Karl Pearson 
405 
a case from Newsholme's Vital Statistics (p. 344) there was not a very remark- 
able divergence. I admit the difficulty of finding the correlation of deviations in 
p s and p t in the general case (n commensurable with m). But this difficulty does 
not exist when m is large as compared with n ; and I venture to think that in 
any case the whole distribution of houses must be compared in the observed and 
theoretical instances. If my analysis be correct — I frankly admit the matter is 
obscure and difficult — the case of enteric fever when treated from this broader 
standpoint indicates that the observed distribution of frequency was not, on 
the hypothesis of equivalent houses, compatible with a mere random scattering of 
the cases. 
Greenwood in his paper on plague distribution has seen the importance of 
comparing the whole of the observed and theoretical distributions. He has found 
the " goodness of fit " of the observed to the theoretical frequency for the whole 
system of villages (not houses) in his data. I feel confident that the " goodness of 
fit " method is the correct method to be applied in such cases, but I doubt if it can 
be applied without weighting the terms of the as, I think, Greenwood has 
omitted to do. In other words, I do not consider that 
is the correct expression to use*. This expression is only true when 
S (p s ) = m, 
but we have to remember that in our present case 
8 (sp s ) = n, 
or the number of cases as well as the number of houses is fixed. Accordingly the 
fundamental relations on which (i) is based, i.e. 
Psft 
a P s a PtPsPt m 
are no longer correct and must be replaced before we can determine the true form 
of x 2 - Greenwood is correct in his view that we must use % 2 , and I think this has 
been overlooked by Troup and Maynard. They on the other hand give the full 
values for p s and a 2 Ps , showing that the first equation of (ii) is incorrect, but they 
do not attempt the second equation of (ii). 
The present writer only offers the approximately correct values of (ii) and the 
correspondingly corrected value of (i). These results apply absolutely to the enteric 
data, but would only apply as an approximation to houses in plague data. It has 
been recently half suggested f that the application of mathematics to medicine does 
* Greenwood's case is not exactly the present case, but the need for weighting the contributions 
to x 2 still I think remains true. 
t British Medical Journal, June 17, 1911, p. 1431. 
