512 Distribution of the Correlation Coefficients of Samples 
of r is, therefore, as we know, either — 1 or +1, and the proportion, in which 
these occur, depends upon p. The ratio of the infinite areas included with the 
asymptotes of the above curve is 
cos~^ p 
cos~^ (— p) ' 
so that the mean value of a number of observations is 
Sll1~^p 
TT 
2 
When 11 = ^ there is still no approach to normality, the curve takes the form 
1 
{6 -2, cot 6 + Se cot^ 0), 
which, when r is positive, increases regularly from its value of when ^ = 0, to 
infinity, to which it approaches as 0 approaches tt. Unless p is actually equal 
to 1, in which case r is also 1 of necessity, the curve has finite ordinates at both 
extremes. For calculating the number of values which should fall within any 
given range, the integral, ^.^^ ^ (1 — 6 cot 6), may be directly tabulated, as has 
been done in forming the accompanying table of "Student's" observations, and 
the corresponding expectations. The values given by Mr Sopor's formula are 
apposed for comparison. 
Table for comparison with p. 114, Biometrika, Vol. IX. 
•905— 1 
■805— -905 
•705— -805 
•605— 
■505— 
•405— 
•305— 
•205— 
•105— 
•005 — 
1-905- 
1-805— 
1^705— 
r605— 
1'505— 
1-405— 
1-305— 
1-205- 
]_-105— 
1— 1^105 
Calculated 
frequency 
202 
124 
88 
65 
49 
37 
30 
24 
20 
17 
14 
12 
10 
9 
8 
7 
6 
5 
Observed 
175-5 
136-5 
84 
66 
55 
45 
24-5 
24^5 
19 
7 
22 
12 
13 
3 
12 
16 
7 
10 
4 
9 
745 
Difference 
e 
15-0 
+ 12-3 
}- 
6-4 
11-6 
+ 7-1 
4-0 
+ 12-7 
+ 5-1 
+ 3-6 
•69 
•09 
1-73 
•74 
3-58 
1- 87 
-80 
10-54 
2- 19 
1-38 
23-61 
H.E.Soper's 
approxi- 
mation 
230-3 
98-9 
72-1 
57^6 
48^0 
40-2 
34-3 
29-7 
25-6 
22-0 
18-8 
16-0 
13^5 
11-2 
9-0 
6-9 
5-1 
3-3 
1-9 
-6 
Difference 
I -17-2 
I +20-3 
I +11-8 
I -15-0 
} 
} 
■21-6 
8-7 
} +12-1 
} + 8-6 
J +10-5 
•90 
3-18 
1-58 
3-52 
9-80 
0-02 
3-06 
9-21 
8-80 
44-10 
84-17 
