Karl Pearson and David Heron 
253 
the severity of attack and immunity due to vaccination on Gaussian bases. This 
is done in Diagram XI, and the corresponding regression lines are drawn. These 
were calculated by supposing the range of the mid or ' abundant ' group the same 
for all arrays of immunity, and the group 25 — -i-5 the same for all arrays of 
severity. It will be seen at once from this diagram that the division into 
vaccinated and unvaccinated is a merely verbal one, that Mr Yule is playing with 
words, not dealing with the realities under class-names*. Immunity and severity 
are continuously changing quantities and they vary in almost linear relationship to 
each other, the greater the intensity of the immunity associated with vaccination 
the less severe is the attack. We do not think the deviations from linearity 
of regression found in this case differ sensibly from the sort of values that 
frequently occur in the case of the regression line found by the product-moment 
method. 
In order, however, that Mr Yule may not attribute this result to the use of the 
Gaussian distribution to find the means, we applied his method of pseudo-ranks to 
determine the means of the arrays and then put in the regression lines on his 
hypothesis that the rank of a huge bracket is a suitable unit in which to measure 
correlation. This is done in the lines A of Diagrams XII and XIII. Although 
the previous scheme is much distorted, we see that on Mr Yule's own hypothesis 
the two variates behind the "vaccination — non-vaccination" and "death — recovery" 
classifications are continuous, steadily increasing in one direction, and that there 
is the correlation of a continuous two-variate system behind them. 
Now let us proceed to do exactly like what Mr Yule has done, namely make 
divisions at vaccinated and non-vaccinated, and between confluent and abundant, 
and dress the table so that there shall be equal numbers of vaccinated and un- 
vaccinated and of the very severe and less severe cases. The table thus transformed 
is Table XXII. The difficulty now is to know what to do with the marginal scales; 
if these in the natural condition were normal, they are hardly likely to remain so 
after selection. But these scales — especially as we have found the correlations by 
the T?-process — are not of first class importance except for exhibition of the results 
graphically. We have therefore retained the old scales and merely calculated the 
means of the arrays condensed at the old means of the old scale of years since 
vaccination (see Diagram XIV). In order that the reader may see how little 
assumption is thus made we have had a second diagram drawn indicating what 
* " The division may also be vague and uncertain : sanity and insanity, sight and blindness, pass 
into each other by such fine gradations that judgments may differ as to the class in which a given 
individual should be entered. The possibility of uncertainties of this kind should always be borne in 
mind in considering statistics of attributes ; whatever the nature of the classification, however, natural 
or artificial, definite or uncertain, the final judgment must be decisive ; any one object or individual must 
be held to possess the given attribute or not." Theory of Statistics, pp. 8, 9. Similar words are used 
by Mr Yule in Biometrika, Vol. n. p. 121: "The judgment must however be finally decisive; inter- 
mediates not being classed as such even when observed." We can hardly conceive statements more 
liable to prejudice the mind of a wavering recorder of actual data. No wonder Mr Sanger said that 
"Mr Yule's work would be the work for Mendelians" ! 
