112 The Con-elation Coefficient of a Polyehoric Table 
Let f and q be any frequencies in a given distribution in which the population 
N is so large that sampling does not alter its composition; then we have the well- 
known results (p. 99) 
VQ 
^ (dpdq) = - 
Now let p and q have a common part c so that 
p = p' + c, 
q= q' + c. 
Then ^ (dpdq) = ^d {p' + c) . d [q + c) 
= ^ {dp'dq' + dp'dc + (^g'(^c + dc^) 
_ _pV_p^_q'c f-. _ c\ 
N N iV ^ V Nj 
iq' + c){p' + c) , ^ 
- ^ 
iv • 
Now the mean product of any two linear functions of jj's and g's, 
S = ^d {ii27i + izPz + ...) .d (Ai^i + k^q^ + ...), 
will consist of the sum of the mean products of terms such as 
isdps . ktdqt. 
But §b • hdqt) = ish^ {dPs ■ dqt) 
where c is the common part of p^ and qt. 
Therefore S = ^ (c.,, - 
Psqt 
N • 
Hence the rule. 
As an example consider S^.^i, 
= ^ {A-^idm-i. + B^j^dm.i — [A^idm^. + i?2i^^'"-i ~ ^^''*2i) 
- ^21 "hi - ^21 "hi + »*11 ^ 
= (^11 + B^i - 1) (-^21 + ^21 - 1) «ii + ^11 (^21 + - 1) n.,i + B^^B^^n^, 
+ ^11 ^21 "12 + 0 . A^i.llzz + 0 . 0 . 
+ ^11^21%3 +0.^421-^^23 +0. 0.^33 
