370 Distribution of Correlation Coefficient in Small Samples 
CT^2 4- ^2 = = 1 — 2 cot-a + 2a cot^a (Ixxxvii), 
2 
ug' = - {cot a + 6 cot'^a + a (1 — 3 cot^a — 6 cot* a)} 
77 
(Ixxxviii), 
/x/ = 1 — 4 cot^a — 6 cot*a + a (6 cot^a + 6 cot^a)* 
(Ixxxix). 
From these formulae were calculated the values given in the following table. 
TABLE III. Samples of Four. 
p, value of 
correlation 
of sampled 
population 
f, mean 
value of 
correlation 
in samples 
M3. 3rd 
moment 
coefficient 
0-,, 
Usual value 
assumed for 
(T,., i.e. 
\/n- 1 
132 
0-0 
0 
0 
■577,3503 
■577,3503 
0 
b800,000 
00 
0-1 
■084,9678 
- •033,6268 
•574,5653 
•571,5768 
■031,429 
b839,929 
58^5418 
0-2 
•170,4532 
- •065,2863 
■566,0965 
•554,2563 
•129,510 
1 •964,665 
15^1700 
0-3 
•257,0089 
- •092,9620 
•551,5835 
•525,3887 
•306,862 
2^ 190,708 
71391 
0-4 
•345,2652 
-•114,5383 
•530,3576 
■484,9742 
■589,510 
2^552,205 
43294 
0-5 
•435,9911 
-•127,7520 
•501,3081 
■433,0127 
1 028,270 
3- 116,256 
30306 
0-6 
•530,1976 
-•130,1567 
•462,6087 
■369,5042 
1 ■728,423 
4-022,982 
2^3275 
0-7 
•629,3378 
-•119,1407 
■411,1087 
■294,4486 
2^940,226 
5-609,288 
1^9078 
0-8 
•735,7362 
-■092,1708 
■340,7311 
■207,8461 
5^428,946 
8-922,221 
1 6435 
0-9 
■853,9806 
-■048,1281 
•236,6586 
■109,6966 
13^184,043 
19^571,006 
1^4844 
0-95 
•920,8889 
-■021,4678 
■157,7942 
■056,2917 
29^8558 
43^4082 
1-4539 
0-98 
•965,7599 
- ^006,4363 
■088,2666 
■022,8631 
87^5994 
130^1935 
1-4862 
0-99 
•982,1321 
- ^002,4358 
■055,4859 
•011,4893 
2033250 
3117316 
1-5332 
1-00 
1 
0 
0 
0 
00 
00 
l-8305t 
values of p near unity 
* These expressions cannot be applied to the case of p = 0. We must return to Equation (ix) and 
put p = 0, finding = or a horizontal line. 
t The ratio pjp^ equals in the limit (27ir2 - 256) (Stt^ - 16)/(97r2 - 80)= = 1-8305. This may te 
1 - ie-, or e = V l - , where e is small. We then find for 
1 - r)l 
3W 
256 \ 
r = p2 _ (1 _ p2 
.37r 
37r-~]{l-p^-)^, (ott- ^^(1 - 
80 
These lead to 
80 
37r - 
and give the above result. 
