ON THE DISTRIBUTION OF THE CORRELATION CO- 
EFFICIENT IN SMALL SAMPLES. APPENDIX II TO 
THE PAPERS OF ''STUDENT" AND R. A. FISHER. 
A COOPERATIVE STUDY 
By H. E. SOPER, A. W. YOUNG, B. M. CAVE, A. LEE 
AND K. PEARSON 
CONTENTS 
PAGE 
(1) Introductory 328 
(2) Properties of the Function U = cos~~'^ {- x)/Vl - x-. (Ordinates of the 
Frequency curves) 329 
(3) On the Determination of the Monicnt-Coeflficients. Series and Difference 
Formulae ............ 332 
(4) On the Determination of the Mode 342 
(5) Determination of Ordinates and Mode by Expansion .... 346 
(6) Equation for IMode and Antimode (;!, = 3) 350 
(7) Tables and Models 351 
(8) On the Determination of the "niost likely" Value of the Correlation in the 
Sampled Population .......... 352 
(9) Special Cases of Frequency for n small: 
(i) Samples of Two . 360 
(ii) Samples of Three 361 
(iii) Samples of Four 369 
(iv) General Case of small Samples, /i>4 . . . . 371 
(10) Approach of the Distribution as n increases to a Normal Character . 371 
(11) Table for determining the Mode of the Frequency Distribution for n of 
considerable size ........... 373 
(12) Table for determining the "most probable" value of the correlation for 
n of considerable size 374 
(13) Construction of Table of (/„ = ' sm"-^(j>d4> 375 
Appendix of Tables A— C 379 
(1) Introductory. In a paper of 1908* '"Student" dealt experimentally with 
the distribution of the correlation coefficient of small samples, and gave empirical 
curves — in particular for the case of zero correlation in the sampled population 
— which have proved remarkably exact. The problem was next considered in 
1913 by H. E. Soperf who obtained the mean correlation and the standard 
deviation of the distribution of correlations to second approximations. Of the 
* Bionietrika, Vol. vi. p. 3U2 et seq. 
I' BiomclriJca, Vol. ix. p. 91 et acq. 
