MISCELLANEA. 
I. On the Distribution of Severity of Attack in Cases of Smallpox. 
By KARL PEARSON, F.R.R. 
Dr Turner finds that the protection provided by vaccination does not diminish as rajiidly 
with age as he would expect it to do. His expectation arises from the fact that current medical 
opinion considers that the immmiity provided by vaccination diminishes rather rapidly with the 
increase of the period which has elapsed since vaccination. I do not propose to consider whether 
protection against death when disease is incurred is really of the same character as immunity 
against an attack. But I should wish to point out that Dr Brownlee has reached a very similar 
conclusion to Dr Macdonell on this point by a very different process, and further that it follows, 
if we simply use the coefficient of association, which makes no a2:)peal to normality of distribu- 
tion. On the other hand, while I am distinctly interested in Dr Turner's theory of a " curtailed 
normal curve," I am compelled to say that I consider it an extremely improbable hypothesis in 
the present application. The group of persons who catch smallpox are a selected group of the 
general population, selected because (i) they have come in contact with the disease, (ii) they 
were at the time of such contact not sufficiently immune against it. This sufficiency of im- 
munity must depend not only on prior vaccination, but on a host of other causes, the virulence 
of the poison they encountered, their particular state of health at the time in question, their 
conduct before and after the risk was run, etc., etc. It is impossible to suppose a rigid line 
drawn at a certain grade and say all below this grade escape this disease, all above will contract 
it. I cannot understand how those ' selected ' to incur the disease differ from any other naturally 
selected group with which we are acquainted. Now biometricians are dealing with selected 
groups every time they measure the variation of a character in a species, but no such truncated 
normal distributions have yet exhibited themselves*. 
Dr Turner says "A curtailed table in measurable characters would result if wc collected 
statistics of height in soldiers, a population from whom all below a certain standard height had 
already been rejected" (p. 497 above). Now this is a case which can be well tested. P'or 
example, height standards exist for both the American and Italian armies. Yet what do we 
find ? That the distribution of the statures of the accepted recruits in both countries, so far 
from forming a curtailed normal distribution is in some cases as close to a complete Gaussian 
* American trotting horses give the nearest approach, there being a time limit to entry in the record 
of trotters. But this is a perfcctly arbitrary line drawn across the trotting population by tlie hand of 
man, and not a selection due to a complex of natural causes, 
Biometrika iv 64 
