An Introduction to the theory of statistics

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Monograph

Identifikator:: 1751730271

URN:: urn:nbn:de:zbw-retromon-127610

Document type:: Monograph

Author:: Yule, George Udny http://d-nb.info/gnd/12910504X

Title:: An Introduction to the theory of statistics

Edition:: 8. ed. rev

Place of publication:: London

Publisher:: Griffin

Year of publication:: 1927

Scope:: XV, 422 S; Ill., Diagr

Digitisation:: 2021

Collection:: Economics Books

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Document type:: Monograph

Structure type:: Chapter

Title:: Part I. The theory of atributes

Collection:: Economics Books

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 alge- braically independent ; no one positive class-frequency can be ex- pressed 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 perfor- mance of more or less lengthy additions. Further, the latter gives the second-order frequencies (4B), (4C), and (BC), which are neces- sary 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 step- by-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

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