Miscellanea 
363 
did not occur with the mildest attacks, or the distribution was not " curtailed " in the manner 
suggested by Dr Turner. Further, I indicated that curtailed distributions did not arise in such 
cases as Dr Turner anticii)ated, e.g. the stature distribution of selected soldiers. In fact most of 
our anthroijometric distributions have been more or less selected, artificially or naturally, and 
they appear as a rule to be as normal as unselected material. 
Dr Turner has replied to my criticism with some interesting further statistics of smallpox. 
He takes the number of pock marks as given by the scheme below : 
Marks 0-100 100-500 500-1300 over 1300 
Frequency 2851 1385 1141 1572 
and suggests that they show a maximum frequency with the mildest cases. He does not, however, 
consider how far they approximate to that curtailed normal population, which as a whole he 
supposes to represent the total population which has run the risk of infection. Taking the four 
groups as they stand, the part of no normal curve whatever will even approximately lit them. 
It may be argued that the failure arises from a considerable number of the mildest cases, 
escaping notice at all. My assistant, Mr E. B. Ross, has therefore taken up the problem, 
omitting the first group altogether. Taking total population to rise by multiples of 10, he 
shows that the only way even to approach Dr Turner's numbers is enormously to increase the 
total population of which the above is to rej^resent the tail, but millions and billions of 
population running the risk of infection will not suffice. In fact the ratio of the bases of 
the two groupings — i'* 2, and the limit to this ratio for the given frequencies treated 
as normal even if the risk-running population were infinite would only be 1-32. As a matter 
of fact the "spot maps" show how small was the population which ran the risk of infection 
even in the London epidemic of 1901-2. Thus whether we include or exclude the group 0 to 
100, Dr Turner's data are wholly impossible even as an approximation to a curtailed normal 
curve. This want of any approach to normalitj' suggests the question of whether the material 
is even approximately homogeneous. Is it possible that the number of pock marks may be 
different according to the extent of acquired immunity 1 Is it not also true that 5 or 10 pocks 
are almost as rare as haemorrhagic cases and the frequency increases from such values up to 
at least 100 j)ocks? In other words the modal severity is not as Dr 1'in-nei''s diagram would 
lead one to suppose at the very mildest cases. If this be so, then the problem hinges on 
whether it is right to suppose severity a linear function of the number of pocks. Non-linear 
functions would not affect the application of fourfold-table methods, but they would affect the 
legitimacy of Dr Turner's argument. 
I think it will be found that unvaccinated cases at least follow fairly closely a normal 
distribution of pocking. Dr J. Brownlee kindly provides me with all the material available from 
the Glasgow Epidemic, 1900-1. We have : 
Sparse 
Abundant 
Confluent 
Haemorrhagic 
Totals 
Cases 
9 
41 
61 
4 
115 
Deaths 
1 
12 
42 
4 
59 
Percentage Deaths + P.E. 
ll-l±7-2 
29-3 ±4-8 
68-9 + 4-4 
100 ±9-8*? 
51-3 
Assuming the distribution normal I find : 
Range of " Sparse " : from — qo to — r417cr; mean of group — 1-868(7, 
„ „ "Abundant" : „ - 1-417 cr to -0-164(r ; „ „ - -694cr, 
„ „ "Confluent" : „ -0-164o-to -i-l-815o- ; „ „ -f- •597(7, 
„ „ "Haemorrhagic": „ -f l-815(r to -f- oo ; „ „ -|-2-208(r. 
'* Deduced by an extension of Bayes' Theorem. 
