Deficiency and Delinquency: An Interpretation of Mental Testing

by James Burt Miner

In considering the curves of development it is desirable first to notice the differences between measurement in equal physical units and measurement in equivalent units of ability or of development. The difference in the point of view of the two forms of measurement is so pronounced that I can hardly hope to make myself clear to those who are not somewhat familiar with such terms as “distribution curves,” “frequency surfaces,” “standard deviation,” and other phrases connected with the theory of probability, which are treated at length in such books as Thorndike's “Mental and Social Measurements” and Yule's “Introduction to the Theory of Statistics.” We often, by mistake, regard the growth of an inch in height, for example, as always representing an equivalent unit of growth. This will lead us into rather serious misconceptions unless we are careful, for it is perfectly evident that the growth of an inch in height has a very different significance for the three-year-old boy than for the eight-year-old. Half of the three-year-old boys grow about 3 inches during a year while at eight years of age not more than about one in seven grow that much. Moreover it is not always satisfactory to regard the same relative increase in physical size as an equivalent unit of development. To say that a boy 20 inches tall who grows 1-10 in height shows an increase in development equivalent to a boy of 50 inches who grows one-tenth, may be quite misleading. Nearly every 20-inch child grows one-tenth in height in a year while not one in fourteen of the boys who are 50 inches in height may grow at that physical rate. In considering human traits, and especially developmental traits, it would seem to conduce to more significant thought if we gave up at times our habit of thinking in terms of equal or relative physical units and thought instead in terms of more equivalent biological units.

In the measurement of mental ability, moreover, it is exceedingly difficult to utilize equal physical units. Most of the objective units which are commonly called alike are clearly not equal even in the physical sense. “Spelling one word,” for example, is not equal to spelling another “one word;” but only equal to spelling the same word. Out of such units of amount accomplished, it is, of course, not possible to build a satisfactory scale without referring to some other concepts of measurement. Some tests, however, are scored in equal units. When the measurements for example, are in the units of time it takes to perform the same task under the same outward conditions we have the possibility of a scale of equal objective units. Such a scale is approached by the results with the form board test which give the number of seconds it takes children to place blocks of different shapes in their proper openings.

Even the unit of time may be deceptive in name, as it is with the Binet scale. A year of time is, of course, the same physical unit and the task proposed with the Binet scale is always the same, but the other essential with this scale, the children of each age who pass the tests at each age norm, varies decidedly. “Test-age five,” for example, means 44% of the children pass and “test-age eleven” means 88% pass, even with approximately random samples of children of these life-ages. This question of the equality of the Binet age units will have to be considered further, therefore, in connection with the other concept of equivalence used in psychology.

In order to determine equivalent units of activity we find that a number of different concepts have been utilized. With some of the scales for measuring educational products, such as Thorndike's Scale for Handwriting, equal units of merit in handwriting mean differences judged equal by relatively the same proportion of competent judges. This form of unit has not been used, however, in any scale of mental development thus far proposed.

In the measurement of mental ability the most commonly accepted idea of equivalent units is that they are provided by the units of standard deviation for a series of measurements which distribute in the normal form. The meaning of these units may be understood by referring to Fig. 3 which shows Gaussian or normal distributions of abilities of individuals at various periods of life in curves A, B, C, D and E. The straight lines of the measurement scales form the bases of these distribution curves. These graphs represent the normal form of distribution usually expected when any fundamental ability is measured in a random group. If the number of cases at each unit of measurement are plotted by a point placed relatively as far above the scale, used as a base line, as the number of cases found at that unit of the scale, it will be discovered that these points arrange themselves in the form of a symmetrical curve high at the middle and flaring out along the base-line scale. This bell-shaped curve, known as a normal probability curve, shows that the largest number of cases occurs at the middle or average measurement. From this middle point on the scale the number of cases falls off gradually and symmetrically in both directions. Distances along the base line of this distribution surface may then be measured in terms of the standard deviation regarded as unity. This S. D. is the best measure of the scatter of the deviations. It is the square root of the average of the squares of the deviations of the separate measurements from the average of all the measurements. There are approximately four units of the standard deviation between the average and either extreme when the distribution is normal, as in Fig. 3. Only six cases in one hundred thousand fall outside these limits.

