Standard nonparametric prediction intervals for a future order statistic are obtained by taking the interval between two order statistics of the initial sample. We obtain improved prediction intervals by taking the shortest of two or more standard intervals.

From an arbitrary distribution, the variances of odd sized sample medians form a monotonely nonincreasing sequence. The variance of an even sized median, which may even increase with sample size, is always less than or equal to that of the preceeding odd sized median.