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 III. Theory of sampling

Collection:: Economics Books

< THEORY OF STATISTICS. reduced, of course, from the original drawing, one of the squares shown representing a square inch on the original. The actual contour lines for the same frequencies are shown by the irregular polygons superposed on the ellipses, the points on these polygons having been obtained by simple graphical interpolation between the frequencies in each row and each column—diagonal interpola- tion between the frequencies in a row and the frequencies in a column not being used. It will be seen that the fit of the two lower contours is, on the whole, fair, especially considering the high standard errors. In the case of the central contour, y= 20, the fit looks very poor to the eye, but if the ellipse be compared carefully with the table, the figures suggest that here again we have only to deal with the effects of fluctuations of sampling. For father’s stature=66 in., son’s stature= 70 in., there is a frequency of 18:75, and an increase in this much less than the standard error would bring the actual contour outside the ellipse. Again, for father’s stature=68 in., son’s stature="71 in., there is a frequency of 19, and an increase of a single unit would give a point on the actual contour below the ellipse. Taking the results as a whole, the fit must be regarded as quite as good as we could expect with such small frequencies. It is perhaps of historical interest to note that Sir Francis Galton, working with- out a knowledge of the theory of normal correlation, suggested that the contour lines of a similar table for the inheritance of stature seemed to be closely represented by a series of concentric and similar ellipses (ref. 2): the suggestion was confirmed when he handed the problem, in abstract terms, to a mathematician, Mr J. D. Hamilton Dickson (ref. 4), asking him to investigate “the Surface of Frequency of Error that would result from these data, and the various shapes and other particulars of its sections that were made by horizontal planes” (ref. 3, p. 102). 12. The normal distribution of frequency for two variables is an isotropic distribution, to which all the theorems of Chap. V. §§ 11-12 apply. For if we isolate the four compartments of the correlation-table common to the rows and columns centring round values of the variables w,, xy , xy we have for the ratio of the cross-products (frequency of #, #, multiplied by frequency of 2, 2), divided by frequency of », 2; multiplied by frequency of x, 7p), 712 ’ ’ ya u)(# =) Assuming that 2; — 2; has been taken of the same sign as x; — x, the exponent is of the same sign as 7, Hence the association for 398

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