M. Greenwood 
81 
observed result. It may be remarked, however, that had we used a normal curve 
/ 1x16 
with s,D. ^ 15 . jYfy~ ' t ' ie oc ^ s es t mia ted therefrom would have been enormously 
greater. 
In conclusion, I desire to refer to a subject indirectly related to the topic of this 
paper and, I think, of importance. We are familiar with such arrangements of 
material as the following. n 1 persons have been immunised against a certain 
disease and, having contracted the disease, a x have died. Of n. 2 not immunised 
a 2 have died ^— < . The extent of protection conferred is then estimated by 
some coefficient of correlation or association. The trustworthiness of the coefficient 
so calculated is then measured by a comparison between its arithmetical value and 
that of its standard deviation or " probable error." This process has its limitations. 
If, for instance, either Oj or cr 2 be zero, Yule's coefficients of association and colli- 
gation become unity and their standard deviations are indeterminate. 
I would put forward for consideration the possibility that the use of Bayes' 
theorem might here be of value. Thus : 
Let the chance of o 3 or more successes in n 2 after a x successes in n x be p 2 and 
the chance of a x or less successes in n Y trials after a» successes in n., trials be p±. 
Then, since either n x or n 2 might have been drawn first, a measure of the probability 
of the observed result will be — ^ ^ . 
We might indeed adopt a scale of reliability by putting P = f f — ^ ^ , the 
function being such that P increases to unity as Fl r ' 2 diminishes to zero. 
I put forward these suggestions with some doubt, but I cannot help feeling 
sure that in such cases as those I have instanced the ordinary method of testing 
the reliability of a coefficient of association is a dangerous and possibly misleading 
artifice*. 
In conclusion I desire to express my regret that this paper is so imperfect. 
The problems treated require mathematical abilities and training not at my 
command. I have only ventured to write upon the subject because of its practical 
importance and may, perhaps, venture to entertain the hope that my numerous 
mistakes of omission and commission will be leniently treated. 
* [The probability corresponding to the x 2 of the fourfold table can be calculated straight away ; but 
the difficulty arises from our not mentally appreciating grades of probability with the readiness we 
appreciate grades of correlation on a limited scale. Editor.] 
Biometrika ix 
11 
