’ THEORY OF STATISTICS. 
termed the ultimate classes and their frequencies the ultimate 
frequencies. Hence we may say that #t is never necessary to 
enumerate more than the ultimate frequencies. All the others can 
be obtained from these by simple addition. 
Example i.—(See reference 5 at the end of the chapter.) 
A number of school children were examined for the presence 
or absence of certain defects of which three chief descriptions 
were noted, 4 development defects, B nerve signs, C low 
nutrition. 
Given the following ultimate frequencies, find the frequencies 
of the positive classes, including the whole number of obser- 
vations JV. 
(480) 57 (aBC) 78 
(4 By) 281 (aBy) 670 
(480) 86 (aB0) 65 
(48) 453 (By) 8310 
The whole number of observations XN is equal to the grand 
total :  =10,000. 
The frequency of any first-order class, e.g. (4) is given by the 
total of the four third-order frequencies, the class-symbols for 
which contain the same letter— 
(4BC) + (4 By) + (ABC) + (4ABy)= (4) = 871. 
Similarly, the frequency of any second-order class, e.g. (4B), is 
given by the total of the two third-order frequencies, the class- 
symbols for which both contain the same pair of letters— 
(ABC) + (4 By) = (4B) = 338. 
The complete results are— 
N 10,000 AB) 338 
(4) 877 40) 143 
& 1,086 i 135 
0) 286 ABC) 57 
14. The number of ultimate frequencies in the general case of 
n attributes, or the number of classes in an aggregate of the nth 
order, is given by considering that each letter of the class-symbol 
may be written in two ways (4 or a, B or 3, C' or vy), and that 
either way of writing one letter may be combined with either 
way of writing another. Hence the whole number of ways in 
which the class-symbol may be written, z.e. the number of 
classes, is— 
AE SH EC le 
2 
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