This paper considers the interpolation of percentiles when the value of the c.d.f. F(x) is specified at n points. Upper and lower bounds on the percentile are obtained assuming that data is generated by a unimodal density. Sharper bounds are derived when the average of the observations in each interval is given. In addition to improving the standard linear interpolation, our results indicate that much information can be gained by reporting group means as well as frequency counts.