Sec. 2.6 PERCENTILES, DECILES, AND QUARTILES 35 



2.6 PERCENTILES, DECILES, AND QUARTILES 



The standard deviation about the arithmetic mean, the range, the 

 average deviation, and the comparative magnitudes of the median 

 and the mean (all presented earlier) provide useful information re- 

 garding the dispersion of the numerical measurements in a group of 

 data which is being analyzed. However, there are some circum- 

 stances in which it is desirable to divide the ordered array into seg- 

 ments each containing a stated percentage of all the numbers in the 

 set. More specifically, it may be convenient to partition a large 

 body of data into four, ten, or one hundred subgroups, each contain- 

 ing approximately the same number of measurements from the set, 

 and with the subgroups corresponding to successive segments of the 

 array. The subgroups will be called quartiles if four divisions are 

 employed, deciles if there are ten subgroups, and percentiles if there 

 are one hundred subgroups.* The aim in stating the upper limit of 

 the first quartile, for example, is to designate a number such that 

 one-fourth the numbers in the array are less than or equal to that 

 upper limit. 



Although the upper and the lower limits of the quartiles, deciles, 

 and percentiles could be read from a carefully drawn r.c.f. curve if 

 the data are sufficiently numerous, it is desirable to have precise 

 definitions for them. This could be done in a variety of ways, not 

 essentially different, so that certain convenient and reasonably stand- 

 ard definitions will be adopted rather arbitrarily. 



Before general rules and methods for determining the limits on 

 the quartiles, deciles, and percentiles are considered attention is called 

 to the following two arrays and to some general problems inherent 

 in the determination of such subgroups as quartiles: 



Setl. 1,2,3,4,6,8,8,9,10,10,11,12,15,18,18. N = 15. 



Set 2. 1,2,4,7,9,9,11,11,12,15,15,18,20,24,25,27. N = 16. 



Suppose that we wish to divide these sets of numbers into four sub- 

 groups, each containing equally many numbers, if possible, and com- 

 ing as close as possible to equality in other instances. Two facts are 



* It seems to the author that the term percentile should refer to an interval 

 which includes approximately one per cent of all the measurements. However, 

 most textbooks use this term to designate only one end point of what is called 

 a percentile herein. Similar remarks apply to the terms decile and quartile. 

 Since we usually speak of a score being in a percentile rather than at it, usage 

 seems to support the point of view taken herein. 



