58 



Mr. F. Galton. 



[Jan. 21, 



call d, whence b may be deduced as follows : — Suppose an exceed- 

 ingly large family (theoretically infinitely large) of brothers; their 

 quartile would be b. Then if we select from it, at random, numerous 

 groups of n brothers in eacb, the means of the mid-deviates of 

 the several groups would form a series whose quartile is 1/ Vn x b. 

 Hence b is compounded of this value and of d ; that is to say, 



6 2 =<Z2 + 1/ W X& 2 or 62=-Ar^. 



n — 1 



I treated in this way four groups of families, in which the values of 

 n were 4, 5, 6, and 7 respectively, as shown in Table VIII, whence 

 I obtained for b the four values of 1*01, 1*01, 1*20, and 1*08, whose 

 mean is 1*07. 



(2.) Let c be the quartile of a series of brotherly centres whose 

 quartile is unknown and has to be determined, and that the statures 

 of the individual brothers diverge from their several family centres 

 CjCg . . ., with a quartile b, the whole group of brothers thus forming a 

 sample of the ordinary population ; consequently c 2 =p 2 —b 2 . Now in 

 fig. 7, MS represents the deviate in stature of a group of like persons 



Fig. 7. 



who are not brothers, and MC represents the mean of the mid-deviates 

 of their respective families of brothers. It can be shown (see Appendix, 

 Problem 2) that if the position of c varies with respect to M with 

 a quartile = \/p 2 — b 2 , and if S varies with respect to c with a 

 quartile = b, then, when S only is observed, the most probable value 



of CM is such that ^(=w)= p2 ~~ h \ 

 SM v p 2 



or b 2 =j) 2 (l — w). 

 Substituting 1*7 for p, and § for w, 



6=0-98 inch. 



(3.) It can also be shown (see Appendix, Problem 2) that the 

 variability of particular mid-brotherly deviates, C ± C 2 • • ., about C,the 



