12 Life and Letters of Francis Galton 



Anthropological Institute*. He further took the subject as the topic of his 

 Presidential Address at the Anniversary Meeting of that Institute in 

 January, 1886f, having meanwhile again discussed it in a lecture at the 

 Birmingham and Midland Institute entitled: "Chance and its Bearing on 

 Heredity" J. Finally we have the mathematical basis of Galton's work more 

 fully provided in a paper on "Family Likeness in Stature" with an 

 Appendix by J. D. Hamilton Dickson, presented to the Royal Society on 

 January 1, 1886§. None of these papers is exclusive, each has something 

 not in the others, but probably those in the Miscellanea of the Journal of 

 the Anthropological Institute and in the R. S. Proceedings are the more im- 

 portant for those who have not time to read them all. We have throughout 

 to remember that Galton was a pioneer, and could not see matters in the 

 clearer light of to-day when we start from a knowledge of bivariate 

 distribution with its two means, two variabilities and its coefficient of 

 correlation; he did not yet clearly recognise the distinction between a 

 coefficient of regression and a coefficient of correlation. It is difficult for the 

 reader now-a-days to appreciate the paradox which Galton reached from his 

 data and finds it needful to discuss at some length, namely : that the coefficient 

 of regression for the offspring on a midparent is double what it is for the mid- 

 parent on the offspring ||. A further difficulty is that Galton invariably thought 

 in terms of grades, quartiles and the "ogive curve," and this I venture to think 

 is by no means helpful for elucidating correlation, as the reader of the first 

 ten pages of the Royal Society paper will find. It has always been a puzzle 

 to me why Galton called in Mr Dickson and placed before him a somewhat 

 artificial problem in probability the answer to which comes directly ^ from 

 Galton's own two statements. 



* Vol. xv, pp. 246-263. f Vol. xv, pp. 489-499. 



| Reported in the Birmingham Daily Post, December 7, 1886. 



§ Roy. Soc. Proc. Vol. XL, pp. 42-73, 1886. 



|| Since the midparental standard deviation is, when the female is reduced to male equivalent, 



o-j/s/2 in our previous notation, the two regression coefficients are respectively: — — r and 



— — j^ r, that is, r/*J2 and «/2 r, or one twice the other. I think Galton was slightly puzzled 



here, because he had not yet fully realised that the two variabilities not being the same, he must 

 measure each variate in its own unit of variability in order to make both regressions the same. 

 11 Galton had discovered that the offspring of parents of character deviation x vary about 

 (rtr 2 /o-,) x with a standard deviation <r, V(l - r 2 ). Hence if y be the deviation of the n offspring 

 of the n parents of deviation x, and we assume, as Galton, that parental and offspring genera- 

 tions both follow the normal law, the number of offspring of deviation y will be 



N i _i£ 



But n' = ~7= — e 2 <ri 2 , where N is the total population of parents, thus substituting for ri we 

 have n jV" 1 /x> 2rxy y 2 \ 



z =o 7r^ e 2 ( 1 - J,2 )W w *i) 



2jrcr 1 <r i! vl — r 

 as the frequency distribution of offspring and parents, the well-known result, which was not 

 even written down by Mr Dickson ! 



