36 SUMMARIZATION OF DATA Ch. 2 



immediately clear, (a) When N is not a multiple of four, we cannot 

 define four groups each containing one-fourth of N measurements; 

 and (b) repetitions of numbers will pose a problem in some instances 

 because numbers of equal size logically must be in the same subgroup, 

 and yet to put them there sometimes will cause one subgroup to con- 

 tain more than its stated proportion of all the measurements. 



It will be convenient first to describe the method to be used to 

 determine percentile limits because deciles and quartiles can be de- 

 fined in terms of percentiles. The general aim in defining percentiles 

 is to divide the ordered array into 100 subgroups, each of which con- 

 tains one per cent of the numbers in the set, as nearly as this is pos- 

 sible. This result will be accomplished by defining the upper limit 



V 



of the pth percentile to be the 



th number in that 



(N+ 1) 

 100 

 array. For example, if N -- 1290, as in Table 2.01, the upper limit 



of the ninetieth percentile is the [^ (1291) ]th, or the 1161.9th, 

 number in the array or along its scale of measurement. Such an 

 "ordinal" number as 1161.9 will be defined to be the number which 

 is nine-tenths of the way between the 1161st and the 1162nd numbers 

 from the bottom of the array. It is seen from Table 2.42 that there 

 are 1169 numbers less than or equal to 129. With this information 

 it is found that the 1161st and the 1162nd scores in order of size are 

 128 and 129, respectively. Hence, the 1161.9th number along the 

 scale of the ACE scores is 128.9, which, then, is the upper limit of 

 the ninetieth percentile. The lower limit of this percentile is just the 

 upper limit of the eighty-ninth percentile. By definition, this is the 

 [89(1291)/100]th number along the array of the ACE scores. Since 

 89(1291)/100 = 1148.99, the lower limit of the ninetieth percentile 

 is a number which is .99 of the way between the 1148th and 1149th 

 scores from the bottom of the array. The 1148th score is 127, whereas 

 the 1149th score is 128; hence the lower limit of the ninetieth per- 

 centile is 127.99. It follows then that the ninetieth percentile con- 

 tains scores of 128 only. Actual enumeration discloses that there 

 are 13 scores of 128, which is as close to one per cent of 1290 as is 

 possible with integers. Such close agreement with the ideal will not 

 be attained with most of the percentiles, especially in the neighbor- 

 hood of the mean and the median, because there will be repetitions 

 of scores which will cover more than one percentile. It may be better 

 when much of this occurs to be content with the coarser subgroups 

 given by deciles or even quartiles. 



