296 
On Theories of Association 
The values of Mr Yule's 8 for these three tables are 
(i) 16374, (h) 1940, (iii) 100. 
If anything is to be judged from these results, (i) is far more highly associated 
than (ii) and (ii) again than (iii). All are far more highly associated than the 
Datura, which we will call (iv), and then the order of association sinks from (i) to 
(iv) in a marked manner. 
Now here are Mr Yule's coefficients of association put against his o~'s 
8: (i) 16374, (ii) 1940, (iii) 100, (iv) 1-34, 
Q: (i) -130, (iv) -282, (ii) -500, (iii) -996. 
As the one series goes down the other goes up ! What, we ask, can be learnt 
from Mr Yule on the subject of association, when his methods, "sufficient for most 
practical purposes," thus contradict themselves ? 
Here in (i), (ii) and (iii) we have kept the total frequency constant, but 
perhaps the most absurd side of Mr Yule's 8 is manifest if we alter the total N 
of the observations. Suppose Bateson and Saunders had experimented with 
8300 plants instead of 83, then 8 would have been 134 instead of 1'34. The 
association is of course absolutely the same, but how would Mr Yule interpret 
his two S's ? 
We regret having to draw attention to the manner in which Mr Yule has 
gone astray at every stage in his treatment of association, but criticism of his 
methods has been thrust on us not only by Mr Yule's recent attack, but also by 
the unthinking praise which has been bestowed on a text-book which at many 
points can only lead statistical students hopelessly astray. 
(e) Mr Yules Assumption as to Absurdities which must arise if 
Normal Distribution be applied to the " Blind." 
Another interesting fallacy is developed by Mr Yule on p. 638 of his paper. 
He writes : 
" Consider for a moment what the assumption of normality of distribution would imply in 
any case where there is an increase of, say, the blind from one age-group to the next. This 
must imply either (1) a fall in the mean of the assumed variable character, goodness of sight, 
1 suppose — if the standard deviation is constant or falling, or (2) an increase of the standard 
deviation if the mean is constant or rising. If the first occurs, then there must be some people 
in the later age-group who are much more blind than any people in the first, and fewer people of 
first-class sight ; if the second, there must still be some people in the later group much blinder 
than any in the earlier, and there will also be some of much better sight. On the assumption 
that lies at the base of the normal coefficient, you cannot, in fact, effect a change in the 
numerical proportion of A's without phanging them qualitatively at the same time. The 
assumption seems to me absurd, to be equivalent in this case to saying that there are certain 
people entirely deprived of sight in the first age-group, and certain others more than entirely 
deprived of sight in the second. The normal coefficient is accordingly inapplicable, and its 
precise values of no special significance." 
We have rarely come across a more specious fallacy. If it were true it would 
be impossible in practical statistics to represent both a population and a selected 
sub-population by normal curves. Let the original population be N, mean M, 
