Postprint version. Published in Statistica Neerlandica, Volume 47, Issue 4, December 1, 1993, pages 279-283.
Copyright © 1993 Blackwell Publishing. The definitive version is available at http://dx.doi.org/10.1111/j.1467-9574.1993.tb01424.x.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
Given a random variable X with finite mean, for each 0 < p < 1, a new sharp bound is found on the distance between a p-quantile of X and its mean in terms of the central absolute first moment of X. The new bounds strengthen the fact that the mean of X is within one standard deviation of any of its medians, as well as a recent quantile-generalization of this fact by O'Cinneide.