by the number of the measures the average of which you are obtaining: add the quotient to the chosen number." To secure full mastery of every procedure taught, this book will contain many model examples and sets of problems to be worked.

The Application of the Theory of Measurements.

A sense of when and how to use statistical methods is even more important than knowledge of the methods themselves. The greatest benefit, therefore, will come to those who in connection with every principle established in the text, call to mind some concrete case to which the principle should be applied. The insight into the actual use of the theory of measurement thus obtained may be increased by a critical examination of the samples of quantitative studies referred to in Chapter XIII.

The Theory of Measurements and the Special Sciences.

This book, as the title announces, deals primarily with the theory of mental measurements. But with a few exceptions the principles and technique which it presents are applicable to all the sciences which study variable phenomena. So far, indeed, physical anthropology has been the science to take the most advantage of them, and in medicine they will perhaps find their greatest usefulness. The illustrations occasionally, and the problems frequently, come from the biological sciences. If one alters the language and replaces the illustrations from the realms of psychology and social science by similar ones from economics, vital statistics, medicine, physiology, anthropometry or biology, as the case may be, he will find the principles to hold, with an occasional obvious modification to fit the special data. The descriptions of technical procedure similarly may, after a few obvious alterations, be applied to variable measurements in general.

The Intrinsic Interest of the Theory of Measurements.

The author may be permitted to express his hope that those who use the book will regard its subject matter as something more than a means to the end, convenient handling of measurements. One can use ingenuity in manipulating measurements as well as in devising experiments; can use logic in working with measures as well as in working with evidence of a more impressive and dramatic sort.

Skill in expression is nowhere more required than in the task of making quantitative estimates, comparisons and relationships, brief, clear and emphatic. Statistics are, or at least may be, something beyond tabulation and book-keeping. In studying even this most elementary introduction one who is willing to use his higher intellectual powers will find something for them to do.

The Special Problems of Mental Measurements.

In the mental sciences as in the physical we have to measure things, differences, changes and relationships or dependencies. The psychologist thus measures the acuity of vision, the changes in it due to age, and the relationship between acuity of vision and ability to learn to spell. The economist thus measures the wealth of a community, the changes due to certain inventions and perhaps the dependence of the wealth of communities upon their tariff laws or labor laws or poor laws. Such measurements, which involve human capacities and acts, are subject to certain special difficulties, due chiefly to the absence or imperfection of units in which to measure, the lack of constancy in the facts measured and the extreme complexity of the measurements to be made.

If, for instance, one attempts to measure even so simple and mechanical a thing as the spelling ability of ten-year-old boys, one is hampered at the start by the fact that there exist no units in which to measure. One may, of course, arbitrarily make up a list of 10 or 50 or 100 words and measure ability by the number spelled correctly. But if one examines such a list, for instance the one used by Dr. J. M. Rice in his measurements of the spelling ability of 18,000 children, one is or should be at once struck by the inequality of the units. Is 'to spell certainly correctly' equal to to spell because correctly'? In point of fact, I find that of a group of about 120 children, 30 missed the former and only one the latter. All of Dr. Rice's results which are based on the equality of any one of his 50 words with any other of the 50 are necessarily inaccurate, as is abundantly shown by Table I. (page 8).

Economists have not yet agreed upon a system of units of measurement of consuming power. Is an adult man to be scored as twice or two and a half or three times as great a consumer as a ten-yearold boy? If an adult man's consuming power equals 1.00, what is the value of that of an adult woman?

If we measure a school boy's memory or a school system's daily attendance or a working man's daily productiveness or a family's daily expenditures, we find in any case not a single result, but a set of varying results. The force of gravity, the ratio of the weight of O to the weight of H in water, the mass of the H atom, the length of a given wire; these are, we say, constants; and though in a series of measures we get varying results, the variations are very slight and can be attributed to the process of measuring. But with human affairs not only do our measurements give varying results; the thing itself is not the same from time to time, and the individual things of a common group are not identical with each other. If we say that the mass of the O atom is 16 times the mass of the H atom, we mean that it always is that or very, very near it. But if we say that the size of the American family is 2 children, we do not mean that it is that alone; we mean that it is sometimes 0, sometimes 1, etc.

Even a very elaborate chemical analysis would need only a score or so of different substances in terms of which to describe and measure its object, but even a very simple mental trait, say arithmetical ability or superstition or respect for law, is, compared with physical things, exceedingly complex. The attraction of children to certain studies can be measured, but not with the ease with which we can measure the attraction of iron to the magnet. The rise and fall of stocks is due to law, but not to any so simple a law as explains the rise and fall of mercury in a thermometer.

The problem for a quantitative study of the mental sciences is thus to devise means of measuring things, differences, changes and relationships for which standard units of amount are often not at hand, which are variable, and so unexpressible in any case by a single figure, and which are so complex that to represent any one of them a long statement in terms of different sorts of quantities is commonly needed. This last difficulty of mental measurements is not, however, one which demands any form of statistical procedure essentially different from that used in science in general.

CHAPTER II.

UNITS OF MEASUREMENT.

boy has one and

LET us examine first a number of units that have actually been used. It is the custom to measure intellectual ability and achievement, as manifested in school studies, by marks on an arbitrary scale; for instance, from 0 to 100 or from 0 to 10. Suppose now that one boy in Latin is scored 60 and another 90. Does this mean, as it would in ordinary arithmetic, that the second one half times as much ability or has done one and one half times as well? It may by chance in some cases, but the fact that the best one and the worst one of thirty boys may be so marked by one teacher, and during the next half year in the same study be marked 70 and 90 by the next teacher, proves that it need not. The same difference in ability may, in fact, be denoted by the step from 60 to 90 by one teacher, by the step from 40 to 95 by another, by the step of from 75 to 92 by another and even by still another by the step from 90 to 96. Obviously school marks are quite arbitrary and their use at their face value as measures is entirely unjustifiable. A 90 boy may be four times or three times or six fifths as able as an 80 boy.

It is the custom to measure the value of commodities and labor by their money price, but since a dollar in one year is evidently not necessarily equal to a dollar twenty years before, systems of index values have been established to give a better unit. Even these index values as arranged by different statisticians differ somewhat. For a unit of power of consumption Engel takes a child during its first year. He then calls a year-old's power of consumption 1.1; a two-year-old's, 1.2; and so on up to 3.0 for a woman 20 years or over and 3.5 for a man 25 years or over. In the United States investigation of 1890-91 the unit was taken as 100 for an adult man, 90 for an adult woman, 75 for a child 7 to 10 years old, 40 for a child 3 to 6 and 15 for a child 1 to 3. The arbitrary nature of the scale of measurement is apparent.

*The reader unlearned in economic science may neglect this illustration.

The extreme inequalities of the spelling words, treated by Dr. Rice as of equal difficulty, are shown in Table I.

TABLE I.

THE RELATIVE FREQUENCY OF MISTAKES WITHIN THE SAME GROUP OF CHILDREN FOR EACH OF 49 WORDS TAKEN BY DR. RICE TO BE OF EQUAL AMOUNT AS MEASURES OF SPELLING ABILITY.

In the three cases so far the arbitrary opinions or guesses of individuals that such and such are equal have been uncritically accepted. It is as if we should measure length in accord with some one's guess that the distance from San Francisco to Chicago equaled three times that from Chicago to New York and eight times that from New York to Boston.

The risk of accepting subjective opinion even in the cases where it is least liable to error may be illustrated by the variation in judgment, even among competent authorities (graduate students of experience in teaching), as to the relative difficulty of different parts of the following simple tests:

*Piece was scored correct.