2*70 On the Probable Error of Dr Spearman's Correlation Coefficients 
To see what happens when ties are carried to an extreme I determined p from 
the original table of 3000 entries (Biometrika, I. p. 216) and from the same table 
condensed to six groups each way by using a 4" scale of height and '8 mm. scale of 
finger lengths. 
In the first case p = '637 giving r^, and in the second "557 with rp '575. 
There seems therefore in extreme cases to be a tendency for the correction to give 
too low a value of p. 
Correction of R for grouping, 
S(D) 71^-1 . 
In Dr Spearman's original paper R is defined as 1 — r when — - — is taken 
H^l 
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as the average value which S(D) assumes. 
The simplest way to see that this is the average value is to write down all the 
possible D's thus : 
1 1 
2 2 1 
3 3 2 1 
4 4 3 2 1 
(h-1) (h-1) 
n n 
(h,-2) (>i-S) (m-4) 1 
(u-l) (»-2) (h-3) ... 2 1 
Here the two columns on the left are composed of the first n numbers. The 
third column is formed by subtracting the top number of the first column from all 
the numbers in the second column in turn, the fourth by subtracting the second 
number from all numbers which give a positive remainder and so on. 
Thus the numbers in the second column could be arranged opposite the numbers 
of the first column in n ! ways. 
And in (n — 1)! of these arrangements any given pair will occur. 
Hence the average value oi' S{D) will be 
O^i; 
[[1 + 2 + ...+(h.-1)] + [1 + 2 + ...+(»-2)]+... +(1 +2) + 1}, 
.•. average value of S(D) 
1 \n (n-1) (n- 2.3 1.2 ) 
2 2 " ^ + •■■ 2 2 J 
-i <?»A„. 1 |n(» + l )(2n+l) n(n + l)) 
= {2" + 1 - 3} = -g— (xn). 
If we now substitute in the second column ties instead of consecutive numbers 
we can find out what effect ties will have on the average value of S(D). As I can 
