W. Brown 
357 
These marks, together with the ages of the boys and their total marks for 
Geometry, Arithmetic, and Algebra, gave thirteen series of measurements and also 
thirteen corresponding series of "ranks" or orders. Correlation coefficients were 
now evaluated, using both actual measurements and also ranks. In the former 
case, Pearson's full method was used, the measurements were grouped, correlation 
tables were drawn up, and the product-moment formula, = ^S^Hy) ^yas 
applied*. In the case of ranks, two methods were used, (1) the product-moment 
formula, which in the case of ranks reduces to 1 
where (Z = differ 
11 ('/I- — 1) 
of corresponding ranks in the two series, was applied. This gave what 
Pearsonf calls p, or " rank- correlation," and the result was then converted into 
"true variate correlation" by the formula ?- = 2sin^^pj; (2) the formula 
R = l — where = gain in rank of second series on firstj, was applied, 
and again the true variate correlation was obtained by Pearson's formula 
r=2cosJ(l-E)-l§ 
o 
The above-mentioned two formulae giving r in terms of p and R respectively 
were deduced by Pearson on the assumption of "normal" or Gaussian distribution. 
If the correction is not made, in other words if ranks are themselves taken as 
measurements, the distribution obtained is a rectangle, — not obviously a reasonable 
assumption. 
The coefficients (crude values) obtained in these three ways are given in the 
following table : — 
S (xy) ^ . 
r= ' ' rmarks] 
r = 2 sin [ranks] 
r = 2cos|(i-i?)-l[ranks] 
Arith. Alg. 
•79 + -03 
■65 
•65 
Geom. Alg. 
•66 -t- ^04 
•63 
•59 
Geom. Arith. 
■58+ •OS 
•61 
■57 
CD 
•81 + -02 
■77 
■74 
EH 
•69+ -04 
■55 
■57 
CG 
•59 + -05 
•56 
•53 
FI 
•49+^06 
■49 
•51 
GI 
•49 ±-06 
■54 
•55 
DG 
•44+ -06 
•43 
•46 
BG 
•26+^07 
■24 
•24 
* For a specimen correlation table, see Appendix II. 
+ Karl Pearson: "On Further Methods of Determining Correlation," Drajjcrs' Company Besearch 
Memoirs, Biometric Series, iv., London, Dulau & Co., 1907, pp. 11 and 18. 
X C. Spearman: "'Foot-rule' for measuring Correlation," Brit. Journ. of Psychology, Vol. ii. 
Part I, July, 1906, pp. 100—104. 
§ Karl Pearson: Oi). cit. p. 17. Spearman suggested in his "Foot-rule" article the formula 
j-=sin li^ , which he attempts to justify on merely empirical grounds. A full criticism of his work 
on " ranks " will be found in Pearson's article above-mentioned. 
Biometrika vii 46 
