cxxviii Tables for Statisticians and Biometricians [XXV 



Illustration (iv). In a long series of observations on Fathers and Sons the 

 correlation coefficient for span was found to be '454, and the standard deviations 

 were 3"14 and 3"'ll respectively. The regression RI of Son on Father for 

 span = '44966. The standard deviation of R! in samples is 



or a- Bl = -p^= x -882,489. 



Vn-3 



Hence, if we can take the parent population for span to be approximately normal, 

 let us ask whether a sample of 19 pairs of Father and Son giving a correlation 

 of "390 and standard deviations for span : Fathers 3""19 and Sons 2"'98, may 

 be reasonably supposed to have been drawn from this parent population. 



Now R! for the sample = '36432 and <r Rl = '221,1222. 



rp , , '36432 - -44966 



ThuS .221,1222" 



Accordingly x* = '1489,4999, 



1489,4999 

 and -M893J99T16- 



This clearly lies within the first part of Table XXV where the differences are 

 unsatisfactory. We therefore use the auxiliary Table XXV bis. For?i = 19, we have 



MO = 1 '335,4038, S 2 M O = 1 2631, 

 MI = 1-305,4459, S% = 12083. 

 Here S 4 's will be unnecessary. 



6 = '223,509, < = '776,491, 0$ = "028,9255, 

 u e = 1-328,7079 - '000,1077 



= 1-328,6002 = ^(19). 

 But P x (19) = '5 + ^ (19) V#, 



P x (19) = -5 + '304,8770 x 1-328,6002 = '905,060, 



or the chance, if this sample were really drawn from the above parent population, 

 that its regression coefficient would differ as much as or more than it does from 

 the regression in the parent population = '189,880. 



We see therefore that in about 19 in 100 samples the deviation of the regres- 

 sion would be greater than that observed. 



Let us, however, look at this problem in another way, which will illustrate a 

 further application of our present table. 



Illustration (v). In the sample of the previous illustration the first product 

 moment coefficient = p u = '390 x 2'98 x 3'19 = 3'707,4180. What is the chance that 

 a sample of 19 with this p it could have been extracted at random from a parent 

 population with no correlation, but with standard deviations 3""14 and 3"'ll ? 



