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

XIL.—PARTIAL CORRELATION. cy differing from zero in a limited sample. Hence, RB will not tend, on an average of such samples, to be zero, but will fluctuate round some mean value. This mean value will be the greater the smaller the number of observations in the sample, and also the greater the number of variables. When only a small number of observations are available it is, accordingly, little use to deal with a large number of variables. As a limiting case, it is evident that if we deal with » variables and possess only = observations, all the partial correlations of the highest possible order will be unity. 17. Tt is obvious that as equations (11) and (12) enable us to express regressions and correlations of higher orders in terms of those of lower orders, we must similarly be able to express the coefficients of lower in terms of those of higher orders. Such expressions are sometimes useful for theoretical work. Using the same method of expansion as in previous cases, we have 0=2(z12.... 00205. Er = (x Sr a (n=1)) ~bowm. on (a, Task... (n=) — Le vee (n=1) (x, “oa an, (n-1)) That is, b1a.54 “ren (n-1)= D134 sss am + bip.os eae (n=1)" Ono.34 ena (n=1) In this equation the coefficient on the left and the last on the right are of order n — 3, the other two of order n — 2. We therefore wish to eliminate the last coefficient on the right. Interchanging the suffixes 1 for # ana n for 1, we have 00.34 ‘inn == bors eve (B=1)"- + bp1.23 wwe wl) bio ven (B=1)p Substituting this value for 8,44 . (n-1 10 the first equation we have b +b .b b em 12.94... . n' ¥Yn2B....(n-1 n213.... acl). 16 RST Y=b00. wnt (n=1) ( ) This is the required equation for the regressions ; it is the equation biome bran + bins - bua 12 1- bins . bora with secondary suffixes 34 ....(n- 1) added throughout. The corresponding equation for the correlations is obtained at once by writing down equation A6) ford, m-1 and taking the square root of the product (cf. § 13) ; this gives Tiesto nt Tinos i nel) TonaE s ao fee) Tis. ...0-0= Gh Bele 324 bey ~Fonns. os w-0' (1 = T5013. . .. n-1)* £3 24¢

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