are tied values
except for the special case $J=2$. A variation of the percentile bootstrap used
in conjunction with the Harrell--Davis quantile estimator performs well in
simulations. The method is illustrated with data from the Well Elderly 2 study.
