ON THE APPEARANCE OF MULTIPLE CASES OE 
DISEASE IN THE SAME HOUSE. 
By KARL PEARSON, F.R.S. 
1. Introductory. 
This problem is one of surpassing interest both from the standpoint of the 
mathematician and of the public health officer. Contributions to the theory of 
the problem have recently appeared from Dr Greenwood, Jnr*, Drs Troup and 
Maynard-f, and Captain McKendrick \. Each of the papers referred to have thrown 
light on the subject, but cannot be considered as giving a final solution. The 
solution I am proposing in the present paper is at best only an approximate one. 
The fundamental condition requisite for its sufficiency is that the number of deaths 
should be small as compared with the number of houses among which they are 
distributed. This condition is amply fulfilled by the data actually used by Troup 
and Maynard. Hence to such cases my theory applies. It would not apply 
without including higher terms in the approximation to cases in which there was 
a heavy epidemic visiting a large percentage of all houses. It is, however, 
applicable to cancer distributions. 
Troup and Maynard have been the first, so far as I am aware, to reach perfectly 
general mathematical expressions for (i) the probable number, p s , of houses with 
s cases, supposing the same number of inhabitants to each house, and each house 
to be equally likely to be attacked, and (ii) the standard-deviation, <r Ps , due to 
random sampling of the frequency p s under the same conditions. 
They do not go further than this and they do not proceed to find the correlation 
in deviations of p s and p t due to the variations of random sampling. This stage 
seems to me needful, if we are to provide an answer to the general problem : Is the 
observed distribution of houses with multiple cases compatible with the theoretical 
distribution on the assumption that the cases are distributed at random ? Troup 
and Maynard confine their attention to the divergence between the expected and 
observed numbers of houses with two cases, and conclude that for enteric fever in 
* See Journal of Hygiene, Vul. x. p. 410. 
+ See Biometrika, Vol. vnr. p. 39(5. 
J Ibid. p. 413. 
