Quantile-locating functions and the distance between the mean and quantiles

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.