Mathematical Contributions to the Theory of Evolution. 397 



Table of Heredity according- to Galton's Law. 





Individual parent. 



Mid-parent. 



Order. 

















Correlation and regression. 



Correlation. 



Regression. 



1 



2 

 3 

 4 



5 

 6 



0-3000 

 0-1500 

 -0750 

 -0375 

 -01875 

 -009375 



-4243 

 -3000 

 0-2121 

 0-1500 

 0-1061 

 -0750 



0-6 

 0-6 

 0-6 

 0-6 

 0-6 

 0-6 



5th 







0-6 



Remarlcs. — The correlation of the individual first parent is to be taken as the 

 mean of the four possible parental correlations due to differences of sex, if these 

 are not sensibly equal, and a like rule holds for the individual sth parent. The 

 individual parental regression is based on the assumption that the variability of 

 offspring and parent are the same. In dealing with the mid-parent, female 

 deviations must first be reduced to male by multiplying them by the ratio of male 

 to female variability. 



his views on regression, and it is the latter which, judging from both 

 theory and observation, I now hold must be discarded.* 



(7) Mr. Galton's law gives us the partial regression coefficients 

 when all the mid-parents are known. It is desirable to deduce from 

 the theory of multiple correlation the values of the partial regression 

 coefficients when we take 1, 2, 3, 4,. . . . mid-parents only. When q 

 mid-parents are taken let the partial regression coefficients be e 1? , e 2? , 

 € sq, e 4s, . . • • € qq ; then again we have for the mean of the offspring Jc : 



k = C lfJ — k l -\-€ 2q — ft 2 + + e qq— kg (™) > 



where <r is the standard deviation of the offspring and a p of the ^>th 

 parental generation. Comparing this with the regression formula 

 immediately under (viii) we have 



<w=-^f =-2^,by(y). 



* I do not agree with the last column of Mr. Cralton's table giving the variability 

 of arrays. For single correlation the variability (standard deviation) of an array 

 = a +/1 — r 2 , where r is the correlation and not the regression. With equal varia- 

 bility of all generations, r in the case of the individual parent may be replaced by 

 the regression. But the correlation is not equal to the regression in the case of 

 mid-parents, because the variability of the mid-parent by (v) is increasingly less than 

 that of the offspring. 



