26 Life and Letters of Francis Galton 



There is another suggestion in the Royal Society paper which has 

 ultimately been followed up to great profit, namely that the variability 

 within the family could be ascertained by considering the difference in 

 the same character of pairs of brothers. Let R be the multiple correlation 

 coefficient of an individual on all his ancestors or his correlation with 

 his "generant," then since two brothers have the same ancestry the 

 variability in a family of brothers is a-Jl—R 2 , where o- is the standard 

 deviation of brothers. Now if x x and x, be the characters in a pair of 

 brothers, for example their statures, we have \ (x 1 + a: 2 ) for their mean and 

 \ (x l — x,y for their standard deviation squared, or so-called variance. If this 

 be taken for a large number of pairs, then it may be shown that 



Mean variance for pairs of brothers = £ o- 2 (1 — r) = J cr 2 (1 — R 2 ), 



where r is the simple correlation of brothers*. 



These results have really been given as early as 1886 by Galton. He 

 does not use R, and instead of standard deviations, speaks of quartile values, 

 i.e. probable errors. He writes b for our *67449 <r </l - R 2 , p for our '67449 cr, 

 and our r is his regression of brother on brother or his w. Thus in his symbols : 



Mean (probable error) 2 of pairs of brothers = \p i (1 — w) = \ 6 2 . 



These results are given on pp. 58-59 of the R. S. Proceedings memoir, 

 and demonstrated by methods which appeal only to the most elementary 

 conceptions. When we come to actual numerical values, Galton finds a series 

 of values for b (the probable deviation in a group of brothers) which ranges 

 from //- 98 to 1 "38 — a result which might be anticipated from the rather 

 heterogeneous nature of his material. If for the reasons already stated we do 

 not trust to the "Special" data only, but use also the R. F. F. results, the mean 

 value (Table, p. 59) found by the various processes for b is 1 //- 179. For p 

 I find from Galton's table on his p. 69, l //, 684, and thus deduce for R the 

 value "7140, comparing not badly with the value "7284 obtained recently for 

 brothers from probably better dataf. Clearly with these values for p and b 

 that for w, the regression of brother on brother or the correlation of brothers, 

 is "5096 and not § = '6667 as Galton assumed it, trusting to his "Special" 

 data; this is a result agreeing far better with later determinations of 

 fraternal heredity J. 



The whole paper is a most remarkable one, not only for the wealth of 

 new ideas it contains, but for the insight it shows Galton had into many 

 problems which have only been recently, or are only at present, under 



* Biometrika, Vol. xvn, pp. 130-1. t Ibid. p. 138. 



% A further point worth recording occurs on p. 58 of the R. S. Proc. memoir. Suppose 

 samples of size n are taken from a normal distribution. Then the mean square standard deviation 

 of these samples, /!,, is given in terms of the standard deviation squared, 2 2 , of the sampled 



>fi \ ji — 1 



population, by /Zj = 2 2 . Galton puts this in probable deviation form as d? = b 2 and 



putting n = 4, 5, 6, 7 applies it to find b 2 from mean square probable deviation (in his termino- 

 logy the quartile) of brothers in families of different sizes. Thus anticipating more recent work 

 on small samples. 



