F. M. Turner 
497 
Thus in Fig. 1 the abscissae represent degrees of immunity, which in the 
case of those who contract the infection are represented by the degree of severity 
Fig. 1. 
of the resulting disease. In those who escape infection the degree of immunity is 
greater than that required to prevent the disease appearing. It may be much 
more than this, but the excess is latent, and we have no means of observing it. 
The ordinates represent frequencies, the line ab divides those who escape infection 
from those who do not ; the line cd divides the deaths fi'om the recoveries. 
The area abdc represents the number of recovered cases, and xcd the 
number of deaths. It is obvious that, if those who escape infection exceed half of 
those who are exposed, the frequencies of the different degrees of severity diminish 
without exception, as the degree increases. Probably the actual frequency curve 
among the vaccinated is of this type. 
Assuming for the above reasons that the normal distribution applies to Table C 
we see that the same cannot hold also for Table D, though all the above calculations 
have been based on the latter assumption. It is therefore important to examine 
what is the effect of neglecting one column in a sixfold normal table and treating 
the remaining figures as if they conformed to a normal distribution. 
Such a table I shall call a " curtailed " normal distribution, and the value of 
r obtained from the full table I shall call the true value. That which is obtained 
from the curtailed table by the use of any formula I shall call the " apparent " 
value and shall denote by r'. 
As a first example I take the table on p. 214 of Biometrika, Vol. I., which 
is shewn by Macdonell to be a close fit to the normal correlation table. Drawing 
a line under " head breadth 15"0 " I have kept all figures above the line and 
discarded all those below. The resulting table is therefore a curtailed correlation 
table, analogous to our smallpox tables drawn from the inmates of a smallpox 
hospital, all those who remained unaffected by the disease having escaped 
observation. A curtailed table in measurable characters would result if we 
collected statistics of height from soldiers, a population from whom all below a 
certain standard height had been already rejected. 
Using product moments, I have found in the above case r = •2446, whereas 
the true value r = -4016 ± 'OlO. 
Biometrika iv 63 
