With the aid of Table XXXII. one can turn measurements by relative position into measurements in units of + or ☛ almost as fast as one can read.

For instance, of 800 schoolboys,

The table tells us at once that, in so far as the distribution of the ability in the group in question follows the type of distribution described above,

There is still another possibility of turning measures by relative position into units of amount and so making them available for common scientific usage. In certain cases it may be justifiable to suppose that the least noticeable difference is a constant quantity for any one trait for any one observer; in simpler words, that if I say that John, James and Peter are to me indistinguishable, say, in literary merit, but that Henry and William are a shade better, and that George and Fred are a shade better than Henry and William, the actual difference between JJP and HW equals that between HW and GF. In so far as this were true we could, with a large group of individuals varying continuously from the low to the high extreme, make groups on the basis of the least noticeable difference and call the steps of ability from group to group always the same.

The measures are then identical in form with those by ordinary units of amount. The only difference is that the amount of the quantity at the starting-point of the whole group (4) and the amount of the step from one subgroup to the next (K) are unknown except from the things measured themselves and are undefinable except in terms of them. The question, 'How much are A and K?' can be answered only by pointing to the achievements of the lowest group and saying, 'That is A,' by pointing to the differences between that group and others and saying, 'This much difference is K, this much 4K, this much 20K and so on.'

The hypothesis that the least noticeable difference in a trait is for the same observer a constant quantity has not been tested sufficiently to decide how far its use is justifiable, but there can be no doubt that some modification of the principle involved will sometime be a valuable resource of the theory. of mental measurements.

For the sake of simplicity, only the case of individuals measured by their relative position in a group has been discussed in this chap

ter. Everything in the chapter applies equally well to measures of the different trials of one individual.

PROBLEMS.

21. Turn into statements in units of the A. D. of the distribution, measured + and — from the average, the measures by relative position given below; first, on the supposition that the form of distribution is a rectangle; second, on the supposition that the form of distribution is of the normal type (use Table XXXII.); third, with no supposition about the form of distribution, but on the hypothesis that the measures represent a grouping by the least noticeable differences and that these differences are equal:

A, B, C, D, E and F are marks running from high to low. Of some 200 and over high-school students, 2 per cent. received A, 22 per cent. B, 44 per cent. C, 25 per cent. D, 6 per cent. E, and 1 per cent. F.

22. Which supposition is the more likely? Why?

23. Using Table XXXI., calculate the measure in terms of units of amount (1) of the highest four per cent. of a group normally distributed; (2) of the six per cent. just above the mode; (3) of the three per cent. from the end of the seventeenth down, i. e., of per cents. 18th, 19th and 20th. Verify the results from the entries for these groups in Table XXXII.

24. On the hypothesis that the distribution of darkness of eyes is normal, use Table XXXII., and transmute into terms of units of amount the following measures by relative position :

It is possible to use the table for a finer scale than to a single

per cent. by interpolating. But it is hardly worth while.

*From Galton's 'Natural Inheritance.'

CHAPTER VIII.

THE MEASUREMENT OF DIFFERENCES AND OF CHANGES.

THE chief questions that concern the measurement of differences in the mental sciences arise in the case of comparisons of groups and measurements of changes. Instead of any general abstract treatment of the measurement of differences, therefore, I shall present the special applications of it to these two problems. Only a very brief outline of the problem as a whole will be given as an introduction.

The difference between any two amounts of the same kind of fact may be measured. The amounts may be :

1. Two single figures, each standing for a general tendency, e. g., averages, medians or modes.

2. Two single figures, each standing for a variability, e. g., A. D.'s, o's or P. E.'s.

3. Two single figures, each standing for a difference itself.

4. Two single figures, each standing for a relationship.

5. Two total distributions, each standing for a general tendency plus the deviations from it.

The general tendency may be to the possession of a certain amount of variability, of difference or of relationship, as well as of a thing or quality. It will, however, commonly be the latter.

The classification above could, of course, be extended ad infinitum with such complexities as: "The measurement of the difference between two variabilities, each being of the amounts of relationship between the amount of difference between (1) 10-year-olds and 11year-olds in motor ability and (2) 10- and 11-year-olds in sense discrimination."

The difference between two single figures will be measured (a) by the gross difference; (b) by the per cent. the difference is of the amount of one of them.

The difference between two total distributions will be measured fully by comparing them item by item; the measurement may be summarized in various ways.

The difference between two facts, each of which is measured by its relative position in a series, may be measured most satisfactorily by transmuting the series and then using regular methods, most quickly by the gross or percentile difference between the two, rated as members of the same series.

The Comparison of Groups.

The common custom of comparing groups by comparing their averages is inadequate because for both practical and theoretical purposes the meaning of a difference between two averages depends upon the variabilities of the groups. The mere fact, for example, that in the A test (see page 46) the averages for 12-year-old boys and for 12-year-old girls were respectively 41 and 46, might mean (1) that the lowest ranking girl was above the highest ranking boy, i. e., that boys and girls were in this trait totally distinct species or (2) that only 5 per cent. of girls were better than the highest ranking boy, or even (3) that no girl was equal to the highest ranking boy. It might mean, in fact, all sorts of conditions, some of which are pictured in Figs. 71 to 76.

It is of no great advantage to estimate the difference in a per cent. rather than a gross amount. One group may in ten different tests have always an average twenty per cent. higher than the other, and yet the differences in ability may really be equal in no two of the ten cases. For, since in mental and social traits there are rarely absolute zero points at which to start the scale,* the meaning of each percentage will depend upon the number chosen as the starting-point in measuring. We can always make a difference so expressed seem less by starting the scale at 10 or 40 or 100 instead of at 0 or 4 or 10. And the same percentage in a case where the variability of the trait is great will always mean for practical purposes a less difference than it does in a case where the variability is small.

For instance, if the A test is scored by the number of A's marked, the percentage superiority of girls to boys is 12.2; if by the number marked more than the lowest 12-year-old record, it is 18.5; if by the number of A's omitted, it is 8.5. Clearly the figure depends on an entirely arbitrary factor.

What is needed for the comparison of groups is some measure * See Chapter II., pp. 15 and 16.