Deficiency and Delinquency: An Interpretation of Mental Testing

by James Burt Miner

For our problem of units and scales of measurement, an asymmetrical distribution sets a very difficult problem. It may be that this very difficulty has been one of the main reasons for slowness in recognizing the drift of the evidence. In order to set forth the difference in the conception of measurement when distributions become asymmetrical I have presented this hypothesis in connection with the curves of development in Fig. 5. It will be noted that if the distributions of mental capacity vary in symmetry, the units of standard deviation change in significance from one form of distribution to another. Minus 2 S. D. may exclude very different portions of groups differently distributed, while it would always exclude the same proportion if the distributions had the same symmetry, or skewness.

Under conditions of variable symmetry there is a sense in which the same relative physical score in units running from zero ability to the best ability would always have an equivalent objective meaning, but this might not express equivalent development conditions at different ages. For example, with shifting forms of distribution, to say that a child of six years had reached three-fifths of the best development for his age on an objective scale might give no significant indication of how nearly he was keeping pace with those three-fifths of the best ability of another age. Neither would his position in units of the deviation of ability at his age give this information without knowledge of the form of the distribution of ability at his age. With varying forms of distribution at different stages of development this would afford an insurmountable difficulty.

Fig. 5. Hypothetical Development Curves (Changing Forms of Distribution)

With unknown or varying types of distribution it is desirable to utilize percentiles as equivalent units for comparing individuals at different stages of development. They differ somewhat from ranks in an order of noticeable differences. With an indefinitely large group, such ranks would mark off only those cases which were indistinguishable in merit. These units would be numbered in order from the highest to the lowest in ranks of just distinguishable merit, a different number of individuals conceivably occurring at the single steps. Psychologically the percentiles are somewhat less significant because they are not conceivable in steps of just noticeable differences. Percentiles have less value in comparing abilities in the same distribution, but have decided advantages when comparing corresponding abilities in different distributions. Except at points where merit is indistinguishable, they signify that a certain proportion of a group is ahead in the struggle for existence. They are thus units of relative rank. Moreover, they are directly translatable into units of the deviation in case the form of the distribution of ability has been determined. This is a special advantage if the forms of distribution turn out to be normal or even uniform.

In using percentiles it is to be remembered that equal differences between percentiles are not comparable in the same distribution except in the sense of the same extra proportions of the group to be met in competition. A change in the degree of ability from the lowest percentile to the lowest 2 percentile would be very different from the change in the degree represented by the 50 percentile to the next percentile above. Differences in the ability of individuals ranking near each other in the middle of the same percentile series would be distinguished with difficulty while it would be easy to make such discriminations at the extremes.

The special value of the percentile units in measurement of ability lies in the comparison of individuals of corresponding position in corresponding groups in which the ability may not be assumed to distribute alike. The concept that 995 out of every 1000 randomly selected individuals at his age are ahead of a particular individual in the struggle for existence has very definite and significant meaning which is quite comparable from one period of life to another regardless of the form of the distribution. We shall return to this question of equivalent units in distributions of unlike symmetry when we compare the definitions of the borderlines of deficiency in terms of intelligence quotient, coefficient of intelligence, standard deviation and percentage. Corresponding percentages of corresponding groups have a more useful definite significance of equivalence than any other units of measurement of mental ability available when the forms of distribution vary at different stages of development or are uncertain, as seems to be true with tested abilities.

B. The Curves of Mental Development.

When we endeavor to make our ideas of mental development more definite, we are assisted by thinking of the various stages in graphic form. This is especially true when trying to think of the position of the deficient individuals, relative to the average individuals and to genius.

In diagrammatically presenting these concepts in Fig. 3 and Fig. 5 we do not wish to assume that all the principles on which the developmental curves have been plotted have been decided. If they make clearer the points still under discussion and direct the discussion to specific features so that more data may be brought to bear upon the empirical determination of their characteristics, they will serve a useful purpose. For our present ends, we shall consider only certain features which have a bearing upon the interpretation of developmental scales and the quantitative definition of the borderline.