Lucy Whitaker 
53 
so that the odds are considerably against such an experience. The mean vakie of 
q is --0469 ± '0959 and the standard deviation of q is •5127 + "0678, both results 
compatible with q indefinitely small and a standard deviation = -4714. The main 
problem, however, of the legitimacy of applying the Poisson-Exponential to such 
series cannot be answered by data involving only total frequencies of 9 to 14 
cases in the individual series. 
Bortkewitsch examines the matter from another standpoint. He clubs the 
results given for each application of the Poisson-Exponential together and 
examines the observed totals against the sums of the calculated totals. Thus 
calculating the 11 Poisson-Exponential series* and adding them together 
Bortkewitsch finds for observed and calculated deaths : 
TABLE VIII. 
Accidental Deaths in 11 Trade-Societies. 
Number of Deaths 
0 
1 
3 
O 
4 
5 
6 
7 
9 
10 
11 
13 
13 & over 
Totals 
Observed Frequencies 
Sums of 11 Exponentials 
5 
3-7 
9 
9-6 
14 
13-9 
13 
1.5-2 
14 
14-3 
16 
12-3 
7 
9-8 
7 
7-3 
8 
5-8 
2 
3-3 
I 
2-0 
1 
1-2 
1 
0-7 
1 
0-6 
99 
99 
Single Binomial 
3-8 
9-5 
13-9 
1.5-6 
14-8 
12-4 
9-6 
6-9 
4-8 
3-1 
2-0 
1-2 
0-7 
0-7 
99 
If we attempt to fit a single binomial to the observed line of totals, we obtain : 
TO ^4-3636, a= = 7-5849 
leading to the negative binomial : 
99 (1-7382 - -7382) -^ii. 
Here: g = - -7382 + -1829t, » = - 5-9111 ± "1391, 
or the constants are significantly substantial with regard to their probable errors. 
The resulting frequencies are given in the last line of the table above. The reader 
* The values of the means and standard deviations for the eleven societies are : 
711 
m 
a 
m 
(T 
13 
7-889 
1-969 
23 
6-222 
2-485 
40 
2-889 
2-424 
14 
2-5.56 
1-343 
27 
1-889 
0-994 
42 
4-55G 
2-061 
12 
2-556 
2-217 
29 
5 -889 
2-885 
55 
4-333 
1-633 
20 
4-333 
2-211 
41 
5-111 
2-079 
1 
All these means are less than 10, which is the limit reached by Bortkewitsoh's Tables for the Poisson- 
Exponential. Bortkewitsch says he has taken the societies for which " the statistics indicated the 
smallest numbers of such accidents." This is not very clear. It is certain that a society with a mean 
number of accidents =100, if it consisted of 200,000 members, would be more suitable for application 
of the exponential, than one with a mean of 8 if it only contained 10,000 members. Both Bortkewitsch 
and Mortara confine their results to means less than 10, and seem to indicate that "smallness" has 
been determined by the absolute frequencies, but clearly it is relative frequency with which we have to 
deal. The use of such a term as Dos Oesetz der hleinen Zalilen for the Poisson-Exponential seems open 
to serious objection, if it be associated with " (;t " an absolutely small number, and not with smallness 
of " g." 
t For 5 = 0, the probable error would be ±-0959 and accordingly q is very divergent from the 
Poisson-Exponential value of zero. 
