Summary: We consider the problem of estimating the \(p\)-quantile for a given functional evaluated on solutions of a deterministic model in which model input is subject to stochastic variation. We derive upper and lower bounding estimators of the \(p\)-quantile. We perform an a posteriori error analysis for the \(p\)-quantile estimators that takes into account the effects of both the stochastic sampling error and the deterministic numerical solution error and yields a computational error bound for the estimators. We also analyze the asymptotic convergence properties of the \(p\)-quantile estimator bounds in the limit of large sample size and decreasing numerical error and describe algorithms for computing an estimator of the \(p\)-quantile with a desired accuracy in a computationally efficient fashion. One algorithm exploits the fact that the accuracy of only a subset of sample values significantly affects the accuracy of a \(p\)-quantile estimator resulting in a significant gain in computational efficiency. We conclude with a number of numerical examples, including an application to Darcy flow in porous media.