188 On Theories of Association 
After equalising the classes of this table we have the following one : 
TABLE V. 
Vaccination Grade and Intensity of Disease. 
Severity of Small Pox. 
Haemorrhagic 
Confluent 
Abundant 
Sparse 
Very 
Sparse 
Totals 
Vaccinated. 1 ^„ ^ 
Years since \ 9r .~ 
vaccination I ^ \ c 
v Over J/5 
Un vaccinated 
0 
20-12 
48-08 
84-40 
47-40 
9-27 
20-32 
35- 07 
36- 66 
98-68 
36-78 
41-39 
44-74 
33 23 
43-86 
68-19 
60-59 
40-54 
23-11 
7-57 
85-76 
57 '58 
31-57 
22-60 
2-49 
200 
200 
200 
200 
200 
Totals 
200 
200 
200 
200 
200 
1000 
Now here are some of the changes that take place in Q, owing to this 
equalisation of the classes : 
TABLE VI. 
Vertical Division 
Horizontal Division 
Old Q 
New Q 
Confluent-Abundant 
Vaccinated-Unvaccinated 
- -7220 
- -7070 
Abundant-Sparse ... 
Vaccinated-Unvaccinated 
- -8599 
- -8945 
Confluent- Abundant 
25 — 45 and over 45 
-•5714 
- -7522 
Confluent- Abundant 
10—25 and 25—45 
- -5411 
- -8163 
Abundant-Sparse ... 
10—25 and 25—45 
- -4469 
- -7742 
Haeniorrhagic-Confluent . . . 
10—25 and 25—45 
- -5711 
- -7798 
Haemorrhagic-Confluent . . . 
25 — 45 and over 45 
- -5449 
- -5861 
Haemorrbagic-Confluent . . . 
Vaccinated-Unvaccinated 
- -1009 
- -1371 
The range of values found for Q for a single table will be considered later ; 
it is sufficient here to indicate how largely the bulk of them are changed by 
equalisation of the groupings. If further evidence be needed for the radical 
changes which invariably accompany Mr Yule's conception of a " natural " 
grouping, we may note that the Yulean deduced from pseudo-ranks is before 
equalisation — "3099, but after equalisation — - 5375. These two equalisations of 
class frequency show that it is impossible to predict a priori how the relationship 
of the two variates will be changed by the process ; in the first the relationship 
was lowered by 59 °/ 0 and in the second case raised by 73 °/ 0 of its value as 
estimated by the Yulean* ! It is clear that only a fictitious type of equalisation 
has been used by Mr Yule. His independence of selection is only an artificial 
one ; sub-groups within his categories are not equalised but only equally selected ; 
and further such selection must neither fall short of nor exceed the arbitrary 
division he has selected for his class. If for example we agree to equalise each 
* Consider as a last illustration the Table VII below. The correlation is -37. The reader will find 
