FOUNDATIONS OF THEORETICAL 
STATISTICS. 
363 
Whence it appears that the further loss of efficiency is only 
a 8 
10,800 ' 
We may conclude, therefore, that the high agreement between the optimum value of 
a- and that obtained by Sheppard’s correction in the above example is characteristic 
of grouped normal data. The method of moments with Sheppard’s correction is highly 
efficient in treating such material, the gain in efficiency obtainable by increasing the 
likelihood to its maximum value is trifling, and far less than can usually be gained by 
using finer groups. The loss of efficiency involved in grouping may be kept below 
1 per cent, by making the group interval less than one-quarter of the standard deviation. 
Although for the normal curve the loss of efficiency due to moderate grouping is very 
small, such is not the case with curves making a finite angle with the axis, or having at 
an extreme a finite or infinitely great ordinate. In such cases even moderate grouping 
may result in throwing away the greater part of the information which the sample 
provides. 
3. Distribution of Observations in a Dilution Series. 
An important type of discontinuous distribution occurs in the application of the 
dilution method to the estimation of the number of micro-organisms in a sample of 
water or of soil. The method here presented was originally developed in connection 
with Mr. Cutler’s extensive counts of soil protozoa carried out in the protozoological 
laboratory at Rothamsted, and although the method is of very wide application, this 
particular investigation affords an admirable example of the statistical principles 
involved. 
In principle the method consists in making a series of dilutions of the soil sample, 
and determining the presence or absence of each type of protozoa in a cubic centimetre 
of the dilution, after incubation in a nutrient medium. 
The series in use proceeds by powers of 2, so that the frequency of protozoa in each 
dilution is one-half that in the last. 
The frequency at any stage of the process may then be represented by 
n 
m — —, 
2 X 
when x indicates the number of dilutions. 
Under conditions of random sampling, the chance of any plate receiving 0, 1. 2. 3 
protozoa of a given species is given by the Poisson series 
