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 II. The theory of variables

Collection:: Economics Books

1° THEORY OF STATISTICS. chapter, on all the observations made, so that no single observation can have an unduly preponderant effect on its magnitude ; indeed, the measure should possess all the properties laid down as desir- able for an average in § 4 of Chap. VII. There are three such measures in common use—the standard deviation, the mean deviation, and the quartile deviation or semi-interquartile range, of which the first is the most important. 2. The Standard Deviation.—The standard deviation is the square root of the arithmetic mean of the squares of all deviations, deviations being measured from the arithmetic mean of the observations. If the standard deviation be denoted by o, and a deviation from the arithmetic mean by z, as in the last chapter, then the standard deviation is given by the equation N70 of = 2(27) : : : a (TY To square all the deviations may seem at first sight an artificial procedure, but it must be remembered that it would be useless to take the mere sum of the deviations, in order to obtain a measure of dispersion, since this sum is necessarily zero if deviations be taken from the mean. In order to obtain some quantity that shall vary with the dispersion it is necessary to average the deviations by a process that treats them as if they were all of the same sign, and squaring is the simplest process for eliminating signs which leads to results of algebraical convenience. 3. A quantity analogous to the standard deviation may be defined in more general terms. Let 4 be any arbitrary value of X, and let & (as in Chap. VIL. § 8) denote the deviation of X from 4 ; <.e. let E=X-4. Then we may define the root-mean-square deviation s from the origin 4 by the equation 1 See ro RUE, . 2 = 3(8) (2) In terms of this definition the standard deviation is the root- mean-square deviation from the mean. There is a very simple relation between the standard deviation and the root-mean-square deviation from any other origin. Let M-4=d. ‘3) so that E=x +d. Then £2=0p2 + 2x.d + d?, 2(£2) = 3(«?) + 2d.3(x) + N.dA. 34 \*

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