194 
On Theories of Association 
of x lying between x and x + Bx and of y lying between y and y + By. This is the 
true measure of independence of the two variates. It is always possible to test 
the independence of two variates — i.e. the probability of their independence — ■ 
by aid of the mean square contingency and the use of Palin Elderton's 
Tables*. But there is a whole class of continuous variates for which the 
regressions are linear and the correlation coefficient is zero, but which are yet 
heteroscedastic, i.e. the arrays of the ?/-variate for a given *--variate are not similar 
frequency distributions. Any frequency surface, with two planes of symmetry, one 
perpendicular to the axis of measurement of each variate, is representative of this 
class. Let DEFG be the oval contour line within which, for a given population N, 
all the frequency lies. This may be the actual curve in which the frequency surface 
cuts the plane of xy, or if the surface asymptotes to that plane, the contour within 
which all individuals of the N observed actually lie. If the distribution were 
Gaussian, this contour would be an ellipse. Generally let us suppose it any non- 
re-entering oval curvef. 
Let the frequencies in the four classes made by divisions parallel to the axes 
of the variates, i.e. to DOF and EOG, be represented by 
d 
, which we have 
used throughout in preference to Mr Yule's more cumbrous ~~ g J ^g ~' 
Then Mr Yule's coefficient of association Q is (ad - be) / (ad + be) ; it is 
unnecessary at this point to consider Mr Yule's coefficient of colligation &>, 
which is only a special function of Q. 
All along EOG taken as one dividing line (the other will be perpendicular 
to it), Q = 0 ; all along DOF also, Q is zero. All along the arcs EF and DG, 
Q = + l; all along the arcs FG and BE, Q = —\. These values are indicated by 
numbers in the diagram. If the frequency surface be non-re-entering, then when 
the axis of the rectangular dividing planes is taken anywhere in the quadrants 
EOF and DOG, Mr Yule's Q is positive and varies from 0 to 1 ; if this axis be 
* Pearson, Drapers' 1 Research Memoirs, " On the Theory of Contingency," p. 6 (Dulau and Co.), and 
Biometrika, Vol. i. p. 155. 
t If the curve be a re-entering one, the rapid value changes of Mr Yule's coefficients are still more 
remarkable! 
