266 
On Theories of Association 
we have found a safe guide to determine how far we introduce risk in using 
tetrachoric r t . It will be seen that the unsatisfactory values of r t occur when this 
smallest quad ran tal frequency is less than 2"5°/ 0 . On the side of Q or r t) towards 
their mean, a line is drawn equal to 2"5 times the probable error ; in all but one 
case of r t this reaches the mean ; in nine cases of Q it does not, and the deviations 
in six of these cases are excessive. There can, we think, be no doubt that tetra- 
choric r t is iu this first of Mr Yule's selected cases far more stable than his 
coefficient of association. 
(B) Skew Table of Number of Mendelian Couplets in Father and Son 
(see Table XVI on our p. 222). 
Tlie following table gives tetrachoric rt and Q with their probable errors for 
a second series selected by Mr Yule. 
Vertical division taken between 
0—1 
1—2 
2—3 
3-4 
0—1 
r ( =-39 + -OI7 
§=-49±'018 
r t =-37 + -017 
§=•51 ±-023 
»- t =-35 + '030 
§=•63 ±"048 
r< =-33 + -086 
§=•75 ±-153 
1—2 
r J = -37 + -OT7 
$=•51 ±-023 
r t =-40 + -017 
§ = •52 ±-019 
r, = -39+-025 
§ = ■61 ±-029 
r ( = -37 + -066 
§=•72 ±-087 
2—3 
r 4 =-35 + -030 
§=•63 ±-048 
n =-39+-025 
§ = •61 ±-029 
^=•41 + -033 
§='69 ±-034 
r t = -40+ -073 
§=-79± -068 
3-4 
r ( =-33 + -086 
§=•75 ±-153 
r t = -37 + '066 
§=■72 ±-087 
r t =-40+-073 
§='79 ±-068 
rj=-43 + -122 
§ = •90 ±-072 
It is interesting to note the high values reached by the probable errors of 
tetrachoric r, for the small quadrant frequency divisions. Both systems of 
probable* errors are based on the assumption that the total frequency is 4096. 
The following results were obtained : 
Weighted Mean 
Weighted 
Standard Deviation 
Tetrachoric r t 
Association § 
•3809 
•5627 
•0176 
■0840 
Of course this table of discrete couplets is not one to which we should think 
of applying tetrachoric ?>, but Mr Yule has applied it ; however, the weighted 
* In using probable errors at all here for a basis of comparison, we assume that the table may be 
considered to have arisen from actual observations. 
