374 Distribution of Correlation Coefficient in Small Samples 
Hence f = p + -116,729, for p = -2852 
= 4019, say. 
We cannot, however, be certain that this is correct to more than two figures. 
Equation (xlvi) gives us r = -4038. 
We will therefore start with r = -4030, say, as the basis of a more elaborate 
approximation, or p^^ = -1149372 say. 
Hence calculating and I.^, and using the difference formula we find 
^1 = 
1-6972,3599, 
h = 
1-0465,6745, 
1-2110,7453, 
1-0879,1327, 
h = 
1-0715,7031, 
h = 
1-1443,9624, 
h = 
1-0262,1201, 
h = 
1-2145,9528, 
h = 
1-0236,1262, 
1-2980,8251. 
Thus the equality of Zg and I^q has not been reached, so that we could hardly 
anticipate (xlvi) giving a very good result. Using (xli) we find 
e = + -0006,0228, 
a sufficiently small correction, leading to pg^ = -1155,3948. and r = -40512, correct 
to the fourth figure. Table VIII for n = 9 gives r in error by about 0"8 %. 
(12) Table for determining the "most probable'' value p of the correlation in a 
sampled population from the hnowledge of the correlation r in a sample of size n, ivhen 
n is considerable and it is legitimate to distribute ignorance equally. 
The required value is 
= , _ Ml _ A.3(r) _ Jgjr) 
^ «-l (n-l)2 {n-lf 
when r is positive ; if r be negative, p has the same value as for r positive, but with 
opposite sign. 
The above formula using Table IX will give p correct to five figures if n = 25 or 
over, and correct to four figures if n = 10 or over. 
It appears best to interpolate not for the separate A-functions, but for the total 
value to be subtracted from r to find p. Thus, suppose we require to find p for 
r =^ -6781 and for n = 16. We have for r = -65 
^ = -65 - -0127,3515, 
and for r = -70 p= -70 - -0121,8204. 
Therefore a difference of — -0005,5311 corresponds to a rise of -05 and accordingly 
one of - -0003,1085 to a rise of -0281. Thus 
p = -6781 - -0124,2430 
= -6657, accurately. 
