Lucy Whitaker 
43 
than if we apply the ordinary process of mean nq, standard deviation \/npq, and 
Sheppard's table for areas to the frequencies. It will be noted that this amounts 
to using Sheppard's correction on the crude second-moment and slightly shifting 
the central ordinate towards the side of greater frequency. This is the Gaussian 
curve used in Table I. 
The object of the present section of our work is to indicate how far it is 
legitimate to use the Poisson-Exponential up to cell frequencies of the order 30 
in a population of about 1000* and how far we then reach a state of affairs, which 
for practical purposes may be described by ordinary tables of the Gaussian. It 
will be seen from Table I that the Poisson-Exponential even for = 10 and 30 is 
not extremely divergent from the Binomial. 
In Plate VII the transition of the exponential histograms of frequency towards 
the Gaussian form is indicated for cell-frequency = 1, 5, 10, 15, 20, 25 and 30 ; in 
the cases of 10 and 30 the corresponding Gaussian curves are drawn. 
It will be seen that with due caution the Poisson-Exponential may be reason- 
ably used up to frequencies of about 30 in the 1000, and that after that it would 
be fairly satisfactory to use the areas of the Gaussian curve as provided in the usual 
tables. 
(6) In order to table the results of the Poisson-Exponential for easy use, it 
seemed desirable to turn them into percentages of excess and defect. For example 
take the distribution for a frequency 5. It is : 
Per cent, of Cases in which : 
0 
•006,737,945 
a defect of 5 occurs : 
0-674 
1 
•033,689,725 
„ 4 or more „ : 
4^043 
2 
•084,224,310 
„ 3 or more „ : 
12-465 
3 
•140,373,850 
„ 2 or more „ : 
26-503 
4 
•175,467,310 
„ 1 or more „ : 
44-049 
5 
•175,467,310 
the true value „ : 
17-547 
6 
•146,222,755 
an excess of 1 or more „ : 
38-404 
7 
•104,444,825 
,, 2 or more „ : 
23-782 
8 
•065,278,015 
3 or more ,, : 
13-337 
9 
•036,265,564 
,, 4 or more „ : 
6-809 
10 
■018,132,782 
„ 5 or more „ : 
3-183 
11 
•008,242,173 
,, 6 or moi e „ : 
1-370 
12 
•003,434,238 
,, 7 or more „ : 
0-545 
13 
•001,320,860 
„ 8 or more „ : 
0-202 
* Of course in the Poisson-Exponential itself the total frequency plays no part ; it is only useful in 
testing the validity of the approximation. 
6—2 
