270 Prof. K. Pearson. On a Criterion which may [Mar. 4, 



Let us apply this to the case of stature of parents and sons in man. 

 Here I have deduced from my " Family Data Eecords n * 



Pfe = 0-5140, Pmc = 0-4938, p mf = 0-2804 (xiv). 



We find : 



X^/o- c 2 = - 0-1028 + fo-5579 - - 0-5359 JL\ 2 ( xv ). 



\ a-f a- m J 



Hence we have the impossible value 32cr c J - 1 <r c for the variability 

 of the arrays of offspring due to mean parents. Generally we 

 deduce — 



Values of ^xy/^e for Arrays. 



Father : xjoy. 



Mother : y\(x m . 



Array: S^/ov. 











(0 -32a/(-1) ?) 



+ 1 





0-46 



+ 1 



-S 



1 -05 



+ 1 



-2 



1-60 



+ 2 



" 2 



2-19 



We are dealing here with measurements on upwards of 1000 

 families ; the probable errors, therefore, of p/ c and p mc hardly allow of 

 our supposing the first term on the right in (xiv) to be zero. But if we 

 do make this hazardous assumption, we see that as we pass from pairs 

 of mean parents to fathers of 6' 1" and mothers of 4' 10" we should pass 

 from an array of offspring of no variability to one twice as variable as 

 the general population. Further, for any given male there would exist 

 a female relatively only very slightly taller than he is, who would have 

 offspring with him of sensibly no variability. In view of this result 

 we may safely assert that the hypothesis of "father's offspring" and 

 " mother's offspring " cannot apply to stature in man under any 

 conditions in the least approximating to Dr. Boas's assumptions in the 

 case of cephalic index. 



(6) Our conclusions may be summed up as follows : — The variability 

 of the array of offspring due to a group of parents of definite character 

 can be satisfactorily used as a criterion between various theories of 

 inheritance. In particular if the variability of the array be plotted to 

 the character of the parent — 



(a) On the hypothesis of the " Ancestral Law " a horizontal straight 

 line is the resulting curve. 



(b) On the generalised Mendelian theory the result is a parabola 

 with horizontal axis. 



* < Bioxnetrika,' vol. 2, pp. 373 and 378. 



