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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Chapter

Document type:: Monograph

Structure type:: Chapter

Title:: Part I. The theory of atributes

Collection:: Economics Books

V.—MANIFOLD CLASSIFICATION. : For suppose that the sign of association in the elementary tetrads is positive, so that— (4 wh ”) (Amir Brtr) > (dn B nD (4,.B nt1) ! (1) and similarly, (An1Bo)(Ami2Busr) > (Amie Bu) (Amir Busr) / i Then multiplying up and cancelling we have (An Bp)(Ami2Bus1) > (Ams Br) (AmB) ” : (3) That is to say, the association is still positive though the two columns 4,, and 4,,,, are no longer adjacent. (2) An wsotropic distribution remains isotropic in whatever way ut may be condensed by grouping together adjacent rows or columns. Thus from (1) and (3) we have, adding— (4B) [(dns1Brsr) HE (Ams2Bns1)] > (AnBri)[(Adms1 Br) ot (4ms2Bn)); that is to say, the sign of the elementary association is unaffected by throwing the (m+ 1)th and (m+ 2)th columns into one. (3) As the extreme case of the preceding theorem, we may suppose both rows and columns grouped and regrouped until only a 2 x 2-fold table is left ; we then have the theorem— If an isotropic distribution be reduced to a fourfold distribution wn any way whatever, by addition of adjacent rows and columns, the sign of the association in such fourfold table is the same as in the elementary tetrads of the original table. The case of complete independence is a special case of isotropy. For if (AnBo) = (An) (BLN for all values of m and =, the association is evidently zero for every tetrad. Therefore the distribution remains independent in whatever way the table be grouped, or in whatever way the universe be limited by the omission of rows or columns. The expression ‘complete independence ” is therefore justified. From the work of the preceding section we may say that Table IL. is not isotropic as it stands, but may be regarded as a dis- arrangement of an isotropic distribution. It is best to rearrange such a table in isotropic order, as otherwise different reductions to fourfold form may lead to associations of different sign, though of course they need not necessarily do so. 12. The following will serve as an illustration of a table that is not isotropic, and cannot be rendered isotropic by any rearrange- ment of the order of rows and columns. 69 oa

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