The distribution of the sample correlation coefficient with one variable fixed

Abstract

For th e usual st raight·line model, in whi c h the ind ependen t variable ta kes on a fixed, kno wn se t of va lues, it is shown th at th e sample co rre latio n coeffi cie nt is di stributed as Q with (n-2) degrees of freedo m and noncentralit y O=({3 /CT) V~(Xi -X)2. The Q variate has bee n defin ed and studi ed e lse· wh ere by Hogben et al. It is noted th at the square of th e correlation coeffi cient is di stributed as a nonce ntral beta variab le. Key Words: Anal ysis of variance, calibrali on, co rrelation coeffi cie nt, degrees of freedom, di s-tribution , fix ed vari abl e , nonce ntral beta variabl e, noncentrality, Q variate.