110 On the Probable Error of the Congelation Coefficient 
It will be seen that the differences of observed and calculated values are for 
the most part several times the probable errors. Estimated in this way case (3) 
p = '66, ?i = 4, is the worst fit, the difference of the mean being five times and that 
of the standard deviation nine times the probable errors. 
From the values of f and a r found above are calculated 
1 
2 
s 
4 
5 
a/ 
3 
7 
3-6140 
9-9905 
48-325 
. X-1 . 
0 
0 
0 
2-5259 
•8124 
•6831 
The frequency distributions of Type I to fit the numbers of samples taken in 
the experiments and the values of f and a r calculated will be of the form 
y = y 0 {\ -x) m - (1 -l- x) m 2 
when referred to the absolute origin and unit of measurement of r, and the 
constants* will be 
l 
2 
3 
k 
5 
»h=4(X-i)U-r)-l 
Wi 2 = i(X-l)(l+f)-l : =' 
3'o- 2mi+ms+1 • r ( mi + 1 ) , r (> /i2+ i) 
0 
0 
372-5 
2 
2 
703-12 
- -46844 
1 -08243 
202-25 
•6557 
6-3348 
95-131 
7-1825 
38-1425 
•0033889 
When the above frequency curves are plotted they appear as shown on the 
diagrams pp. 112, 113 and are seen to be in fair consonance with the frequencies 
observed in the experiments and shown by the rectangles upon the same diagrams. 
They are perhaps as good an expression of these frequencies as could be found 
amongst the type of frequency curve assumed. The case of p = 0, ?i = 4, for which 
theory prescribes a horizontal straight line is seen to be very nearly so in the 
experiment, apart from individual fluctuations. In p — 0, n = 8, the curve well fits 
the deviations from zero correlation observed. In p — - 66, n = 4>, the as^'mptotic 
nature of the distribution towards the value unity which the fitted curve fore- 
shadows is borne out in the samples of four drawn. With larger samples from the 
same class of material the displacement of the mode and the skewness of the 
distribution resulting from the assumed types are corroborated in the tests. 
At the same time it must be admitted that there are considerable differences 
* For the special case of p = 0, eqns. (ix), (xl) and (xliii) show that the curve is y = y 0 (1 - A' 2 )*' w-4 '> 
the form suggested for this case by " Student," Biometrika, Vol. vi. p. 306. 
