An Introduction to the theory of statistics

Bibliographic data

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

Usage license:: Get license information via the feedback formular.

Chapter

Document type:: Monograph

Structure type:: Chapter

Title:: Part II. The theory of variables

Collection:: Economics Books

214 THEORY OF STATISTICS. whole, to be safer, for it eliminates the assumption that the errors in # and in y, in the same series of observations, are uncorrelated. An insufficient though partial test of the correctness of the assumptions may be made by correlating #, — 2, with ¥1—¥,: this correlation should vanish. Evidently, however, it may vanish from symmetry without thereby implying that all the correlations of the errors are zero. 8. Mean and Standard-deviation of an Index.—(Ref.11.) The means and standard-deviations of non-linear functions of two or more variables can in general only be expressed in terms of the means and standard-deviations of the original variables to a first approximation, on the assumption that deviations are small compared with the mean values of the variables. Thus let it be required to find the mean and standard-deviation of a ratio or index Z = X,/X,, in terms of the constants for X, and X,. Let [7 be the mean of Z, M, and J, the means of X; and X,. “Then lin 2) zy ¥ 7-533) rarx(+ 3) Expand the second bracket by the binomial theorem, assuming that »,/M, is so small that powers higher than the second can be neglected. Then to this approximation 1M, 1 1 3 | I== 77 - I) == 7 ) . That is, if r be the correlation between x; and #,, and if v, = o,/M,, vy =0o/M,, . Y I= a! — 70,0, + Vy?) 9) If s be the standard-deviation of Z we have 1_/7X\2 CL Joe Nl] S47 7% 7) 1 M2 ( xy \? z Ne “7A 3 03 Expanding the second bracket again by the binomial theorem, and neglecting terms of all orders above the second, 1.0.2 Z\2 7 7,2 2 mae a iNT Sr =i -¥ 2A] +5) (1 I, +378) M2 = 7 1 + 2,2 — drow, + 30,2)

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