Karl Pearson 
15 
(8) I am able to illustrate by another example kindly worked by Mr E. C. 
Rhodes, the accuracy of this binomial curve. He has considered the case of an 
indefinitely large population containing 10 ^ of a given character, and worked out 
the Type III cTirve by both the geometrical relationship and the equality of 
moments to determine the frequency distribution of samples. 
TABLE II. 
Table of Fercenta[/e Frequencies up to giueii Distances from the Mean. 
In Excess of Mean 
In Defect of Mean 
Up to 
Gaussian 
Actual 
IS* Curve 
C|- Curve 
Up to 
Gaussian 
Actual 
B Curve 
C Curve 
•5 
9-40 
9-25 
8-84 
9-05 
- -5 
9-40 
9-25 
9-29 
9-01 
1-5 
26-24 
24-66 
24-01 
24-52 
- 1-5 
26-24 
27-34 
26-76 
27-55 
2-5 
38-30 
35-42 
34-71 
35-40 
- 2-5 
38-30 
41-19 
40-25 
41-09 
3-5 
45-22 
41-85 
41-39 
41-96 
- 3-5 
45-22 
48-99 
48-03 
48-57 
4-5 
48-40 
45-18 
44-89 
45-35 
- 1-5 
48-40 
51 -85 
51-13 
51-40 
5-5 
49-57 
46-70 
46-68 
46-72 
- 5-5 
49-57 
52-37 
52-00 
52-10 
6-5 
49-91 
47-31 
47-38 
47-53 
- 6-5 
49-91 
52-37 
52-17 
.52-27 
7-5 
49-99 
47-53 
47-72 
47-73 
- 7-5 
49-99 
52-37 
52-17 
.52-27 
8-5 
50-00 
47-61 
47-78 
47-73 
- 8-5 
50-00 
52-37 
52-17 
52-27 
9-5 
50-00 
47-63 
47 -80 
47-73 
- 9-5 
50-00 
52-37 
52-17 
52-27 
10-5 
50-00 
47-63 
47-83 
47-73 
- 10-5 
50-00 
52-37 
52-17 
52-27 
* B is the Type III curve determined by the geometrical relation, 
t C is the Type III curve determined by equality of moments. 
It will be seen that the use of mduients gives a better result than the 
geometrical property. But Type III curve obtained either way is far closer 
than the Gaussian to the actual distribution. See Diagram II. 
(9) It would appear, therefore, that starting from an enlarged view of Bayes' 
Theorem, and approaching the matter from the standpoint of Laplace, we reach, both 
theoretically and of course in any case of actual numbers, a better result than 
the normal curve by using Type I or Type III for our hypergeometrical series, i.e. 
by using tables of the incomplete B- or F-functions rather than the table of the 
probability integral (areas of the Gaussian or normal curve). It must be insisted 
upon that Laplace reached the probability integral as an approximation to the sum 
of terms in a certain hypergeometrical series, and any better approximation to the 
sum of those terms has greater logical validity than L;iplace's integral. Other 
deductions of the probability integral are, so to speak, after-thoughts. Its essential 
origin lies in Laplace's endeavour to solve the fundamental problem in statistics. 
The so-called Gaussian or normal curve was first introduced into statistics as a 
rough and ready solution for the sum of a certain number of terms in a definite 
hypergeometrical series, and the sacrosanct character of the " probability integral " 
and the " probable error " in the eyes of many physicists and astronomers is of 
the character of a dogma ; it is based on authority rather than reason. 
