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

344 THEORY OF STATISTICS. to test whether we may treat the distribution as approximately normal (cf. also § 16 below). (As regards the theory of sampling for the median and per- centiles generally, cf. ref. 15, Laplace, Supplement II. (standard error of the median), Edgeworth, refs. 5, 6, 7, and Sheppard, ref. 27: the preceding sections have been based on the work of Edgeworth and Sheppard.) 10. Standard Error of the Arithmetic Mean.—Let us now pass to a fresh problem, and determine the standard error of the arithmetic mean. This is very readily obtained. Suppose we note separately at each drawing the value recorded on the first, second, third . . . . and nth card of our sample. The standard-deviation of the values on each separate card will tend in the long run to be the same, and identical with the standard-deviation o of « in an indefinitely large sample, drawn under the same conditions. Further, the value recorded on each card is (as we assume) uncorrelated with that on every other. The standard-deviation of the sum of the values recorded on the nm cards is therefore a/n.oc, and the standard-deviation of the mean of the sample is consequently 1/nth of this; or, o On ="In . (5) This is a most important and frequently cited formula, and the student should note that it has been obtained without any reference to the size of the sample or to the form of the frequency- distribution. It is therefore of perfectly general application, if oc be known. We can verify it against our formula for the standard-deviation of sampling in the case of attributes. The standard-deviation of the number of successes in a sample of m observations is a/m.pg: the standard-deviation of the total number of successes in n samples of m observations each is there- fore a/mm.pq: dividing by n we have the standard-deviation of the mean number of successes in the = samples, viz. mpg [n/n agreeing with equation (5). 11. For a normal curve the standard error of the mean is to the standard error of the median approximately as 100 to 125 (¢f. § 4), and in general the standard errors of the two stand in a somewhat similar ratio for a distribution not differing largely from the normal form. For the distribution of statures used as an illustration in § 6 the standard error of the median was found to be 0:0349: the standard error of the mean is only 0-0277. The distribution being very approximately normal, the ratio of

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