70 The Opsonic Index — A Medico-Statistical Enquiry 
fashion what the reliability is for counts of 100 instead of 25. The case chosen 
for illustration, however, is one which bears most hardly upon this question of the 
reliability of the opsonic index. The tubercle powder used for making the emul- 
sions invariably consists of old, dead and dried bacilli. It is scarcely to be expected 
that the reactibility (if I may use the term) of the organism under these circum- 
stances would stand very high. The degree of variability therefore which one 
could hope to get with such a reagent would not be very extended, and this is 
borne out by experience. The records of the Inoculation Department of St Mary's 
Hospital, London, now afford a very considerable body of data which would I think 
well repay the analysis of some kindly mathematician. The records concerning 
the tuberculo-opsonic index are particularly large, and although by taking this 
index as the basis of our argument we state the case for reliability in the most 
depreciatory manner, still its supreme importance requires that it should be con- 
sidered before all others. The rule at St Mary's Hospital is to take from different 
pei'sons in the laboratory several normal sera against which to compare the test 
sera for the day. We have therefore here material for testing variability of 
normal and test serum indices. Suppose we have, corresponding to any given 
batch of test sera, three normal sera 1, 2, and 3, then the average of any one of 
these normal sera distributions may be supposed equally valid as supplying the 
normal unit for the estimation of the index. Taking the three normal sera by 
themselves we might calculate a set of "normal" serum indices by dividing each 
normal average by the other, that is to say 1 by 2, 1 by 3, 2 by 1, 3 by 1, 2 by 3 
and 3 by 2. In the same way we might obtain " test " serum indices by dividing 
the average for each " test " serum by the averages of the several normal sera in 
turn. By doing this systematically in every case and collating results we should 
obtain two frequency distributions, and be in a position to state what is the degree 
of variability amongst normal and test sera, at least as tested by means of an 
emulsion of old and dried tubercle bacillus. Further if we assume that degree of 
variability is the basis of the differentiation of an opsonic serum from a normal 
serum then the degi ee of variability of the corresponding averages or ratios of 
averages may give us some idea at any rate of the limits within which their 
differentiation becomes justifiable. The objection to the procedure is: that we 
introduce into the " normal " index frequency distribution both indices, and their 
reciprocals. It would probably be better then to use some other method which 
would not have this effect. It is essential too, in selecting any one normal average 
for use in the determination either of normal or of test indices, that it should be 
altogether a random selection. The plan adopted to obviate the first difficulty 
and to comply with the requirements of random selection was to take only one 
normal average as divisor. Amongst the two or three normal averages from which 
I had to select in any given batch of observations, if I took the first recorded in 
the one case, then I took the second in the next and the third in the next, or if 
there was no third went back to the first. This method was followed for both 
" normal " index and " test " index distributions. In order to compare the varia- 
bilities in question, one forms the frequency distributions and then the frequency 
