1V| Introduction \\\ 



In (ii) N is the size of the sample and n' the number of variatcs ar 1( a^, ... *-, 

 wilier multiple correlation with a? , the (n' + l)th variate, has to be tested. 



Now it is clear that if we write n =n'+ 1, (ii) becomes (i). Accordingly any 

 table based on (i) to give properties of T/ 2 , will give corresponding proper; 

 it \\( enter it with n = total number of variates involved, i.e. the variate x 9 with its 

 // multiply correlated variates. 



From the above curves we deduce 



fj 2 or $ 2 = Modal value of?; 2 , or of R?= ^ - (Hi), 



n 1 * 



if- or R 2 = Mean value of i) 2 , or of R 2 = ~ - (iv), 



<V = Standard Deviation of i) 2 = .. V2ip(l-if) (v), 



vN + 1 



a-jf = Standard Deviation of R 2 = - i J2R 2 (1-R*) ( vi). 



VJV + 1 



Since rj 2 and R 2 really stand for mean i) 2 and mean R 2 , or for (T; 2 ) and (R*), they 

 must not be confused with the mean 77, or 77, squared, and the mean R, or R, squaredf. 



Now Table IV gives the mean 7? 2 or R 2 , which for simplicity in printing we 

 write 7 2 and R 2 , and the value of the corresponding standard deviations, which 

 we will denote by cry and <TR*. 



We now take a quantity \ defined by 



Observed T? 2 Mean ?; a rj 2 rj* 



or alternatively by 



Standard Deviation of 7? 2 

 R 2 -R 2 



and consider what must be the value of \, or of 



in order that the tail cut off the distribution curves of TJ* or R 2 may be equal to 

 y^th or y&jths of the total frequency. The two values, \i and X a , are provided by 

 Woo's table. Thus \i and X 2 measure the deviations from the mean, ^ and 

 R 2 , at which the chances of an rj 2 or R 2 occurring are respectively less than 

 PI = '01 or P 2 = "02, supposing no correlation exists. When a value of rp (or .R 2 ) 

 gives a value of X greater than \i, then we may reasonably predict the existence 

 of correlation. When T? 2 (or R 2 ) gives a value less than Xj, then we must remain 

 doubtful as to the existence of any correlation. Had the distribution of rf or R 2 been 



* If the size of the sample be large, as in all satisfactory work it must be, ^r > ^- , the value 

 in use before the publication of the papers of Fisher and Hotelling. 



t A brief table indicating the differences between rj and V(ip) and (n) 3 and (17") is given in Biometrika, 

 Vol. xxi. p. 3. 



B. II. / 



