H. E. SoPER, A. W. Young, B. M. Cave, A. Lee, K. Pearson 861 
and the distribution consists of 
COS" 
P) 
at r 
, , cos-i p , ^ ^ 
= + 1 and at /■ = — 1 
or at what is the same thing I (1 + r) and I (1 ~ r). 
The moments are 
Ml' = H-d = "f'' H-i = fJ-i = ^, 
and accordingly 
)U,2 = 1 - ^^ = - 2^ (1 - H-i = - r-) (1 + 3r-)...(lxxiv). 
Hence = 4r7(l - r^), = (1 + 3r2)/(l - r^). 
and accordingly 
- /3i - 1 = 0. 
TABLE I. Samples of Tivo. 
p, value of 
correlation in 
sampled 
population 
r, mean 
correlation 
of samples 
Number of 
positive cor- 
relations per 
1000 samples 
Number of 
negative cor- 
relations per 
1000 samples 
00 
0 
1 
0 
1 
500 000 
500-000 
01 
•063,7686 
•997,965 
•016,332 
1^016,332 
531^884 
468^116 
0-2 
•128,1884 
•991,750 
•066,827 
b066,827 
564 094 
435906 
0-3 
•193,9734 
•981,007 
•156,387 
1-156,387 
596^987 
403^013 
0-4 
■261,9798 
•965,007 
■294,764 
1-294,764 
630-990 
369010 
0-5 
•333,3333 
•942,809 
•500,000 
r500,000 
666^667 
333-333 
0-6 
•409,6655 
•912,236 
•806,686 
r806,686 
704-833 
295-167 
0-7 
•493,6334 
•869,670 
r288,724 
2^288,724 
746-817 
253-183 
0-8 
•590,3345 
•807,164 
2^139,534 
3^139,534 
795-167 
204-833 
0-9 
■712,8674 
■701,299 
4^ 133,056 
5^133,056 
856-434 
143-566 
1-0 
l^OOO.OOOO 
•000,000 
oc 
1,000-000 
-000,000 
The distributions are two lumps given by the last two columns, and are accurately 
given by Pearson's skew frequency distributions for the relation — — 1 = 0 
(see Phil. Trans. Vol. 216 A, p. 433). 
(ii) Samples of Three, n = 3. 
1 
dU 
where 
as before. Hence 
y'l _ ^2 dx' 
X = rp and U = cos~^ (— x)/Vi — x^ 
(■+1 1 r+p 
l^p = r^y^dr = — I x^y^dx 
^ -1. P • -p 
7Tp 
P+1 
-pVp' 
_ dU 
72 dx 
dx. 
* Since we can determine at sight whether any pair is positively or negatively correlated, this 
gives a method of determining p by simply counting the number of + 1 and - 1 correlations in the 
arrays, sav mp and ;n„ , then p = cos tt ( — — ) and the probable error of the determination is 
Vto„, + m„J 
-67449 
VT 
Vn ' N \ N 
N being the number of pairs used. Cf. "Student," Bioitietrika, Vol. vi. p. 304. 
B.ometrika xi 
24 