The studies of biological traits suggest that a unit of the standard deviation is the most important measure we have for equivalent degrees of any trait which distributes normally. It measures the same portion of the total distance from the lowest to the highest ability on any objective scale so long as the distribution of measurements is in the normal form. It thus affords the best interchangeable unit from measurements at one life-age to those at another, provided that the distributions keep close to the form of the normal probability curve. This is the assumption on which practically all the developmental scales have been based. The difference in ability between an individual at the average and at -1 S. D. (standard deviation) below the average is equivalent to that between the last individual and one at -2 S. D. The same distances along the base line of different distribution surfaces measured in terms of their respective deviations set off equivalent portions at each age so long as the distributions are normal. For example individuals measuring between -2 and -3 S. D. in any distribution in Fig. 3 are equivalent in ability to those lying between -2 and -3 S. D. in any other of these normal distribution surfaces. Later we shall consider equivalent units when the form of the distribution of ability is not normal or is unknown.

We may now compare the relations of the units in the physical scale, shown at the left of the figure, to units of the scales for adults or for the immature of any age, expressed in units of the standard deviation from the averages of these groups. Relative ability measured on the physical scale or any one of the distribution scales in Fig. 3 will be found identical since they all start from the same zero point and the distributions are all normal. But the ability of an individual in one distribution can hardly be compared with that of an individual in another distribution in a biologically significant way by their actual positions on the physical scale. A physical unit, does not measure the same sort of fact of development in a scale for the immature that it measures in the scale for adults or that it measures in another dynamic scale for the immature. This can be seen when a physical unit is compared with the amount of standard deviation which it measures in the different scales. Moreover, the correspondence of relative distances on the physical scale and any one of these other scales will not hold the moment the distributions do not start from the same point or are unsymmetrical.

It does not seem seriously wrong to suppose that there are some individuals at any age who have no more mental ability than the baby of the poorest mental ability at birth. At any rate our intelligence scales are hardly fine enough to measure the difference in intellectual capacity between the dullest adult idiots and the dullest idiot babies. We shall, therefore, here assume that mental capacity extends to zero at each age. The importance of this will be evident when we consider the question whether the distributions of ability are symmetrical around the average point at each age. Postponing for the present the discussion of unsymmetrical or skewed distributions, we may consider the several meanings of stages of development.

In applying the concept of the probability curve we should distinguish between individuals who have attained their mature mental capacity and those who are still maturing. The former would be represented by a random group of adults (Distribution E, Fig. 3) the latter by a group of nine-year-olds (Distribution C). If we say, for example, that a child has reached a certain stage of development we might have in mind the final distribution of mature capacity or the distribution of capacity among those of his particular age or of all ages. When we compare stages of development we must, therefore, be careful to indicate the distribution surface to which we are referring.

An increase in development may refer to at least five different things depending upon the scale of measurement to which reference is made. Besides an increase measured by the physical scale, the scales for adults, for the immature or for all ages, to which we have already referred, it may mean an increase judged by the distribution of increases which individuals of the same life-age and capacity make in the same period of time. This last meaning may be the most significant, although it has never been used. It has reference to a distribution surface of increases such as is represented in Distribution F, Fig. 3. This is intended to show the increases in one year of all two-year-old children who had average ability at 2 years, on the assumption that at 3 years these children would on the average equal the average of all three-year-olds. It is clear that when these increases are measured in objective units the latter have a still different significance from that assigned to them in connection with other scales. An increase of one objective unit here might represent twice the standard deviation, while it only represents 0.2 of the standard deviation in another distribution.