It is well known that the sample $p$-quantile is asymptotically normally distributed. This paper presents an estimate for the accuracy of the normal approximation. The special intention of this paper is to obtain favorable bounds for moderate samples.
"On the Accuracy of the Normal Approximation for Quantiles." Ann. Probab. 2 (4) 741 - 744, August, 1974. https://doi.org/10.1214/aop/1176996617