mate answer to this question will be the discovery of units of measurement in every school subject, and the construction, by scientific methods, of scales that can be applied as the foot-rule is now applied, regardless of time, or place or persons. The best possible ratings of individuals by relative position are only temporary expedients that must some day give way to ratings by means of standard scales. The nearest approach to such a scale, and a perfect illustration of the method, is E. L. Thorndike's "Handwriting," Teachers College Record, March, 1910. The Courtis Standard Tests in Arithmetic also furnish a means of comparing the achievement of one school with that of another, and the work of one year with that of another. We are not likely to continue to spend billions of dollars on education and be satisfied with guessing at results. Measurements of results with quantitative precision will be made as soon as people know enough to demand such measurements.
Lacking the necessary units and scales, we may even now ask whether the differences among individuals in mental capacities are explainable by any simple causes and amenable to any single type of description. They are not, if we are to accept the tables and figures just presented as correct records of the abilities of college students. But fortunately we are not dependent on such unscientific data. Psychologists have recently given us many rigorously scientific studies of the distribution of mental traits.
These studies have shown that in any group of individuals representing a single species, the distribution of any trait not greatly influenced by natural selection appears to be that of a chance event. The surface of frequency for that trait approaches that of the probability integral. It is like the cross-section of a pile of sand dumped from a cart. The most convenient way to represent tables of frequencies is by means of diagrams in which distances along a base line represent the different quantities, or units of measurement, and the heights of columns erected upon it represent the frequencies of the several quantities. Fig. 9 presents several illustrations, D representing the results of a memory test. By such graphic representations rather that algebraic formulae, the answer to our question and the evidence for it can be made clear even to one unfamiliar with the mathematical properties of the surface of frequency of a chance event.
Fig. 9, A, gives the distribution, or surface of frequency, of the type to which we assume that all distributions of mental traits conform. Fig. B is the same type of distribution with a coarser separation into grades. This type is called the normal surface of frequency. It describes, for example, the distribution of accidental errors in scientific observation. Thorndike's numerous measurements show a remarkable uniformity in the distribution of mental traits among individuals. Fig. 9, D, showing the memory span for digits in 123 American women stu-