M. Greenwood and J. D. 0. White 
383 
While an analysis of counts of 75 would not enable us to settle the type 
distribution, yet in the face of such figures it is not justifiable to assume that 
the distributions are normal. 
But if the distribution of the variates round the mean of a sample is not 
Gaussian, then to assume that the curve representing the locus of means obtained 
from such samples is normal, and to use the ordinary notation for "probable 
errors," would be equally unjustifiable. 
In discussing a kindred subject in a recent number of this journal*, Pearson 
remarks — " A further point not yet discussed, but of some importance, is the 
significance of the ' probable error ' at all when we are taking the standard 
deviation of standard deviations in the case of very skew material. How far does 
■ the distribution of standard deviations follow a normal curve ? " 
In a valuable meraoirj-, " Student " has investigated the distribution of the 
means of small samples derived from a normal population, and has shown that 
such distribution is not Gaussian. The problem has not yet, we think, been 
studied in the case of skew material. Theoretically it is difficult because the 
analysis becomes troublesome when we have to deal with a case in which, among 
other things, the odd moments do not vanish ; the problem is nevertheless of such 
importance that we hope to publish later some theoretical considerations. 
However this may be, whether we can determine the curve of means with the 
help of algebraic analysis alone, or whether we are compelled to fall back on 
directly empirical methods, an essential preliminary is a knowledge of the 
frequency distributions of phagocytic cells with respect to the number of bacteria 
contained in them, as determined on statistically adequate samples of the material 
from which the small working-samples are extracted. When we have this know- 
ledge, the question as to whether the opsonic method be worthy of confidence will 
be, not solved, but capable of solution. 
We hope that this vital point will be clear to the non-statistical reader. Briefly 
recapitulating — The reliability of the opsonic method turns on whether the mean 
of the sample measured with the control serum, and the mean of the sample 
measured with the patient's serum represent adequately the values which would 
be obtained if a very large number of cells had been counted in each case. We 
require some measure of the variation in each mean value which may be considered 
to depend, not upon technical errors, but upon the fact that only a small sample of 
the population is examined. Until this question is settled, manipulation of the 
ratios of the means of samples (i.e. opsonic indices) does not seem to be desirable. 
Such statistical tests as are ordinarily applied to the cases of small random samples 
are based on assumptions which, in this instance, are arbitrary. 
We shall therefore first investigate the frequency distributions of phagocytic 
cells with respect to the content of tubercle bacilli, determined by the analysis of 
* Biometrika, Vol. vi. p. 117. 
t Biometrika, Vol. vi. pp. 1 — 25. 
