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Correlation and 4pplication of Statistics to Problems of Heredity 15

predict the probable value of any third variate from a knowledge of two others*. The working of the Forecaster is almost obvious on examination of the diagram, but for the benefit of those who come for the first time to the subject of regression I give Galton's own words

"The weights M and F have to be set opposite to the heights of the mother and father on their respective scales; then the weight sd will show the most probable heights of a son and daughter on the corresponding scales. In every one of these cases it is the fiducial mark in the middle of each weight by which the reading is to be made. But, in addition to this, the length of the weight sd is so arranged that it is an equal chance (an even bet) that the height of each son or each daughter will lie within the range defined by the upper and lower edges of the weight on their respective scales. The length of sd is 3 inches = 2f f ; that is, 2 x 1.50 inch.

"A, B and C are three thin wheels with grooves round their edges. They are screwed together so as to form a single piece that turns easily on its axis. The weights M and F are attached to either end of a thread that passes over the movable pulley D. The pulley itself hangs from a thread which is wrapped two or three times round the grove of B and is then secured to the wheel. The weight sd hangs from a thread that is wrapped in the same direction two or three times round the groove of A, and is then secured to the wheel. The diameter of A is to that of B as 2 to 3. Lastly, a thread wrapped in the opposite direction round the wheel C, which may have any convenient diameter, is attached to a counterpoise.

"It is obvious that raising M will cause F to fall, and vice versd, without affecting the wheels A, B, and therefore without affecting sd; that is to say, the parental differences may be varied indefinitely without affecting the stature of the children, so long as the mid-parental height is unchanged. But if the mid-parental height is changed, then that of sd will be changed to 2 of the amount.

"The scale of female heights differs from that of the males, each female height being laid down in the position which would be occupied by its male equivalent. Thus 56 is written in the position of 60.48 inches, which is equal to 56 x 1-08. Similarly, 60 is written in the position of 64-80, which is equal to 60 x 1.08+."

The last words indicate what is, I think, an important point : Galton obtains the female from the male stature by multiplying by the constant factor 1-08. This he obtained as the ratio of the male to the female mean value, and he practically assumes this ratio to be the same for all other statures.

In a certain sense I think this is, at least theoretically, a retrograde step from his suggestion of 1877. He then took the transmuted female mean to be the male mean plus the female deviation increased in the ratio of male to female variability. This appears to be theoretically a better process of transmutation. Practically the two methods will only agree, if the ratio of the two variabilities is equal to the ratio of the two means, i.e. if the so-called coefficients of variability of the two sexes are equal. This is approximately but not absolutely true for a number of human characters.

There are of course several other conditions which must be fulfilled to make Galton's definition of midparent valid, and some of these he discusses. In the first place the parents must mate at random with regard to the character dealt with, i.e. there must be no sexual selection in the form of assortative mating with regard to stature, tall must not tend to marry tall,

* It would only be needful to adopt scales in accordance with the constants of the bivariate regression formula.

t In this paper Galton uses the symbol f for the quartile deviate. + Journ. Anthrop. Institute, Vol. xv, p. 262.