Sec. 2.5 CALCULATION OF fi AND <r FROM TABLES 31 



It can be seen in Tables 2.51 and 2.52 that when d — for the 

 class with the largest frequency (/) the resulting arithmetic involves 

 smaller numbers than for the other methods. It should be noted, 

 again, that all three of the methods illustrated give exactly the same 

 answers; the only differences lie in the ease of computation. 



TABLE 2.52 



Illustration of a Simplified Method for Computing /jl and <r from a 

 Frequency Distribution Table with Equal Class Intervals 



H = (z for class with d = 0) + ~^ (7) = 12.5 + (l/25)(5) = 12.7. 



a = {1) hjf.d*) - [S(/-d)]V gq) = (5) /27- 1 a)V25 , 51Q 

 V S(/) V 25 



The derivation of the formula shown for a- is more difficult than 

 that for (x, as might be expected, but it can be obtained by elementary 

 algebra, formula 2.13, and by expressing each z in terms of the one 

 for which d = 0. This derivation will be left as an exercise for the 

 ambitious student. 



The methods just described can be applied to obtain satisfactory 

 approximations to the arithmetic mean and the standard deviation 

 of the ACE scores in Table 2.01, a task which clearly would be quite 

 laborious if formulas 2.13 and 2.11 were to be employed directly on 

 the 1290 numbers in that table. The d are taken as zero for the class 

 with a frequency of 277 (Table 2.42) instead of the class with / = 278 

 because they are essentially the same size and the distribution is a 

 bit non-symmetrical (or skewed) in the direction of the lower ACE 

 scores. The general result is to have smaller d's with the larger fre- 



