We consider estimation of a quantile based on random sample from a discrete distribution. We derive small and large sample properties ofthe sample quantile, and consider two different types of confidence sets for a population quantile. Either type of a confidence set has the feature that its nominal level is conservative; we propose to enlarge the confidence in this set by examining the data. We also propose aheuristic procedure for obtaining a singleton confidence set. The obtained results are quite different from the well-known results on the estimation of a quantile from a continuous distribution.