116 The Correlation Coefficient of a Polychoric Table 
a, h, etc.) for any I, m will be the same throughout. Hence we may write the 
complete coefficient of as 
All 
Xll Xl2 A13 
+ V P^^ 
A12 
All A12 A13 
Vii+ Viz + •• 
Xl2 Xl3 
+ etc. 
Pi 
^Xir" xi2^" xi3^" V 
In the case of an enneachoric table for example we have, TiCgt being = 1, 
^11 p (^12 p ^^21 p ^22 
d-'^ll ■ d-' 12 d-' 21 
Xll Xl2 Xl3 X22 
^11 p ^"12 p I C'21 p C22 
Xll X12 Xll X22 
o r r 
^12 p I '-'21 p I ^22 
&-f^l2 ~r 6^21 
(98). 
+ 
+ 
22 J 
p V 
d-C^22 I 
'^ll p 
Xll 
etc. 
(99). 
- „ ... 6P22) 5^ 
X12 X21 X22 / 
the P's being at once written down from the table on page 114. 
Or more generally thus : 
Since the P of any cell W;^ with reference to any cross point {st) is invariable it 
may be written generally as imPst- This notation gives up the recognition of the 
equality of the P's in any given quadrant but gains in generality. The quadrantal 
suffix and sign may be supplied by inspection. We have then the following lemma: 
~ llP St- llP s't' 11 + 21P st-2lP s't' '^21 + ■•• + 12Ps«-12Ps'S' '''12 + ••• 
— 2^ ihnP st-lmP s't''>^lm) 
Im 
The standard deviation of r,, may then be developed as follows: 
(S(7,,) dr = 2 (Cstdr,,) 
8Ps*> 
.(100). 
Xst 
.(101). 
\Xst J \XstXs't' I 
i^h'^ p 2 _L OV ^s''' P P 
I " I Im^ St T im^ st - Im^ s't 
\Xst/ st Xst Xs't' 
= S 
Im 
Im 
2 ZmPs^l 
.si \Xst 
More fully written this is 
.(102). 
(S(7s()^CT^^ — ('-^'^ iiPii + iiPi2 H — ~ 11P13 + 
vXll X12 Xl3 / 
C12 p I C'la 
12^12 i- la-' 13 
P13 + 
W12 + 
.(103). 
