1 The Halving Method for Sample Quartiles * ANWAR H. JOARDER Dept of Mathematical Sciences, King Fahd University of Petroleum and Minerals Dhahran, Saudi Arabia 31261, Email: anwarj@kfupm.edu.sa An attempt is made to put the notion of sample quartiles on a mathematical footing in the light of ranks of observations, and equisegmentation property that the number of ranks below that of the first quartile, that between the consecutive quartiles, and that above the third quartile are the same. Ranks of sample quartiles provided by the proposed Halving Method, based on hinges, does satisfy the property. 1. Introduction There are many methods available for calculating sample quartiles in different elementary text books on statistics without any explanation. The most popular one, called Popular Method hereinafter, is described here. The rank of the ) 3 , 2 , 1 ( = i i th quartile is given by 3 , 2 , 1 i , 1)/4 ( = + = + d l n i (1.1) where l is the largest integer not exceeding 4 / ) 1 ( + n i . Then the Popular Method uses the following linear interpolation formula for the calculation of sample quartiles ) 1 ( ) ( ) ( ) 1 ( ) ( ) 1 ( ) ( + + + − = − + = l l l l l i x d x d x x d x Q , ) 3 , 2 , 1 i ( = , (1.2) where ) ( l x is the l -th ordered observation ( Ostle, Turner, Hicks and McElrath, 1996, 38). However, students and instructors alike are curious to know why the formulae for quartiles in (1.1) contain the quantity 1 + n . Why not n or ? 1 − n Though the formulae for the median in the literature appear to be different, they all are equivalent. It is given by 2 / ) 1 ( 2 + = n Q th observation. In case n is odd, 2 / ) 1 ( + n will be an integer so that the median will be an observation with integer rank. If however, n is even, 2 / ) 1 ( + n will lie between 2 / n and 1 2 / + n . Then the median can be calculated by the use of linear interpolation. Because of the success of the quantity ) 1 ( + n in equation (1.1) to find the median, the idea of proportional weight given by (1.1) or (1.2) has possibly been popular to find other quartiles by the above method. * To be Published in International Journal of Mathematical Education in Science and Education, 2003, London, UK]