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. variation on the standard-deviation of simple sampling is quite small, for, as calculated from equation (4), #=—(18 x 982 — 900) s=130/n/n as compared with 133/s/n. 13. We have finally to pass to the third condition (c) of § 8, Chap. XIIL, and to discuss the effect of a certain amount of dependence between the several “events” in each sample. We shall suppose, however, that the two other conditions (a) and (0) are fulfilled, the chances p and ¢ being the same for every event at every trial, and constant throughout the experiment. The problem is again most simply treated on the lines of § 5 of the last chapter. The standard-deviation for each event is (pg)! as before, but the events are no longer independent: instead, therefore, of the simple expression 0? =n.pg, we must have (cf. Chap. XL. § 2) o2=npq+2pq(rg +r t «oo Togt ooo) where, 7,4, 7,4, etc. are the correlations between the results of the first and second, first and third events, and so on—correlations for variables (number of successes) which can only take the values 0 and 1, but may nevertheless, of course, be treated as ordinary variables (¢f. Chap. XI. § 10). There are n(n —1)/2 correlation-coefficients, and if, therefore, 7 is the arithmetic mean of the correlations we may write a? =mnpg[l +7(n—-1)]. ; . (Bb) The standard-deviation of simple sampling will therefore be increased or diminished according as the average correlation between the results of the single events is positive or negative, and the effect may be considerable, as o may be reduced to zero or increased to m(pg)t. For the standard deviation of the propor- tion of successes in each sample we have the equation s2 = +r(n-1)] . (8) It should be noted that, as the means and standard-deviations for our variables are all identical, » is the correlation-coefficient for a table formed by taking all possible pairs of results in the n events of each sample. 286

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