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

124 THEORY OF STATISTICS. geometric mean is always determinate and is rigidly defined. The computation is a little long, owing to the necessity of taking logarithms: it is hardly necessary to give an example, as the method is simply that of finding the arithmetic mean of the logarithms of X (instead of the values of X) in accordance with equation (11). If there are many observations, a table should be drawn up giving the frequency-distribution of log X, and the mean should be calculated as in Examples i. and ii. of § 9 and 10. The geometric mean has never come into general use as a repre- sentative average, partly, no doubt, on account of its rather troublesome computation, but principally on account of its some- what abstract mathematical character (cf. § 4 (c)): the geometric mean does not possess any simple and obvious properties which render its general nature readily comprehensible. 23. At the same time, as the following examples show, the mean possesses some important properties, and is readily treated algebraically in certain cases. (a) If the series of observations X consist of » component series, there being IV, observations in the first, &V, in the second, and so on, the geometric mean G of the whole series can be readily expressed in terms of the geometric means @;, G@,, etc., of the component series. For evidently we have at once (as in § 13 (®))— N.logG=0N,log G+ Ny, logG+ .... +N, log@,. . (12) (6) The geometric mean of the ratios of corresponding observa- tions in two series is equal to the ratio of their geometric means. For if X= 1/ Xo, log X =log X; —log X,, then summing for all pairs of X;’s and X's, G=0G,/G, : --9(13) (c) Similarly, if a variable X is given as the product of any number of others, z.e. if X= Xa vid. 20, xX, X, .... JX, denoting corresponding observations in 7 different series, the geometric mean G' of X is expressed in terms of the geometric means @;, Gi. =. 0. Gof X,, X,, . ..*. X,, by the relation = 0, CN 0, . (14) That is to say, the geometric mean of the product is the product of the geometric means. sg

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