70 
On Errors of Random Sampling 
permeation of medicine by quantitative methods, it is widely felt that this proce- 
dure is questionable and that the " probable error " of the result must be found. 
The medical writer who has attained this level accordingly refers to a text-book 
and tests his proportions upon the basis of a " normal " curve of errors with the 
binomial standard deviation Jnpq. 
The specialist in mathematical statistics is aware that the time-honoured 
theory of the "probable error" rests upon certain assumptions of a quite definite 
character not adequately fulfilled in the imaginary case described. Warnings as 
to this are given in the better text-books, and are indeed unnecessary for those 
who care to read the proofs of the usual formulae. 
We must, however, bear in mind that not every medical man has either the 
time or the training requisite for the comprehension of mathematical analysis and 
many will be inclined to consult a book which, while giving formulae without 
proofs, contains explicit instructions as to their practical employment. Such a 
book as, for instance, Professor Davenport's Statistical Methods, seems admirably 
adapted to the needs of the laboratory worker. On p. 14 (2nd edition) he will 
find the following sentence :— " The probable error of the determination of any 
value gives the measure of unreliability of the determination; and it should 
always be found." The statement is commendably clear but, unfortunately, quite 
incorrect in many cases which come under the notice of the medical inquirer. 
The present memoir is an attempt to make the limitations of the process 
recommended by Professor Davenport arithmetically obvious to the medical 
reader, and to provide the latter with some assistance in the exceptional cases. 
To the trained mathematician or biometrician I have nothing to offer which is 
novel and little which is of interest, while the medical reader may find some 
difficulties in following every step of the inquiry. I hope these difficulties have 
been reduced to a minimum, but a risk of falling between two stools has to be 
faced by any writer dealing with a subject not new in itself but relatively so in its 
applications. My biometric colleagues will recognise the difficulties of the task, 
and are alone competent to determine the measure of success or failure achieved. 
II. 
The chance of an event happening is p and of it failing, q (p + q = 1). What is 
the "probable error" of pin successes in m trials'? 
In the problem stated the probability for the occurrence of an event and the 
independence of the happenings are supposed to be known. This either means 
that they have been ascertained by long experience or that their values (i.e. the 
value of p and the zero correlation between the results of successive trials) are 
defined by an hypothesis which we desire to verify. Accordingly the distribution 
of successes in m trials is given by the expansion of the binomial (p + q) m . If m 
be moderately large and p"~q small, the ordinates of a "normal" curve with 
Standard Deviation *Jrnp~q are a close approximation to the terms of the binomial 
