=

26. Groups III. and IV. are approximately normally distributed. Group III. has Median = 10 and A. D. 4 and Group IV. has Median 12 and A. D. = 3. exceed the median for Group III.?

=

What per cent. of Group IV. will

What per cent. of Group III.

will exceed the median for Group IV.? (The table on page 60 affords the further data necessary.)

FIG. 80.

-The height of any one of the lines at its left-hand extreme measures the change in stature of one boy from 12 to 14; its height at the right hand extreme measures the change from 14 to 16.

27. If we know the average wealth of 100 men in 1900 to be $5,000 and in 1905 to be $10,000, what do we know about the changes that have taken place?

28. Recall any arguments based on the application to individuals. of some change true of them only as a group. Where else have we in this book met a similar fallacy?

THE MEASUREMENT OF RELATIONSHIPS.

THE difficulty of measuring mental and social relationships is, of course, due to their variability. The relation of the weight of a gas at constant temperature and pressure to its volume we assume to be always the same, but the relation of intellect to morality is almost never the same; the relation of the force of gravity to the product of the masses of the two bodies is constant, but the relation of ability in school to efficiency in life is very variable. The problem is thus to represent the total tendency shown by many different individual relationships.

Case I.

The relationship of changes in the amount of one thing to changes in the amount of another thing, when the things are physical, is shown by a series of corresponding values of the two things reckoned from zero points in both cases, each pair of values being represented by two constants. It is expressed mathematically by the equation which represents the way in which the amount of the one thing depends upon the amount of the other.

The following case may serve as an illustration:

n = the index of refraction of air.

d the density of air.

=

p (a quantity subject to the control of the experimenter) = Cd. N (a quantity measurable by the experimenter) = C2(n − 1).

C1 and C2 are constants.

The experiments consisted in varying p and measuring the related changes in N. The results are as follows:

If each of these pairs of related values is turned into an equation of the form N= xp, the results are:

Obviously, a single equation N = 31.68p expresses very closely the relationships found for different values of p.

The measurements of relationship here are, of course, not absolutely free from variability. For instance, the 10.163 came really from 7 measurements with an average deviation of .012. But the

variability is here small and presumably due entirely to variations in the instruments or observers.

There

If the pairs of values are plotted as in Fig. 81, the slope of the line shows the relationship. The equation N31.68p expresses very closely the slope of this line referred to its coordinates. N/p is thus constant. (n 1)/d equals N/p times some constant. fore, (n-1)/d itself equals a constant. The relation between the index of refraction of air and its density is then such that (n − 1)/d = k or n = kd + 1.*

Case II.

When changes in the amount of a mental trait are to be related to changes in the amount of a physical trait, the series will be of pairs, of which one will be a constant and a quantity measured from a zero point and the other a variable and often a quantity with no ascertained zero point. The following case may serve as an illus

tration:

Ebbinghaus in studying the relation between the lapse of time and memory found that if a series of syllables was memorized and then 24 hours allowed to pass, there was required to rememorize the series 73.6 per cent. as much time as was originally needed. In another test, however, the result was 60.4 per cent., and he quite properly announces not only the average of all the numerous varying results, but also each separate one. So also for the time taken after intervals of 19, 63 and 525 minutes and 2 and 6 days. In the statement of the relationship which follows (in Table XXXV.), the 'time saved in learning' quite evidently is a variable. One may note the wisdom of the investigator in measuring the change, not in the ambiguous units of so many words lost, but in 'per cent. of original time taken to relearn,' a system of units with an intelligible zero point.

If we plot the pairs of values as in the previous illustrations, the result is Fig. 82, which shows the general tendency of the relationship and at the same time its lack of uniformity.

In such cases it is common to replace the tables of frequencies for the mental trait by their averages. This procedure never fully

*The figures in this illustration are quoted from a report by Henry G. Gale of a research 'On the Relation between Density and Index of Refraction of Air.' Physical Review, January, 1902.

describes the relationship and, unless the distributions are symmetrical about a central mode, may misrepresent it. At all events, the total fact of the relationship should always be presented, as well as its abbreviated and more convenient form. In so far as the zero point from which the mental trait is measured is unknown, it is necessary to replace all face values y, y1, y, etc., of the mental traits measured by k+y, k+ y1, k+ y2, etc. The formulation of any algebraic expression for the relationship is thus less simple.

* From Herm. Ebbinghaus, ‘Über das Gedächtniss,' pp. 93–103.