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

XIIL.—SIMPLE SAMPLING OF ATTRIBUTES. RN We have therefore M =p, and oi=p-p*=pq. But the number of successes in a group of # such events is the sum of successes for the single events of which it is composed, and, all the events being independent, we have therefore, by the usual rule for the standard-deviation of the sam of independent variables (Chap. XI. § 2, equation (?)), o, being the standard- deviation of the number of successes in z events, oa=npq . : : L(Y) This is an equation of fundamental importance in the theory of sampling. The student should particularly bear in mind that the standard-deviation of the number of successes, due to fluctuations of simple sampling alone, in a group of mn events varies, not directly as n, but as the square root of n. 6. In lieu of recording the absolute number of successes in each sample of n events, we might have recorded the proportion of such successes, 7.e. 1/nth of the number in each sample. As this would amount to merely dividing all the figures of the original record by 7, the mean proportion of successes—or rather the value towards which the mean tends to approach—must be p, and the standard-deviation of the proportion of successes s, be given by S=cifnt=pgin . . . . (2 The standard-deviation of the proportion of successes in samples of such independent events varies therefore inversely as the square root of the number on which the proportion is calculated. Now if we regard the observed proportion in any one sample as a more or less unreliable determination of the true proportion in a very large sample from the same material, the standard-devia- tion of sampling may fairly be taken as a measure of the unreliability of the determination—the greater the standard- deviation, the greater the fluctuations of the observed proportion, although the true proportion is the same throughout. The reciprocal of the standard-deviation (1/s), on the other hand, may be regarded as a measure of reliability, or, as it is sometimes termed, precision, and consequently the reliability or precision of an observed proportion varies as the square root of the number of observations on which it is based. This is again a very important rule with many practical applications, but the limitations of the case to which it applies, and the exact conditions from which it has been deduced, should be borne in mind. We return to this point again below (§ 8 and Chap. XIV.). 7. Experiments in coin tossing, dice throwing, and so forth have been carried out by various persons in order to obtain ex- ne 257

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