cxxxvi Tables for Statisticians and Biometriciam [XXV 



Take the following series of values from " Student's " original paper : 

 Additional Hours of Sleep gained by the use of hyoscyamine kydrobromide. 



Now it is clear that s M _ p =l-17 is much less than \/(r70) 2 + (l'90) 2 , which it 

 should be, were u and v independent. Actually worked out on these ten cases the 

 correlation is over '79. Is it likely even on ten cases that the correlation would 

 exceed numerically "79, if it were really zero in the parent population ? 



The curve of distribution is (see p. cxxv) 



y = y Q (l-r z ) 3 . 



We have therefore to enter our table with # = r 2 = '6241, and as ('* 3) =3 

 with n = 9, we find that the chance of such a correlation coefficient from a 

 population of zero correlation lying outside the limits '79 is between '006 and 

 "007. There is small doubt therefore that u and v in the sampled population are 

 correlated, probably highly correlated, as the influence of any sleeping draught 

 whatever is a characteristic of the individual. " Student," in applying his test to 

 the difference, has noted this fact as accounting for the low value of the probable 

 error of the difference. 



But what, I think, it is desirable to emphasise is that this correlation may 

 exist in most of the examples to which " Student " applies his test, either owing 

 to the influence of the same individual, or of the same year, etc. Accordingly the 

 denominator of "Student's" ratio will be subject to large variation owing to the 

 presence of this correlation in s u - v , the correlation itself being subject to large 

 variation in small samples of such sizes as 10. Meanwhile the most probable value 

 of the numerator m u m v will not be zero owing to the influence of this correlation. 



Now "Student" takes x' = 1-58/M7 = 1'35, a;' 2 = 1-8225, and accordingly the 

 transformed x = # /2 /(l + x' z ) = '6457, while n = 10. 



