L—NOTATION AND TERMINOLOGY. 13
The ultimate frequencies form one natural set in terms of which
the data are completely given, but any other set containing the
same number of algebraically independent frequencies, viz. 27
may be chosen instead.
15. The positive class-frequencies, including under this head the
total number of observations &, form one such set. They are algebraically
independent ; no one positive class-frequency can be expressed
wholly in terms of the others. Their number is, moreover,
2", as may be readily seen from the fact that if the Greek letters
are struck out of the symbols for the ultimate classes, they become
the symbols for the positive classes, with the exception of afy
. +. . for which # must be substituted. Otherwise the number
is made up as follows :—
Order 0. (The whole number of observations) . : 1
Order 1. (The number of attributes noted) . : n
Order 2. (The number of combinations of n things 2 together) ph
Order 3. (The number of combinations of n things 3 together) aol) Fed
and so on. But the series
n(n—-1) n(n-1)(n-2)
l+n+ 1.9 {55 Th eieite
is the binomial expansion of (141) or 2", therefore the total
number of positive classes is 2".
16. The set of positive class-frequencies is a most convenient
one for both theoretical and practical purposes.
Compare, for instance, the two forms of statement, in terms of
the ultimate and the positive classes respectively, as given in
Example i,, § 13. The latter gives directly the whole number of
observations and the totals of 4’s, B’s, and (’s. The former gives
none of these fundamentally important figures without the performance
of more or less lengthy additions. Further, the latter gives
the second-order frequencies (4B), (4C), and (BC), which are necessary
for discussing the relations subsisting between 4, B, and C, but
are only indirectly given by the frequencies of the ultimate classes.
17. The expression of any class-frequency in terms of the
positive frequencies is most easily obtained by a process of stepby-step
substitution ; thus—
(@B) =(a)- (aB)
=N-(4)-(B)+(4B) . ?)
(afy) = (ap) - (aBC)
=N - “4 - (B) + (4B) - (aC) + (a BC)
=X (4) -(B)-(C) + (4B) + (AC) + (BC) = (4BC) (4)
(c
