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

VIIL.—MEASURES OF DISPERSION, ETC. 143 Again, as in § 13 of Chap. VIL, it is convenient to note, for the checking of arithmetic, that if the same arbitrary origin be used for the calculation of the standard deviations in a number of component distributions we must have (LE) =2NED +3) + LL +3(ED. (8) 12. As another useful illustration, let us find the standard deviation of the first #7 natural numbers, The mean in this case is evidently (¥+1)/2. Further, as is shown in any elementary Algebra, the sum of the squares of the first VV natural numbers is NHN +1)(2N +1) La 2 4 The standard deviation o is therefore given by the equation— C=F(NV+1)2N+1)-L(N +1) that is, ol=7(N2-1) . (9) This result is of service if the relative merit of, or the relative intensity of some character in, the different individuals of a series is recorded not by means of measurements, e.g. marks awarded on some system of examination, but merely by means of their respective positions when ranked in order as regards the character, in the same way as boys are numbered in a class. With & individuals there are always N ranks, as they are termed, whatever the character, and the standard deviation is therefore always that given by equation (9). Another useful result follows at once from equation (9), namely, the standard deviation of a frequency-distribution in which all values of X within a range +7/2 on either side of the mean are equally frequent, values outside these limits not occurring, so that the frequency-distribution may be represented by a rectangle. The base / may be supposed divided into a very large number & of equal elements, and the standard deviation reduces to that of the first NV natural numbers when & is made indefinitely large. The single unit then becomes negligible compared with J. and consequently 12 l= 12 : . (10) 13. Tt will be seen from the preceding paragraphs that the standard deviation possesses the majority at least of the properties which are desirable in a measure of dispersion as in an average (Chap. VIL § 4). It is rigidly defined ; it is based on all the observations made ; it is calculated with reasonable ease ; 1t lends itself readily to algebraical treatment ; and we may add, though the student will have to take the statement on trust for the present, that it is, as a rule, the measure least affected by fluctuations of

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