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

XIV.—REMOVING LIMITATIONS OF SIMPLE SAMPLING. 287 It should also be noted that the case when r is positive covers the departure from the rules of simple sampling discussed in $ 9-10: for if we draw successive samples from different records, this introduces the positive correlation at once, even although the results of the events at each trial are quite independent of one another. Similarly, the case discussed in §§ 11-12 is covered by the case when 7» is negative : for if the chances are not the same for every event at each trial, and the chance of success for some one event is above the average, the mean chance of success for the remainder must be below it. The cases (a), (6) and (c) are, how- ever, best kept distinct, since a positive or negative correlation may arise for reasons quite different from those discussed in § 9-12. 3 14. As a simple illustration, consider the important case of sampling from a limited universe, e.g. of drawing n balls in succession from the whole number in a bag containing pw white balls and gw black balls. On repeating such drawings a large number of times, we are evidently equally likely to get a white ball or a black ball for the first, second, or nth ball of the sample : the correlation-table formed from all possible pairs of every sample will therefore tend in the long run to give just the same form of distribution as the correlation-table formed from all possible pairs of the w balls in the bag. But from Chap. XI. § 11 we know that the correlation-coefficient for this table is — 1 [(w—1), whence n-1 0? =n.pq(1 oo 1) w—n = 9% 5 If n=1, we have the obviously correct result that o = (p9)}, as in drawing from unlimited material : if, on the other hand, n=w, o becomes zero as it should, and the formula is thus checked for simple cases. For drawing 2 balls out of 4, ¢ becomes 0-816 (npg); for drawing 5 balls out of 10, 0-745 (npg)t; in the case of drawing half the balls out of a very large number, it approxi- mates to (0-5.npq)}, or 0-707 (npg)t. In the case of contagious or infectious diseases, or of certain forms of accident that are apt, if fatal at all, to result in whole- sale deaths, r is positive, and if n be large (as it usually is in such cases) a very small value of » may easily lead to a very great increase in the observed standard-deviation. It is difficult to give a really good example from actual statistics, as the conditions are hardly ever constant from one year to another, but the following will

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