Goodness of Fit of Frequency Distributions. 371 



Now consider the partial correlation coefficient of x s 

 and x s < for constant # l5 x 2 . . . a' 2 _i for all the q—1 variates 

 except x s and x s >. We may put these constant variates 

 zero, and it is clear that we have to find the correlation 

 of x s and oe s > when we take samples of: size n s + n s < + n q from 

 the m S9 m s ', and m q classes only. 



But in this, if %' s , £V be the standard deviations, we 

 must have 



* m g + m s , + m q \ m s + m s , + w ? / 



- / ., n s \ n s (n»' + n q ) ,... v 



= n g I 1 — _ , — 1 = - , =; . . . (in.) 



\ n s + n S f-rnq/ n s + n s >-{-?i q v y 



and, similarly, 



^ /2 _ n a i(n 8 + n q ) 

 s ' n s + n s <-\-n q 



Further, if p ss < be the required correlation, 



ra s + m S ' -f m y ??i s + m s > + m q 

 n s n s > 



ns + n s > + n q 

 Thus 4/T - 



^n s + n q s/n s ,-\-n q 

 But by the general theory of multiple correlation, 



A S8 > 



A,,/ /A~~ A,y 



cr s0V A/ V o-2A X erjfA' 



or A gg , = _^^__ //l^X/J- 1\ 



o- 5 avA \/?i, + w 9 V^'+Wg V Vt? s n q )\n s > n q y 



using the values found for (ii.). Thus we deduce 



A„,/ 1 



i ^ 



,ovA w. 



(v.) 



Accordingly we have found the coefficient of the term x s m 8 > 

 in the power of the exponential, which may now be written 



z = z Q e~i^, (vi.) 



2 02 



