Journal of Probability and Statistical Science 2(1), 95-109, Feb. 2004 A Comparison and Contrast of Some Methods for Sample Quartiles Anwar H. Joarder and Raja M. Latif King Fahd University of Petroleum and Minerals ABSTRACT A remainder representation of the sample size n = 4m + r (r = 0, 1, 2, 3) is exploited to write out the ranks of quartiles exhaustively which in turn help compare ranks for quartiles provided by different methods available in the literature. The criterion of the equisegmentation property that the number of integer ranks below the first quartile, that between the consecutive quartiles, and that above the third quartile are the same, has been used to compare and contrast different methods. Four segmentation identities can be obtained for each method of quartiles which show clearly the number of observations in each of the four quarters if the observations are distinct. The Halving Method and the Remainder Method have been proposed for the calculation of sample quartiles. The quartiles provided by each of these two methods satisfy the equi- segmentation property if the observations are distinct. More interestingly, in these two methods r also represents the number of quartiles having integer ranks. Keywords Quartiles; Remainders; Modulus; Quantiles. 1. Introduction Quartiles, deciles, percentiles or more generally fractiles are uniquely determined for continuous random variables. A p th quantile of a random variable X (continuous or discrete) is a value such that p x p x X P p ≤ < ) ( and p x X P p ≥ ≤ ) ( . Let X be a continuous or discrete random variable with probability function and the cumulative distribution function If the distribution is continuous, then nd since . Therefore, for the continuous case, ) ( x f () Fx p = ( PX = ) ( P ). x ≤ 0 ) = = p x X p x X P p = < ) p x F p = ) ( . ( a x X P p ≤ ( The quartiles , 50 . 0 2 25 . 0 1 x Q x Q = = and 75 . 0 3 x Q = for a continuous random variable with cumulative distribution function F are defined by ) ( x , 25 . 0 ) ( 25 . 0 = x F and 50 . 0 ) ( 50 . 0 = x F _________________________ Received August 2003, revised October 2003, in final form November 2003. Anwar H. Joarder is an Associate Professor and Raja M. Latif is an Assistant Professor in the Department of Mathematical Sciences at King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia 31261; emails: anwarj@kfupm.edu.sa and raja@kfupm.edu.sa . © 2003 Susan Rivers’ Cultural Institute, Hsinchu, Taiwan, Republic of China. ISSN 1726-3328