’ 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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