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

Ea THEORY OF STATISTICS. conditious under which, samples are drawn, or that some essential change has taken place during the period of sampling. We may represent such circumstances in a case of artificial chance by supposing that for the first f; throws of = dice the chance of success for each die is p;, for the next f, throws p,, for the next f, throws pg, and so on, the chance of success varying from time to time, just as the chance of death, even for individuals of the same age and sex, varies from district to district. Suppose, now, that the records of all these throws are pooled together. The mean number of successes per throw of the n dice is given by n MU = (apy +1oPy + 13Ps + ltl ge ) = 1.0, where V=23(f) is the whole number of throws and p, is the mean value 2(fp)/N of the varying chance p. To find the standard- deviation of the number of successes at each throw consider that the first set of throws contributes to the sum of the squares of deviations an amount Alnengy + ney =o’) n.p,q, being the square of the standard-deviation for these throws, and n(p, -p,) the difference between the mean number of successes for the first set and the mean for all the sets together. Hence the standard-deviation o of the whole distribution is given by the sum of all quantities like the above, or No? =n2(fpqg) + n* 2f(p — Py) Let o, be the standard-deviation of p, then the last sum is N.n2, and substituting 1 — p for ¢, we have ol = np, — np; — no> + Noy = npg + n(n —1)as . . + AD) This is the formula corresponding to equation (1) of Chap. XIII : if we deal with the standard-deviation of the proportion of successes, instead of that of the absolute number, we have, dividing through by 7? the formula corresponding to equation (2) of Chap. XIIL., viz.— e Loto ol Bn + Sa : (2) 10. If » be large and s, be the standard-deviation calculated from the mean proportion of successes p), equation (2) is sensibly of the form S2= 5 = oo 789

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