AN EXPERIMENTAL DETERMINATION OF THE PROB- 
ABLE ERROR OF DR SPEARMAN'S CORRELATION 
COEFFICIENTS. 
[Being the paper read to the Society of Biometricians and Mathematical 
Statisticians, December 13th, 1920.] 
By " STUDENT." 
In the British Journal of Psychology, Vol. ii. p. 96, 1906 *, Dr Spearman suggested 
two methods of determining correlation, based on replacing actual measurements by 
ranks. 
As an illustration we may take the following purely imaginary example : 
TABLE I. 
Individual 
Height 
Length of 
middle linger 
mm. 
Kank in 
height 
Rank in 
length of 
finger 
A 
6' 0" 
12-8 
2 
1 
B 
5' 3" 
11 -.5 
4 
3 
C 
5' 7" 
10-0 
3 
4 
D 
6' 1" 
12-4 
1 
2 
Instead of correlating the figures in the second and third columns of the above 
table Dr Spearman proposed to use the figures in the fourth and fifth columns, 
and to determine one or other of two coefiicients : of these the first {p) gives the 
ordinary correlation coefficient between the figures representing the ranks, and the 
second (i?) was described as a ' footrule ' for correlation, i.e. a rough instrument 
which could be used by the unskilled. Dr Spearman also proposed to use R in 
cases where it was thought advisable to weight mediocre observations more heavily 
than extremes. 
The method of determining p and R was to take the difference D between the 
numbers representing the ranks, e.g. for A in Table I 
Z) = 2-l = l. 
* [Dr Spearman's results were first given in a paper entitled "The Proof and Measurement of 
Association between Two Things" in the American Journal of Psychology, Vol. xv. pp. 72 — 101, 1904. 
The dogmatic statements as to the accuracy of his methods in that paper are, 1 think, erroneous, and 
he does not lay adequate stress on the fact that correlation of ranks is not a correlation of variates and 
may differ very considerably from it. The suggestion of considering the correlation of ranks is due to 
A. Binet and V. Henri : see La Fatigue Intellectuelle, Paris, 1898, p. 252, also VAiiiicc Psychologique, 
T. IV. p. 1.55, Paris, 1898. Their process is very obscure and they also do not appear to have realised 
that the correlation of variates is not that of ranks. Ed.] 
