A Study on Computing Sample Quantiles of Continuous Distributions

When sample data is given, we estimate the population quantiles by computing the quantiles of the data. In statistics textbooks there are two methods of computing sample quantiles. Method 1 uses a single order statistic or a simple mean of two adjacent order statistics as a sample quantile, and method 2 uses a weighted mean of two adjacent order statistics as a sample quantile. In this paper, we compared these two methods for the case of several continuous distributions by simulation. We considered some cases of uniform, normal, exponential, chi-square, and beta distributions, and for each distribution we performed 10,000 times of simulation of drawing a random sample of sizes 20, 30 and 50. In each time of simulation, we computed the difference between the population quantile and the sample quantile obtained by each method. We compared the two methods by two criteria: one is the mean square error over 10,000 times of simulation, and the other is the frequency of giving smaller difference from the population quantile than the competing method. As a result of simulation, method 1 turned out to be much more frequently superior to method 2 by criterion 2.