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

345 THEORY OF STATISTICS. standard-deviation, and so on, or if we draw the successive samples from essentially different parts of the same record, the standard error will be greatly increased. For suppose we draw k, samples from the first record, for which the standard-deviation (in an indefinitely large sample) is oy, and the mean differs by d, from the mean of all the records together (as ascertained by large samples in numbers proportionate to those now taken); &, samples from the second record, for which the standard-deviation is o,, and the mean differs by d, from the mean of all the records together, and so on. Then for the samples drawn from the first record the standard error of the mean will be o/,/n, but the distribution will centre round a value differing by d; from the mean for all the records together: and so on for the samples drawn from the other records. Hence, if 0, be the standard error of the mean, XV the total number of samples, Ji Nat, = (iD) + 3k). But the standard-deviation o, for all the records together is given by N.o2 = 2(ka®) + Z(kd?). Hence, writing 2(kd?) = N.s;,, oh mir iol (9) n n This equation corresponds precisely to equation (2) of § 9, Chap. XIV. The standard error of the mean, if our samples are drawn from different records or from essentially different parts of the entire record, may be increased indefinitely as compared with the value it would have in the case of simple sampling. If, for example, we take the statures of samples of # men in a number of different districts of England, and the standard-deviation of all the statures observed is o,, the standard-deviation of the means for the different districts will not be a,/x/n, but will have some greater value, dependent on the real variation in mean stature from district to district. (b) If we are drawing from the same record throughout, but always draw the first card from one part of that record, the second card from another part, and so on, and these parts differ more or less, the standard error of the mean will be decreased. For if, in large samples drawn from the subsidiary parts of the record from which the several cards are taken, the standard- deviations are oy, 0, . . . . On and the means differ by d;, dos LK

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