154 
On Polyclioric Coefficients of Correlation 
Hence the distance between the means is 
•6935 o-j, - -3910(75 
= {-0935 - -3910 X 1-1196/1-5155} ay 
= -4046 o-y 
= 1'0866, if we introduce the value of ay. 
This and the corresponding values are recorded in the fifth column of 
Table XVI. It will be seen that these values approximate to those in the third 
column, the greatest differences being in the small first and last arrays. 
Of course in actually working with material solely given in broad categories we 
use the value '4046, treating ay as our unit of measurement. The means of the 
columnar arrays can be found with great ease and with considerable approximation 
by this method. 
If we now proceed to take the mean of our means duly weighted with their 
frequencies, we find it to be — '0510, — not a very serious divergence from zero. 
However, we subtract it* from the means in the fifth column of Table XVI, 
multiply the squares of the remainders by the corresponding frequencies, sum and 
divide by the square of ay. Thus we obtain 
r818,8034_ 
"7-211,9103 ~ 
or: 7? = -502148. 
If we divide by the class index correlation of the a^variate, i.e. '962,329 f, we 
obtain 
7) = -5218, 
which correlation ratio we may take to be the correlation coefficient and compare 
with our polychoric coefficient -5204 (p. 142). Clearly although our means as found 
by the hypothesis of normal distribution of the columnar arrays agree only approxi- 
mately with the polychoric means of the third column of Table XVI, they lie 
practically on the same regression line, as Diagram IV indicates. We conclude, 
therefore, that in this case as probably in many like cases, it is quite adequate to 
obtain the means of the columnar arrays by treating them as normal distributions, 
then determining their correlation ratio and correcting it for the class index. The 
corresponding regression line with the means of the columnar arrays indicated 
will be for many purposes an adequate graph showing the general nature of the 
correlation. 
The general purpose of this paper has now been fulfilled ; it has been shown 
how a general polychoric coefficient covering all the data provided in a given 
contingency table may be found, and how a graph may be drawn representing such 
a table effectively. At the same time such a process is very laborious and probably 
will not be lightly undertaken or only in cases of grave uncertainty. The method 
* Correlation ratio without subtraction ='52'2'2. 
t See p. 142. 
