2G6 On the Prohahle Error of Dr Spearman's Correlation Coefficients 
the first way would rank 1, 2, 2, 4 while the second would rank 1, 2J, 4 if the 
second and third of four individuals constituted a tie. 
Now the first would give different results according as we read the scale forwards 
or backwards and also alter the mean of the set of numbers so I have only tried 
to use the mid-rank method, for which I have found the correction which follows. 
Correction of p for ties. 
If D = X — y, when x and y are any two variables measured from their means, 
then 
D" = X- + y- — %xy. 
Summing for all n samples and dividing by ii 
n .1 ., . 
crJ" + o-/ - 1 2o-^o";, (vii). 
If now X and y are the first n numbers, then 
1 p „ -, ^ , /sum of 1st n numbersX^ 
a/ = <jy- = - X sum 01 squares oi 1st n numbers — ( j 
(« + 1)(2h + 1) (n + l)= 
6 4 
— 1 
12 
Substituting in (vii) we find 
p = r, -' 
.(viii). 
6 n 7/ 6 
n (n^ — 1) 
6 
.(ix). 
71 
6 
Now suppose that thei'e is on the x side a tie of t in number from q to q + t — 1. 
Using the mid-rank method we substitute for each of the numbers 
. , ■ 2q + t-l 
q, q + 1, ... q + t — 1 their mean ■ . 
Hence in finding a/ the mean is unaltered but in the sum of the squares 
q"- + (q + lf + ... + (q + t-iy]s replaced by H^l+llL^^ 
4 
Hence a.J is smaller by 
