Correlation and Application of Statistics to Problems of Heredity 13
The most noteworthy point, however, is this, that Galton having the correlation table before him of the statures of 928 offspring and of their midparents proceeded after smoothing the frequencies to determine the contour lines and found them to be
(i) a system of concentric and similar ellipses about the common mean of the filial and midparental statures.
(ii) the regression straight lines were conjugate diameters to the two axes of stature.
He also determined from his contours the ratio of the axes of this ellipse system, and the inclination of the major axis to the horizontal. The ellipse, which served as type, is given in the accompanying diagram (see Fig. 4, p. 14), and the observed values on this ellipse and the values computed from Mr Dickson's Formulae are*
Galton from Contours From Dickson's Formulae
Regression Slope 1 in 3 6 in 17.5
Major to Minor Axis 10 to 5.1 %/_7 to %F2 or 10 to 5.35
Inclination of Major Axis 25° 26° 36'
It is needless to say that Galton was delighted with this accordance. He wrote' as follows with regard to it
"I may be permitted to say that I never felt such a glow of loyalty and respect towards the sovereignty and magnificent sway of mathematical analysis as when his [Mr Dickson's] answer reached me confirming, by purely mathematical reasoning, my various and laborious
statistical conclusions with far more minuteness than I had dared to hope, for the original data ran somewhat roughly, and I had to smooth them with tender caution $."
We ought on no account to overlook the fact that the theory of linear regression and the associated homoscedasticity were evolved by Galton from his sweet-pea experiments, confirmed by his stature measurements, and resulted practically in the form of the normal surface for two variates with its elliptic contours, before the mathematical theory of correlated errors was known to him. . It is one of the most striking lessons in what may be achieved by a patient analysis of even crude observations. Yet without being discouraged in our own attempts at similar discoveries, we do well to remember that only an exceptional mind has the insight to discriminate between the essential and the non-essential in a mass of statistical data, and to select those two principles which illuminate the manner in which a population reproduces itself stably by aid of heredity-and what is more in so doing to pave the way to the solution of many other problems of a wholly different character. Fig. 5, p. 16, shows the regression line of offspring on midparent for the case of stature; it is, I think, the second regression line ever drawn, and Galton indicates by the line at 45° exactly how much the offspring fall behind the stature of their individual midparent. He added to this regression diagram, a picture of his " Forecaster of Stature "-which might equally well be used to
Journal of the Anthropological Institute, Vol. xv, p. 263. t Ibid. p. 255. $ Ibid. p. 255.