204 Opsonic Index — " Mathematical Error and Functional Error" 
himself? Have we not here an instance of how disastrous is the specialisation 
of modern science, which so completely prevents a distinguished specialist from 
knowing the history of branches of science outside his own little field ? The 
modern theory of errors did not arise from " such an exercise as counting the 
number of pips on a series of cards," where " no skilful functioning is required," 
but from the needs of a science like vaccine therapy, where skilful functioning 
was essential. 
The insistence of Sir Almroth Wright on the distinction between variations 
due to random samples and variations due to observational skill is perfectly 
legitimate and of vital importance, but his argument that the one and not 
the other is capable of exact treatment shows not only limitation of his know- 
ledge of statistical methods, but has led him into the following statement which 
will form the text of the present paper : 
I have satisfied myself, and all my fellow-workers have satisfied themselves, arid I am glad 
to say a very large and increasing number of bacteriological workers all over the world have 
satisfied themselves, that when the " functional error " has been reduced, as it can be by practice 
and patience, to small dimensions, and when, in connection with tubercle, the customary counts 
of 100 or more leucocytes are made, the "mathematical limit of error" of the opsonic index 
is such as need not seriously be taken into account. In view of this, I suggest that those critics 
who have put forward figures showing enormous working errors in opsonic estimations may 
have supplied the world data with regard to their own functional errors, instead of — as self- 
esteem assured them — data with regard to errors inherent in the opsonic method (p. 29). 
" I have satisfied myself," writes Sir Almroth, — but the day of authority in 
any branch of science has gone by, and we ask legitimately and instinctively 
for the data from which this satisfaction was extracted. 
It is surely a question of statistics and not of individual satisfaction. Possibly 
Sir Almroth Wright may be correct, possibly he may not be. A priori no state- 
ment can be made. What we need is a sufficiently long series to establish a 
definite result by one in whom " the ' functional error ' has been reduced, as it 
can be by practice and patience, to small dimensions." A comparison of that 
series with those obtained by other observers — i.e. those critics whom Sir Almroth 
Wright suggests have enormous functional errors- — would at once enable us to 
see the relative value of functional and sampling errors. No doubt Sir Almroth 
Wright has undertaken an investigation of this kind ; but, until he has published 
it and submitted it to the test of trained statisticians, can he be surprised if 
they are not wholly convinced by the mere expression of his satisfaction that 
the difference of functional error is the source of those " enormous working 
errors " which so far appear in all the scientific statistical data available for the 
estimation of opsonic index variation ? 
(2) Sir Almroth Wright has taken up the ground that " the enormous 
working errors in the opsonic estimations " are due to functional error*. I take 
* By the very use of the word "error" he introduces bias. The observations show large variations, 
the question before us is whether these are due to personal equation or to random sampling. 
