19. In each of the following cases, determine the mode and the

variability of the distribution around it.

and the variability around it.

Calculate also the average

=

20. In the report † from which Case II. is quoted the 4.0 4 or less and the 8.5 = 8.5 or more. If these facts had been announced in the problem, which measures only could have been calculated?

* Roberts' 'Manual of Anthropometry' is the source of these figures. † New Zealand Official Year-Book, 1901, p. 231.

CHAPTER VII.

THE TRANSMUTATION OF MEASURES BY RELATIVE POSITION INTO

TERMS OF UNITS OF AMOUNT.

IF a group of individuals are ranged in order according to the amounts which they severally possess of a trait, we can, even when ignorant of what the amounts are for each and all of the individuals, assign to each the amount of his deviation from the average, provided the form of the group's distribution is known.

For instance, let 100 boys rank with respect to scholarship as shown in Table XXIX., and let the form of distribution be that of Fig. 68.

TABLE XXIX.

100 Boys a, b, c, ETC., RANKED BY RELATIVE POSITION.

3 b, c, d are the next highest ranking and are indistinguishable.

If we build up approximately the distribution of Fig. 68 by a series of 40 rectangles of equal base, the result is Fig. 69. Call the low extreme A and the length of base of each of the rectangles

K. Then the upper extreme is at A+ 40K. The approximate distribution in terms of these units is given in Table XXX. The frequencies may, of course, be reckoned on the basis of any arbitrary In Table XXX., the total area is taken to be 1,680.

unit.

The highest ranking boy, a, who was the top 1 per cent. of the group, will in our figure occupy the top 1 per cent. in the table, the highest 16.8 of the frequencies. His ability then is from A + 40K part way into A + 34K. The abilities of the next three, b, c and

A+5K

FIG. 69.

A+40 K

d, will occupy the next 50.4 of the frequencies and be included between the limits A + 34.7K and A + 30.4K. So on with the next six and the rest. The limits for each group are shown in Fig. 70. The average ability of each group may be calculated roughly

*

*

By a subdivision of the surface into finer rectangles the precision of these averages could have been increased.

from the facts obtained in this way. Thus the highest boy, being represented by 0.5 (A + 39K), 2 (A + 38.5K), 2.5 (A + 37.5K), 4 (A + 36.5K), 5.5 (A + 35.5K) and 2.3 (A + 34.5K), has as an average A+ 36.5K.

A table can thus be formed as follows:

Boy a has as his ability A + 36.5K;

Boys b, c, d, have as their ability A+ 32.2K;

Boys e, f, g, h, i, j, have as their ability A + 28.0K;

Boys k, l, m, n, o, p, q, r, s, t, have as their ability A+ 23.8K; etc.

FIG. 70.

These measures can further be turned into distances from the mode or median or average of the distribution instead of from its lower limit A. They can be put in terms of any measure of the variability of the scheme, or of any part of it instead of K. For the distribution given in Table XXX. can be used in every way like one with known quantities in place of the A and K. For instance, the best boy is + 26K from the mode, or, in units of the 75 percentile mode measure of variability, is + 3.38.

The scholarship of every boy in the group is thus represented in definite quantities of some unit of amount of difference from some standard. This unit itself is definable as the difference between this person and that person. The standard is similarly definable as the scholarship of such and such a person.

By this method the obscurest and most complex traits, such as morality, enthusiasm, eminence, efficiency, courage, legal ability, inventiveness, etc., can be made material for ordinary statistical procedure, the one condition being that the general form of distribution of the trait in question be approximately known.

If now one has a group of individuals ranked by their relative position in the group, his first task before he can transmute the series of relative positions into a series in units of amount is to ascertain the form of distribution. This may be done (1) by measuring objectively in units of amount enough sample individuals, or (2) if the trait cannot be measured in units of amount, by inferring the form of distribution from that of similar traits which can be.

1. Suppose one had 2,000 ten-year-old boys measured with respect to intellect by relative position.* If now one measured 200 of them objectively with tests scorable in units of amount, he could properly transmute the 2,000 on the basis of the type of distribution of the 200.

2. Suppose one had 1,000 individuals measured with respect to delicacy of discrimination of sound by relative position. (It is wellnigh impossible to measure sensitiveness to sound in objective units which another observer can duplicate, because of the influence of size of room, resonance, etc.) It is fairly certain from studies of the delicacy of discrimination of length, weight, etc., that delicacy of discrimination of sound is distributed in something approximating sufficiently to a probability surface, with range of from + 30 to − 30, to prevent calculations on that basis from being more than a little wrong on the average. We may, therefore, transmute the 1,000 measures by relative position into units of amount, on the hypothesis that such is the form of distribution. So also with school marks if intellect in general is found to follow the probability type of distribution.

The labor of transmutation for cases which follow the probability type of distribution is rendered almost nil by the use of tables.

If the probability surface of range + 30 to 30 is divided up into 100 equal areas representing the 100 successive per cents. from the highest to the lowest of the total group, and the average distance from the average in terms of σ is calculated for each per cent., the result is Table XXXI.

If now we ask, 'What will be the average ability of the highest 6 per cent.?' we have only to add the figures for the first 6 per cents. and divide by 6 (the result being, of course, 1.99). Similarly to get

* Such measures, at least approximately correct, would in fact be easy to obtain through school marks, teachers' opinions, personal conferences, etc.