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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Chapter

Document type:: Monograph

Structure type:: Chapter

Title:: Part III. Theory of sampling

Collection:: Economics Books

273 THEORY OF STATISTICS. form a correlation-table between the true proportion p in a given universe and the observed proportion = in a sample of n observa- tions drawn therefrom. What we have found from the work of the last chapter is that the standard-deviation of an array of =’s associated with a certain true value p, in this table, is (pg/n)t; but the question may be asked —What is the standard-deviation of the array at right angles to this, <.e. the array of p’s associated with a certain observed proportion =? In other words, given an observed proportion w, what is the standard-deviation of the true proportions? This is the inverse of the problem with which we have been dealing, and it is a much more difficult problem. On general principles, however, we can see that if n be large, the two standard-deviations will tend, on the average of all values of p, to be nearly the same, while if » be small the standard- deviation of the array of =’s will tend to be appreciably the greater of the two. For if #=p +38, 8 is uncorrelated with p, and therefore if o, be the standard-deviation of p in all the universes from which samples are drawn, o, the standard- deviation of observed proportions in the samples, and os the standard-deviation of the differences, ol =o} +0}. But o} varies inversely as ». Hence if » become very large, os becomes very small, o, becomes sensibly equal to a, and therefore the standard-deviations of the arrays, on an average, are also sensibly equal. If n be large, therefore, [m(l—=)/n]} may be taken as giving, with sufficient exactness, the standard-deviation of the true proportion p for a given observed proportion 7. But if » be small, os cannot be neglected in comparison with a, oy, i8 therefore appreciably greater than a, and the standard-deviation of the array of #’s is, on an average of all arrays, correspondingly greater than the standard deviation of the array of p’s—the state- ment is not true for every pair of corresponding arrays, especially for extreme values of p near 0 and 1. Further, it should be noticed that, while the regression of = on p is unity—a.e. the mean of the array of ’s is identical with p, the type of the array—the regression of p on = is less than unity. If we as- sume, therefore, that a tabulation of all possible chances, observed for every conceivable subject, would give a distribution of p ranging uniformly between 0 and 1, or indeed grouped symmetri- cally in any way round 05, any observed value 7 greater than 0-5 will probably correspond to a true value of p slightly lower than , and conversely. We have already referred to the use of the inverse standard error in § 13 of Chap. XIII. (Case IL, p. 269). If we determine, for example, the standard error of the difference Tw

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