Karl Pearson and David Heron 
299 
have realised that more people in the more blind grades or more people in the 
better grades of sight are not the same thing as some people " much more blind " 
or some people of "much better sight" than any in the earlier aged groups. 
Mr Yule's alternatives are not real alternatives, he makes no reference to a shift 
in mean and a decrease in variability; such a combination involves only a reduction 
of the numbers in the grades of good sight, a very reasonable result with increasing 
age, and an increase of the numbers in the grades of bad sight, also a very 
reasonable hypothesis. Towards the 'tails' of both age curves, theoretically 
there would be fractional units, while in actual observations there would be 
isolated units at relatively wide intervals (cf. Galton's " Difference Problem * "). 
What the distribution of such units might be, could not be a priori predicted. 
But it is quite possible for two distributions with slightly separated means and 
slightly different variabilities to give quite reasonable fits to Gaussian curves 
and yet the distribution with the greater variability to have no outlying units 
with "much more" of a character than any which exist in the less variable 
distribution. The variabilities are much more closely determined by the bulk 
of cases with moderately large deviations than by the one or two extreme 
outlying individuals. 
In the case we have last discussed the age-group 45-55 has two individuals 
lying outside 4 - 652, and the distance between the means being '192 this gives 
4 - 46S = 4'51o- for the corresponding distance on the 55-65 age curve. Outside 
this limit are 2"5 individuals of this older age curve. Are we to say that that 
half individual represents Mr Yule's necessity for some people in the later age- 
group who are " much more blind " than any people in the first ? In truth some 
people would be much less blind, if they would only stay to express their opinions 
in actual numbers before writing them down. The " minute sifting of numerical 
results " is the foundation of all true statistical inference, and here, as in other 
phases of his recent work, Mr Yule has committed himself to superficial statements 
reached by verbal disquisition which vanishes into nothingness if the touchstone 
of numerical investigation be applied to it. 
There are many other points at which we should like to traverse Mr Yule's 
statements, but we think we have brought forward enough evidence to indicate 
how unreliable are his methods and how biased are his criticisms. 
(10) Summary of General Conclusions. 
In order to sum up the general conclusions reached in this paper, wo must 
state first one or two principles which we accept as almost axiomatic : 
(i) There is no universal method of dealing with an n x n-fold table, except 
the method of mean square contingency, leading to a probability measure of the 
independence of the two characters, unless we know : 
* Biometrika, Vol. i. p. 390. 
