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 II. The theory of variables

Collection:: Economics Books

142 THEORY OF STATISTICS. give a more definite and concrete meaning to the standard deviation, and also to check arithmetical work to some extent— sufficiently, that is to say, to guard against very gross blunders. It must not be expected to hold for short series of observations : in Example i., for instance, the actual range is a good deal less than six times the standard deviation. 11. The standard deviation is the measure of dispersion which it is most easy to treat by algebraical methods, resembling in this respect the arithmetic mean amongst measures of position. The majority of illustrations of its treatment must be postponed to a later stage (Chap. XI.), but the work of § 3 has already served as one example, and we may take another by continuing the work of § 13 (0), Chap. VII. In that section it was shown that if a series of observations of which the mean is M/ consist of two component series, of which the means are J; and J/, respectively, NM=N.M +N, M, XN; and XN, being the numbers of observations in the two com- ponent series, and N=, +, the number in the entire series. Similarly, the standard deviation o of the whole series may be expressed in terms of the standard deviations o; and o, of the components and their respective means. Let M-M=d, M,- M=d, Then the mean-square deviations of the component series about the mean JM are, by equation (4), 32 +d;% and o,%+d,? respec: tively. Therefore, for the whole series, N.o%=N (2 +d2) + Noa 2 +45) o =D) If the numbers of observations in the component series be equal and the means be coincident, we have as a special case— 0? = (oy? + 07?) 6) go that in this case the square of the standard deviation of the whole series is the arithmetic mean of the squares of the standard deviations of its components. It is evident that the form of the relation (5) is quite general : if a series of observations consists of » component series with standard deviations oy, oy, . . . 0, and means diverging from the general mean of the whole series by d,, dy, . . . d,, the standard deviation o of the whole series is given (using m to denote any subscript) by the equation— N.o?=3(N,.0,% + 2 N,.d,2%) (7) is

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