268 On the Probable Error of Dr Spearman's Correlation Coefficients 
Tx would be|^ + 2 + 5 = 7| and if there were no ties in the y ranks p would be 
165-7^-^(1)^) ^ 1571- >S(Z) -) 
\/(165-15) 165 ~ Vl50Tl65 ' 
and if we were to take it as 1 — ^ ^^^ the error would come in the third significant 
Lull) 
place of decimals. 
In determining p for my 375 samples of eight I found that much tied samples 
usually gave low values of p and it occurred to me that although undoubtedly 
equation (x) gives the true value of the correlation of ranks, yet it might be 'that 
the loss of precision due to ties would give low values for the correlation. To test 
this I doubled the width of my unit of grouping first for one variable and then for 
the other so that I got three values of p for each sample : 
(i) Converting the original figures into ranks, 
(ii) Using coarser grouping on one side and the original grouping on the 
other before converting into ranks, 
(iii) Using coarser grouping on both sides. 
An example will make my meaning clearer. 
(1) ■ 
Original figures 
0 
_ 2 
+ 3 
- 1 
+ 1 
+ 1 
+ 3 
0 
+ 3 
0 
+ 3 
_ 2 
+ 3 
+ ■2 
+ 4 
+ 1 
(2) 
X grouped 
coarsely 
Putting + 1 
andOas^, &c. ^ 
+ A 
- n 
+ n 
-H 
+ 
+ i 
+ H 
+ I 
I 
+ 3 
0 
+ 3 
_2 
+ 3 
+ 2 
+ 4 
+ 1 
(3) 
Both grouped 
coarsely 
+ i 
-U 
-n 
+ * 
+2.1 
+ I 
Pairs 
Ti'iplets 
Quartets 
+ 2-1- 
+ * 
+ 2i 
- n 
+2I 
+21 
+41 
+ * 
Ranks 
(1) 
(2) 
51 
8' 
U 
7 
31 
■4 
n 
51 
41 
7* 
u 
n 
4i 
41 
1* 
41 
3 — 
— 1 
2 — 
— 1 
(3) 
4* 
7| 
li 
7* 
41 
^2 
U 
4I 
2 
1 
31 
6| 
3* 
3| 
1 
6i 
1 
1 
Here originally T^^U^ and Ty = 2 and n{n?_V) _ ^ ^ ^ ^ 
After grouping x coarsely = 6 and Ty = 2 „ = 76. 
After grouping both coarsely Tx = Q and Ty = 5^ „ = 72|, 
and p will be found to take the values "833, "869 and "834 in succession. Working 
in this way I obtained three values of p for each of the 375 samples and determined 
the mean, standard deviation and mean {Tx + Ty) for each of the three series of 
375. These results are given in Table II. 
