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. 285 For the mean number of successes we evidently have M=mp, +m,py + maps +... . =n a, Pp, being the mean chance Z(mp)/n. To find the standard-deviation of the number of successes at each throw, it should be noted that this may be regarded as made up of the number of successes in the m, dice for which the chances are 21 9; together with the number of successes amongst the m, dice for which the chances are p, q,, and so on: and these numbers of successes are all independent. Hence 0 =m P1qy + MoPoy + MgPigs + +o = Z(mpq), Substituting 1-p for ¢, as before, and using o, to denote the standard-deviation of p, ol =npyg, — no; v2.3) or if s be, as before, the standard-deviation of the proportion of successes, _Puls_0y $= Si . 4) 12. The effect of the chances varying for the individual dice or other “events” is therefore to lower the standard-deviation, as calculated from the mean proportion Ppp and the effect may conceivably be considerable. To take a limiting case, if p be zero for half the events and unity for the remainder, Po=9,=3, and o,=3, so that s is zero. To take another illustration, still some- what extreme, if the values of p are uniformly distributed over the whole range between 0 and 1, p,=9,=4 as before but = 1/12=0-0833 (Chap. VIIL § 12, p. 143). “Hence §2=01667/n, s=0408//n, instead of 0-5/n/n, the value of s if the chances are $ in every case. In most practical cases, however, the effect will be much less. Thus the standard-deviation of sampling for a death- rate of, say, 18 ver thousand in a population of uniform age and one sex is (18 x 982)}/s/n=133//n. Ina population of the age composition of that of England and Wales, however, the death- rate is not, of course, uniform, but varies from a high value in infancy (say 150 per thousand), through very low values (2 to 4 per thousand) in childhood to continuously increasing values in old age ; the standard-deviation of the rate within such a popula- tion is roughly about 30 per thousand. But the effect of this

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