510 
On the Distribution of the Means of Samples 
The only prejudice to which the operator seems to have been — and thinks he was — subject 
was the idea which impressed itself on his mind during the count that the number of 3 bacilli 
per cell occurred with the greatest frequency ; this idea may have inclined him to welcome 
threes with undue alacrity. It is possible that this may have helped in some degree towards 
producing an excess of threes in the final result. The operator believes that, at least in his 
case, the auditory sense is more impressed with the sound of numbers containing e's and I's, 
3, 5, 9, than with others ; he thus tended to believe that these numbers occurred more frequently 
than was actually the case. Evidently the truth of the matter cannot be exactly known until 
several independent observers have undertaken the same large count. 
We have already mentioned the difficulty regarding clumps ; another point is as to whether 
some gross heterogeneity were produced either by notable differences in various slides or by 
variations in the method of counting on different occasions, diff'erences of which the operator 
was unconscious. In making the frequency tables just considered, there emerged necessarily a 
rough representation of a frequency curve for each five hundred cells. All these curves showed 
a rough similarity and each one a rough regularity. In some cases, however, the regularity was 
appreciably less than in others. On going through the data a second time, the groups of 500 
which showed the greatest irregularities were excluded, on the supposition that they might 
represent a somewhat lower grade of accuracy in counting and might have introduced a gross 
heterogeneity into the total count. The application of this criterion led to the exclusion of 
5000 cells. The remaining 15,000 cells were then analysed as before. 
Tables III, III a and Graph 2 indicate the results of this process. There is a slight improve- 
ment in the fit, but the change is not pronounced. It can hardly be said that the results justify 
the exclusion. 
TABLE III. 
Actual Distinhvtion of 15,000 Cells. 
Bacilli 
Number of 
Bacilli 
Number of 
per Cell 
Cells 
per Cell 
Cells 
0 
1092 
9 
306 
1 
200« 
10 
203 
2 
2585 
11 
126 
3 
2632 
V2 
69 
4 
2096 
IS 
44 
5 
1518 
U 
24 
6 
1035 
15 
12 
760 
16 
6 
8 
484 
Although we should naturally have preferred to obtain closer fits, the result 
does not seriously affect the value of our material for the purposes of the present 
investigation. It is clear that the heterogeneity is not caused by a compounding 
of Gaussian distributions, and that we have to deal with markedly skew variation. 
We sincerely trust that some other workers may be tempted to repeat the experi- 
ment. We now turn to our main inquiry, viz. the problem of sampling. 
The 20,000 cells were arranged in consecutive 25's, 50's and lOO's, and grouped 
as shown in Table IV. In each case the group is an inclusive one, thus 76 — 83 
