THE OPSONIC INDEX—" MATHEMATICAL EEEOE 
AND FUNCTIONAL EKEOE." 
By KARL PEARSON, F.R.S. 
(1) In a recent paper on Vaccine Therapy : its Administration, Value and 
Limitations*, Sir Almroth E. Wright has raised the interesting point of the 
relative influence of what he terms " mathematical error " and " functional error " 
in producing variation in the opsonic index. He very properly points out that 
the variation in the value of this index may depend upon two sources : (i) the 
functional error of the individual observer, and (ii) the error of method. By 
the last term I understand him to mean what we, as statisticians, should 
term the error of random sampling, i.e. the variation produced when random 
samples are taken from a population, the samples consisting of relatively small 
numbers selected without bias of any kind whatever. Sir Almroth admits that 
this last variation must be a subject for mathematical evaluation; at least, I 
so understand him. On the other hand the first variation, "functional error" 
as he terms it, 
is an error which attaches only to methods which involve a certain amount of skilful functioning. 
It attaches to the operator. It has a different value for every operator. It may in the case of 
one and the same operator vary from hour to hour with his physiological efficiency. Its value 
can be diminished by practice and attention. It cannot be evaluated by a mathematician. It 
can be pretty accurately gauged by the operator himself (p. 28). 
Now it is difficult to see wherein the functional error of the vaccine therapy 
observer differs from that of the astronomical observer. His personal equationf 
also varies from hour to hour with his physiological efficiency, and it has a 
different value for every operator. What sort of reply would Gauss and Bessel 
have given to the astronomer who said that his functional error could not " be 
evaluated by a mathematician," but could be " pretty accurately gauged " by 
* Proceedings of the Royal Society of Medicine, Vol. in. 1910. My references are to the pages 
of the offprint. 
f That the "personal equation " is always in flux and is correlated with environment or " atmosphere " 
is well known : see my memoir Phil. Trans. Vol. 198, A, pp. 235 — 299. 
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