264 0)1 the Prohahle Error of Dr Spearman's Correlation Coefficients 
Then _ 
11 {n^ — 1) ^ ' 
6 
and R = l -— (n), 
where ?i is the nnmber in the sample : in the case of R, S (D) denotes the summa- 
tion of positive differences- only. 
Dr Spearman gave an empirical formula connecting R and p, viz. p = sin ^ R^, 
but I do not suppose that he attached any very great importance to this. 
He further gave the probable errors of p and R for the case of no correlation 
•6745 , -4266 
as — , — and — ; — . 
V n sjn 
In his memoir ' On further methods of determining correlation ' Prof. Pearson 
investigated these coefficients for the case of the normal correlation surface and 
found the relations between p and R and r the ordinary correlation coefficient 
to be 
r = 2 sin 
and r = 2cos~(l-ii)-l (iv). 
o 
Pearson further found the standard error of p to be for large samples 
■"^'{l + -086p" -l--013p^ + -002p'^-l- ...j (v), 
and of ?-p, i.e. r determined from p by (iii), to be 
1-0472 {1 + •042r2 + -QOSr^ + •002r6 + . . .1 (vi). 
He did not succeed in evaluating the error of R or of (i.e. of r determined 
by (iv)), but pointed out that just as in the case of r the \]n in the denominator is 
really Vji — 1. He also pointed out that R can only take values between + 1 and 
•4226 
— •S and that Spearman's ^ does not imply that R is more accurate than p or 
•6745 
r with their probable error of ^ - since R itself is smaller than p or r in about 
the same proportion. 
Since that time the use of p and K has become general among psychologists, 
especially in America, where they are preferred to r on account of the ease and 
speed with which they can be determined for small samples. 
For example in a note on correlation in Employment Psychology, by H. C. 
Link*, a book written to urge the claims of Psychology on the devotees of 'Scien- 
* Macmillan, 1919. 
