dren; (2) a division into long- vs. short-time groups on the estimates given, yields means very close together; (3) the unreliability of the resulting figures rises greatly when the already small population is divided. For the last reason also was excluded the method of division into a Palo Alto and a Mayfield-Menlo Park group, with ensuing comparison of comparable coefficients (on the theory that environment in the former case might act as a more powerful similarizer). Other modes of attack are impracticable in the present case because the probable errors of the resulting measures are unknown or inaccessible. Such modes are, briefly,

(a) The arrangement of the tests in order of size of their correlation coefficients between child and less similar parent, on the supposition that environment might operate to make the child closer in ability to both parents, while heredity would perhaps favor only one.

(b) The construction, for each test, of a fraction which would represent approximately the ratio of offspring variability due to inheritance from parents alone to that due to all ancestors; by the Law of Ancestral Inheritance, in the case of inheritance uncomplicated by environment, this should be 12whence departures therefrom should measure relative influence of environment. The derivation of numerator and denominator are somewhat complicated, and are given by R. A. Fisher in Transactions of the Royal Society of Edinburgh, Vol. 52, p. 400, footnote, from which the following is condensed:

f

f

Denominator: Where V-variance, i.e., standard derivation squared, of a sibship, o2=variance of general population, r=fraternal correlation, V/2=1-r; i.e., a reduction of o2 to V suffered by the variance in passing from the general population to a sibship population, is proportional to a reduction from 1 to 1—r, that is, to a reduction of r; and the element of difference between the general and the sibship population consists in the fact that the members of the latter have a com

f

r

f

mon ancestry, the influence of which fact is thus measured by the reduction г; or, putting it somewhat differently, the fraternal correlation is a measure of the effect (on the variance) of the total ancestry.

c.mf

с

Numerator: If o2 be the variance of all children, and that of those having both parents of a fixed level of ability (that is, of a single array), then o2c.mf/2 is the ratio of reduction suffered by the variance in passing from the general population to a population limited by the fixation of the level of parental ability. But σ2 c.mt/02=1-12 σ 1—r2 or the square of the multiple correlation coefficient measures the effect of common parentage on the variance.

с

c.mf'

This is the most interesting and promising of the modes of attack, with the possible exception of one yet to be mentioned; it is not, however, either logically perfect or well understood. Given a larger population and better-known probable errors, the agreement of several of these methods, as measured by high rank-order intercorrelations, should be evidence that some approach is being made to the true order of susceptibility to environmental influence.

A brief notation will be included below on a certain tentative technique useful in attack on a slightly different type of problem; but in the meantime it is of moment to determine at what level the familial correlation coefficients of resemblance lie; and in the process of ascertaining them, valuable facts have come to light respecting the growth and decline of mental function.

III

STATISTICAL REDUCTION OF DATA

The Age Curves and Their Use. These latter facts became apparent in connection with the elimination of the influence of varying ages among the subjects, which was effected by converting the scores into "standard" scores. The standard score implies that the individual's raw score is considered as a deviation from the mean raw score of the group with which he is comparable (in the present case his age-group) and expressed in terms of the standard deviation of the same group. Here we have, however, the peculiar condition that the number of groups is arbitrary, since the population is continuous with respect to age; what we wish to determine is really the curves described by the means and standard deviations of an infinite number of groups arranged in serial order of age. We can approximate this by grouping, computing constants, and estimating the curves from the computed points by graphic methods. The groups chosen varied in number from about 30 individuals through the fast-growing years of childhood to about 60 where the curves were not so steep. The resulting curves are produced herewith (pp. 249-254); to those for Test 10 (chosen because the probable errors of its points were largest) the zones of approximate probable error have been added. An effort has been made to equalize, roughly, the vertical dimensions of the different test-graphs, but in reading them it must be kept in mind that just what equalization means is uncertain.

Both the mean from which the deviation x is measured, and the standard deviation σ, are found to vary in a more or less regular relationship with age. Both show a somewhat sharp rise and a long, gradual decline; and they are roughly proportional to each other.

Comparison of the contours of these curves is instructive,