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 II. The theory of variables

Collection:: Economics Books

THEORY OF STATISTICS. if d denote, as in Chap. IX., the deviation of the mean of an array of X’s from the line of regression, we have by the relation of Chap. IX,, § 11, p. 172 a. (1 LL, 72) =, + 02. (6) Substituting for o,, from (2), that is, of =o =v . (7) But ¢, is necessarily positive, and therefore 7,, is not less than 7. The magnitude of o, and therefore of %2—72 measures the divergence of the actual line through the means of arrays from the line of regression. It should be noted that, owing to the fluctuations of sampling, r and 7 are almost certain to differ slightly, even though the regression may be truly linear. The observed value of #%- 7? must be compared with the values that may arise owing to fluctuations of sampling alone, before a definite significance can be ascribed to it (cf. Pearson, ref. 19, Blakeman, ref. 22, and the formule cited therefrom on p. 352 below). 22. The following table illustrates the form of the arithmetic for the calculation of the correlation-ratio of son’s stature on father’s stature (Table III. of Chap. IX. p. 160). In the first column is given the type of the array (stature of father); in the second, the mean stature of sons for that array; in the third, the difference of the mean of the array from the mean stature of all sons. In the fourth column these differences are squared, and in the sixth they are multiplied by the frequency of the array, two decimal places only having been retained as sufficient for the present purpose. The sum-total of the last column divided by the number of observations (1078) gives o,,,2 = 2058, or ,,, = 143. As the standard-deviation of the sons’ stature is 2:75 in. (cf. Chap. IX., question 3), 7,,=052. Before taking the differences for the third column of such a table, it is as well to check the means of the arrays by recalculating from them the mean of the whole distribution, ¢.e. multiplying each array-mean by its fre- quency, summing, and dividing by the number of observations. The form of the arithmetic may be varied, if desired, by working from zero as origin, instead of taking differences from the true mean. The square of the mean must then be subtracted from 2(f-m2)/N to give o,,,2% If the second correlation-ratio for this table be worked outrin the same way, the value will be found to be the same to the second place of decimals: the two correlation-ratios for this table are, therefore, very nearly identical, and only slightly greater than the correlation-coefficient (0:51). Both regressions, it 206

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