and not abuse, the general practical advice to which it leads, which is: If for any reason you have to make an hypothesis about the form of distribution of any trait in the absence of the facts, the most likely one in the case of anatomical, physiological and mental traits is that the form will be something like the normal probability surface. The probable error of any such hypothesis is least in the case of anatomical traits. Prediction of the form of distribution of economic traits is very insecure. The interpretation of any ascertained form of distribution is difficult, but may prove very instructive.

If in dealing with group measurements you can, without violating any known fact, use the hypothesis that in a homogeneous group not subject to selection on the basis of the trait in question, any mental trait due to natural as opposed to artificial causes, is distributed approximately normally, do so.

The normal surface of frequency (which is that of a quantity due to the chance combinations of n causes, all equal and independent, when n is infinitely large) is, as stated on page 36, the surface enclosed by the normal probability curve,

and the abscissa or base line on which x is scaled.

In this form of distribution the Average, Median and Mode coincide, for y is the same for any given

x as for the same + x, and is greatest when x 0. Constant relations hold between the different measures of variability, viz:

=

Between Av. - σ and Av. + o are 68.2 per cent. of the cases.

66 Av.

=

- A. D. and Av. + A. D. are 59.5 per cent. of the cases.

The frequencies of different deviations from the mode (or average or median) are given in gross in Tables XXII. and XXIII. Detailed tables will be given later.

TABLE XXII.

FREQUENCIES OF DEVIATIONS ABOVE THE MODE IN A NORMAL SURFACE OF FREQUENCY IN TERMS OF A. D. THE FIGURES CAN BE USED

IDENTICALLY FOR MINUS DEVIATIONS.

Between 0 and +.2 A. D. are 6.3 per cent. of the cases.

CHAPTER V.

THE CAUSES OF VARIABILITY AND THE APPLICATION OF THE THEORY OF PROBABILITY TO MENTAL MEASUREMENTS.

THE varying measures of one individual's performances and the varying measures of the individuals in a group were found in Chapters III. and IV. to be often distributed approximately after the fashion of the surface of frequency enclosed by the probability curve and its abscissa. In those chapters brief mention was made of the properties of this surface or type of distribution, acquaintance with which is a great assistance to convenient handling of mental measurements. The recognition of this type of frequency surface, the appreciation of its meaning and that of certain common departures from it, and the use of tables derived from the probability integral in calculations of measurements of traits approximately normally distributed, are all possible, at least to the moderate degree required for ordinary statistical work, without any knowledge of the abstract principles involved. But such knowledge is well worth obtaining for the sake of the additional insight into the meaning of concrete facts thereby given, and even merely for the sake of the additional facility in the use and construction of tables and the common formula. The present chapter will, therefore, contain a very simple introduction to the study of the applications of the mathematics of probability to the theory of the distribution of mental traits. From it the student may proceed to the study itself with the aid of the references given at the end of the chapter. The chapter will also introduce the student to the more general problem of the relation which the nature of the causes determining the amount of a trait hold to the trait's distribution.

1

Let us begin with the consideration of a quantity which is dependent on the action of one cause which is as likely to occur as not, and call the cause a. For example, a may be the action of John's father in giving him a Christmas gift of a dollar.

The condition of affairs resulting will be, of course, no action or α. The quantity in question, John's Christmas money, will be 0 or $1.00. Its distribution will be

Its surface of frequency will be a rectangle, composed of two rectangles of equal base and altitude.

Suppose now that two causes contribute to determine the quantity, a and b, the possible actions of John's father and mother, and that all combinations of these causes are equally likely. The condition of affairs resulting will be, then, no action, a, b or ab, all being equally likely. If now a = a gift of $1.00 and b likewise, the quantity in question, John's Christmas money, will be 0, $1.00, $1.00 or $2.00. Its distribution will be

Its surface of frequency is that shown in Fig. 55. If the conditions are kept the same but the number of causes increased to three, the condition of affairs will be, no action, a, b, c, ab, ac, bc, or abc. If as before a = b = c in magnitude, John will get $2.00 as often as $1.00 and three times as often as nothing or $3.00.

The surface of frequency of the quantity, John's Christmas money, will be four rectangles, as shown in Fig. 56.

Keeping all the conditions the same, let the number of causes be increased to 4, then to 5, and then to 6. The condition of affairs in each case and the resulting distribution-schemes and surfaces of frequency are given in Tables XXIV., XXV. and XXVI., and Figs. 57, 58 and 59.