12S 
Oti Polyclioric Coefficients of Correlation 
no greater assumption than when we exhibit a pressure-volume curve as a straight 
line by using a logarithmic scale. 
Now let the polychoric table be such that in the population N under discussion, 
the sth category of the first variate A contains individuals and the s'th category 
of the second variate B contains n.^' individuals, while the number of individuals 
who combine in the population N the sth category of A and the s th category of B 
is n^g'. 
Now when we proceed to exhibit the categories of the 4 -variate on a normal 
scale, the process will give us two important quantities : 
(a) We shall have the ratio of abscissa to standard deviation at the dichotomy 
between each pair of broad categories. 
If ?!]., ??2-, th., ... Vg., ... be the frequencies of the ^-variate for the several cate- 
gories the values of the ratios of abscissae to standard deviation will be specified as 
— X , A-i, ho, A3, h^, ... kg, ... + CO. 
Here //.,_!, are the values on either side of the category Hg. and if there be 
q categories, n.-^ is bounded by Ji^ or — ac and h^, while is bounded by hg^^ and 
or -I- 00 . The lower /t's will have negative and the upper positive signs and the 
greatest care must be taken to see that the proper signs are given to the values 
of h. Similarly if th(' frequencies of the various categories of the 5-variate be 
v.o, 'H.3, ... n.gi, 
the values of the ratios of ordinates to standard deviation will be represented by 
— X , ki, kc, , k^, ... kg'^i , kg' , . . . kg' , + CO , 
where kg'_i and kg- give the dichotomies on either side of n.g. 
We may consider the coordinate at the back of the variate A when represented 
on a normal scale to be x', the origin being taken at the mean on the normal scale. 
Hence if the standard deviation be cr^., we shall find it convenient to write the 
absolute normal abscissae 
w' = a^x, hg = (jj}g. 
Similarly we take y' for the coordinate at the back of the variate B, measured 
from the mean, and write : 
y' = (T,,y, kg' = ay kg, 
where o-,, is the standard deviation of B. Clearly until a quantitative scale has 
been determined we shall know h, k, x, y but not //, A;', x', y', and <jy. 
(h) We shall determine the ratio of abscissa to standard deviation, or the ratio 
of ordinate to standard deviation of the centroids or means of the groups 
and n.g'. 
Let H,= -^e~-^'^, K, = ~e'^^\ 
then the means of the categories and are determined by 
