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¢