A class of statistics is considered, which are based on order-statistics and have properties analogous to Student's $t$. They can be used to estimate a required quantile of a random variable or to test hypotheses about a quantile; they are simple to compute, and can be calculated when not all sample values are available, e.g. for censored samples. The limiting distributions of these statistics are derived and shown to be independent of the distribution of the underlying random variable. A numerical tabulation of the limiting distribution is included, for the special case when the quantile considered is the median.