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, 1886t, having meanwhile again discussed it in a lecture at the Birmingham and Midland Institute entitled : "Chance and its' Bearing on Heredity" 1. 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 important 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 midparent on the offspring 11. 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 T from Galton's own two statements.
* Vol. xv, pp. 246-263. t Vol. xv, pp. 489-499. $ Reported in the Birmingham Daily Post, December 7, 1886. § Roy. Soc. Proc. Vol. %L, pp. 42-73, 1886.
~~ Since the midparental standard deviation is, when the female is reduced to male equivalent,
in our previous notation, the two regression coefficients are respectively: r and al
r, that is, r/2 and %l2 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 (rat /a,) x with a standard deviation o-, %/(1 -r2). 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 generations both follow the normal law, the number of offspring of deviation y will be
nn' 1 \ _ rv2 ~2.
e 2c22 (1- r2) y Q, X)'.
N 1 1 x2
2~r a2 ^/1 - r2
e 2 c12, where N is the total population of parents, thus substituting for n we
have nN 1 x2 2rxy y2 x-2~ra,a2~1-rte 2(1-r2)(v,2 ?,v2+a22)
as the frequency distribution of offspring and parents, the well-known result, which was not even written down by Mr Dickson !