wk THEORY OF STATISTICS. 
values be assigned to x,,, and all the following deviations, the 
correlation between x, and x, on expanding x, is, as we have 
seen, normal correlation. Similarly, if any fixed values be 
assigned to aj, to #1, and all the following deviations, on 
reducing x, ,, to the second order we shall find that the correla- 
tion between x,; and x, is normal correlation, the correlation 
coefficient being r,,,, and so on. That is to say, using % to 
denote any group of secondary suffixes, (1) the correlation between 
any two deviations x, and x, ts normal correlation ; (2) the correla- 
tion between the said deviations 1S Tp, whatever the particular 
fixed values assigned to the remaining deviations. The latter 
conclusion, it will be seen, renders the meaning of partial 
correlation coefficients much more definite in the case of normal 
correlation than in the general case. In the general case 7. 
represents merely the average correlation, so to speak, between 
Zn; and z,,: in the normal case 7,,; 1s constant for all the sub- 
groups corresponding to particular assigned values of the other 
variables. Thus in the case of three variables which are normally 
correlated, if we assign any given value to x; the correlation 
between the associated values of #; and w, is ry, : in the general 
case rq if actually worked out for the various sub-groups 
corresponding, say, to increasing values of x; would probably 
exhibit some continuous change, increasing or decreasing as the 
case might be. Finally, we have to note that if, in the expression 
(15) for ¢, we assign fixed values, say A, ks ete., to all the 
deviations except a;, and then throw ¢ into the form of a perfect 
square (as in § 4 for the case of two variables), we obtain a normal 
distribution for #; in which the mean is displaced by 
a hi Gy.95. in F193... 
T1234... — ht Piosh.. ng. rn ce Ting... lg TT fen, 
But this is a linear function of A, As, etc., therefore in the case of 
normal correlation the regression of any one variable on any or all 
of the others ts strictly linear. The expressions Tig ....n» 
Gros. ...nf023....m etc. are of course the partial regressions 
bios... . mw ©LC. 
REFERENCES. 
General. 
(1) Bravars, A., “Analyse mathématique sur les probabilités des erreurs de 
situation d’un point,” Acad. des Sciences : Mémoires presentés par divers 
sawants, 11° série, ix., 1846, p. 255. 
(2) GavrroN, Francis, ¢“ Family Likeness in Stature,” Proc. Roy. Soc., vol. xl. 
. 42. 
(3) ha natices, Natural Inheritance ; Macmillan & Co., 1889. 
296