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exact quantitative treatment. Though we cannot equate crimes, we can arrange them in a list according to their magnitude, and measure any one by its position in the list. Similarly St. Augustine, if placed in his proper rank amongst men for piety, is measured as exactly as if given a numerical score. The step from Shakespeare to Middleton in a series of dramatists ranked in order of ability is a definite measure. If a boy moves in English composition from the position of the 500th in a thousand to the position of the 74th in a thousand his gain is measured as clearly and exactly as when we measure the inches he has grown in height. Measurement by relative position in a series gives as true, and may give as exact, a means of measurement as that by units of amount.

Measurement by relative position in scientific studies is of course but an outgrowth of the common practice of mankind. The man in the street measures things not only as being so many times this, but also as being 'the biggest he ever saw' or 'about average size.'

Measures by amount of some unit have been the subject of great development in the hands of physical science, while measures by relative position have been comparatively neglected, though for the mental sciences they are of the utmost importance. The use that has been made of them already by Galton, Cattell and others gives promise that the value of a measure to which the most subtle and the most complex traits alike are amenable will in the future be more appreciated.

In measuring any person or trait by position in a series, the chief desiderata are:

1. That the arrangement of the series should not be the result of any individual's chance bias, i. e., that the arrangement should represent the general tendency of a number of observers.

2. That it should not be influenced by a constant error, by bias common to all, i. e., that there should be, on the whole, as much bias any one direction as in any other.

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3. That it should be on a sufficiently minute scale.

Suppose, for instance, that we wish to find the position of a certain theme among 1,000 English themes written by first-year highschool boys. No one person can, except by accident, be a perfect rater of these, for his momentary impulse or his peculiar ideals or training will overweight certain features. The combined opinion of

ten equally good judges will always be truer than the opinion of any one of them. If, however, all the ten over-emphasized spelling or punctuation or humor, their combined rating would be false. Such a constant error in judgment is avoided as far as possible if judges are chosen at random.*

The value of having the themes arranged on a fine scale is; first, that the finer the scale the more precise the measure, and, second, that if a theme is then misplaced by chance it will not be displaced so far. For instance, if themes were rated simply Good or Bad, a theme near the dividing line, if put on the wrong side, would be put very far to the wrong side, viz., one fourth of the total distance, whereas if they were rated in 20 divisions, one in the middle would, if put to the wrong side, be moved only one fortieth of the total distance. As a practical rule one should divide the series into as many groups as one can distinguish.

Amongst school abilities, achievements in handwriting, drawing, painting, writing English, translation, knowledge of history, geography, etc., are readily measured by serial rating, and the agreement of observers is such that great reliance can be put upon the results. In the case of more general characteristics the service of the method will be greater still, though the readiness and accuracy of the process are less.

Measures by relative position have one grave defect. Ordinary arithmetic does not apply to them. It is not possible to add '17th from top of 1,000 in wealth' to '92d from top of 1,000' as we can add fortune of $1,000,000 to fortune of $790,000. We can not say that the 10th ability from the top in 100 plus the 20th ability from the top in 100 is equal to the 14th plus the 16th. We can not equate different positions in the series with each other as we can different amounts of the same thing.

We can not, that is, on the basis of what has been so far said about measurement by relative position in a series. There are, how

*Of course the constant errors due to the Zeitgeist, the general bias of the opinion of experts at any time, can be overcome only by getting ratings made fifty years apart! And it is always possible for the critic to say that the human judgments which we are invoking here, even if the best of their kind, are fallible; that the future or Deity might in perfect wisdom rate otherwise! This is true enough, but for the humble statistician the best human judgment is all that is needed. And commonly the critic's complaint that the ultimate structure of the universe contradicts a given human judgment really means that he himself does not agree with it.

ever, two possibly valid ways of transmuting a measure in terms of relative position into terms of units of amount. Given a certain condition of the series as a whole, and the statements of position can be expressed in terms of amount and made amendable to ordinary arithmetic. Given the truth of a certain theory of the amount of difference noticeable, and the same result will hold. These possibilities will be discussed in a special chapter on the measurement of mental traits by relative position.

PROBLEMS.

1. Why would the number of men giving instruction in a university not be a fair measure of the amount of teaching done? 2. What are the faults of the following proposed as a measure Birth-rate

of civilization:

Death-rate

?

3. How could you get commensurate units of amount of ability in addition? In what sense could you, after obtaining such units, say that A's ability in addition was twice or three times B's?

4. In giving examination marks, the custom is to measure downward from a standard of perfection. Suggest a better starting point to take.

5. What are some objective units of amount used to measure criminality? What would be the advantages of measuring here by relative position?

6. Group the following measures by whole numbers, first, by using the whole numbers 14, 15, etc., to represent 13.5-14.499, 14.5-15.499, etc., and second by using 14, 15, etc., to represent 14-14.999, 15-15.999, etc.:

18.642, 17.39, 21.45, 14.81, 15.51, 17.23, 19.60, 18.42, 21.7, 15.861, 16.5, 17.92, 14.4, 19.38, 20.6, 20.5, 18.39, 17.489. Which method would you expect to be the easier and least subject to error if one had equal amounts of practice with both? Why? 7. What is the average salary of the group represented by the following statistics? :

8 individuals have salaries above $1,000 and under $1,100

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THE MEASUREMENT OF AN INDIVIDUAL.

ANY mental trait in any individual is a variable quantity. If we measure it a number of times with a fine enough scale of measurement we get not one constant result, but many differing results. The amount of addition John Smith can do in a minute, the number of cubic feet of sand Tom Jones can dig in an hour, the food consumed by Richard Brown in a day, the weekly earnings of a particular factory-these and all facts depending on human mental traits are variable.

A constant can be measured in a single figure, but a variable for its complete measurement requires as many different figures as there are varieties of the thing. Since John Smith can add now 20, now 21, now 22, now 23 digits in a minute, his ability is not any one of these nor the average of them all, but is described truly only as 20 such and such a per cent. of the times, 21 such and such a per cent. of the times, etc. Any single figure would be but an extremely inadequate representation of his ability in addition or of that of any variable trait. The measure of a variable quantity implies a list of the different quantities appearing, with a statement of the number of times that each appeared. Such a list and statement together are called a table of frequencies or a distribution of a trait. The measure of a variable trait is thus its entire distribution or table of frequency. It is common to present a table of frequencies in a diagram in which distances along a line represent the different quantities, and the heights of columns erected along it their frequencies. Thus Figs. 1, 2 and 3 represent at once to the eye the facts given by Tables VI. to VIII. Such a figure is called a surface of frequency; the compound line which, with the horizontal base line, encloses it is called a distribution curve.

Another method of presenting graphically a table of frequencies is to draw instead of the top lines of the columns a line joining the middle points of these top lines. Figures 1A, 2A and 3A repeat Figs. 1, 2 and 3 in this form.

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FIG. 1.-Surface of frequency of the ability of B. F. A. in memory span. ber of letters correctly written and correctly placed, after one hearing of a series of 12. Number of measurements = 40.

FIG. 2.-Surface of frequency of the ability of E. H. in discrimination of length. Number of millimeters error made in drawing a line to equal a 100-mm. line. Number of measurements = = 100.

FIG. 3.-Surface of frequency of the opportunity for work in a trade. Number of members of the Amalgamated Society of Engineers lacking employment. Number of measurements = 31 (years).

FIG. 1A.

FIG. 2A.

- Same as 1, but drawn by joining mid-points of columns.

- Same as 2, but drawn by joining mid-points of columns.

FIG. 3A. -Same as 3, but drawn by joining mid-points of columns.

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