134 On Polychoric Coeficients of Correlation 
The question now arises as to the manner in which we can find /• for the whole 
table most effectively. 
Clearly we might assume the product-moment components from (xiii) and sum 
for all cells. We should have 
since tlie coordinates are measured from the means in terms of the standard 
deviations as units. 
Hence substituting from (xiii) we have : 
r = S ('^' {% T,%> T: + r^, T,%. T; + ...+ Tp%. T; + ...}) (xvi). 
Here Tisa' must be substituted from (x) and we have finally 
^, ZX- To' + r% r; + . ■ ■ + rP% Tj, X^ T p' + ...]\ 
This equation based upon the product-moment method of finding r is clearly 
likely to be very complicated, and although it can be proved that the product- 
moment method is the "best" method of finding ?• when we are dealing with 
a series of quantitatively measured individuals, we have no certainty that it is the 
best method in the present case of broad categories. It may indeed be questioned 
whether another method now to be considered cannot be shown to be better or 
at least equally efficacious. 
Let us consider for a moment what we have in view. We observe n^s- as the 
frequency of the sth-s'th cell ; we find that with a given correlation r the frequency 
of this cell would be ngs- on the assumption that the frequency sui'face is the normal 
frequency surface corresponding to the observed marginal totals. Accordingly the 
most probable value to give to r would be that which made 
X — >^ = — mnnmura, 
6', S' ''ss' 
or, what is the same thing, 
(""—)= minimum. 
, Us.. 
This leads us, differentiating with regard to r, to 
or, writing at length, our equation for r is : 
[ / »,,-Y %T,%>T,' + 2rXr,X'T,' +...+ prP-'XTp'^s'Tp' + . . . [ ^ • • 
\[n) i^ToX'To' + r%r,%-T; + r^XT.^,r: + ... +rP%Tp%-Tp' + ...)4 ^ ^^ 
Neither (xvi) nor (xvii) are very readily solved. Probably the easiest way will 
be to obtain an approximate value of r by existing methods either from a good 
fourfold table, or from contingency, and then evaluate (xvi) or (xvii) for values of 
one well above and one well below this result, so that the real value of r lies 
