An Introduction to the theory of statistics

Bibliographic data

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 II. The theory of variables

Collection:: Economics Books

X.—CORRELATION : ILLUSTRATIONS AND METHODS. 205 In suzh cases it would always tend to give a value for r markedly higher than that given by the product-sum method. The product-sum method gives in fact a value based on the standard- deviation round the line of regression; the method used above gives a value dependent on the standard-deviation round a line which sweeps through all the means of arrays, and the second standard-deviation is necessarily less than the first. We reach, therefore, a generalised coefficient which measures the approach towards a curvilinear line of regression of any form. Let s,, denote the standard-deviation of any array of X’s, and let , as before, be the number of observations in this array (Chap. IX., § 11), and further let Cl=3(n-5. 0/8 . <A) Then o,, is an average of the standard-deviations of the arrays obtained as suggested at the end of the last section. Now let ru = 51 = 1.) _ 2_ 1% 3 Noy" Fi (3) O, Then 7,, is termed by Professor Pearson a correlation-ratio (ref. 18). As there are clearly two correlation-ratios for any one table, it should be distinguished as the correlation-ratio of X on Y: it measures the approach of values of X associated with given values of Y to a single-valued relationship of any form. The calculation would be exceedingly laborious if we had actually to evaluate o,, but this may be avoided and the work greatly simplified by the following consideration. If JZ, denote the mean of all Xs, m, the mean of an array, then we have by the general relation given in § 11 of Chap. VIII (p. 142) No? =3n(s.* + [M, — m,]2). Or, using o,,. to denote the standard-deviation of m, , ok = Coe + Tid (4) Hence, substituting in (3) a = Tn 6 Nzy ee (5) The correlation-ratio of X on ¥ is therefore determined when we have found, in addition to the standard-deviation of X, the standard-deviation of the means of its arrays. 21. The correlation-ratio of X on ¥ cannot be less than the correlation-coefficient for X and Y, and 5,2 -7? is a measure of the divergence of the regression of X on Y from linearity. For or

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