Deficiency and Delinquency: An Interpretation of Mental Testing

by James Burt Miner

The differences between the deviations for ages 7 and 11 or between ages 8 and 10, are more than three times their standard errors, so that we would not be justified in assuming that the standard deviations of the separate ages measured in terms of years of excess are equivalent. There seems to be a tendency for the deviations to increase, at least from age 7 to 10 and 11.

The comparison of the year units on the Binet scale with the diagrams in Fig. 3 shows that if the scale at each life-age shut out the same lowest proportion, say half, of the children of that age, then the year units might be regarded as equal in the sense of equal average growth increments, as Pearson suggests. A child 7 years of age testing VII would be at least one annual average-growth unit higher in mental development than one of 6 years testing VI, and so with each age until the limit of development had been reached. This is the condition approximated closely for children by the new Stanford scale and the corrected Jaederholm data. Since there is little prospect, however, even with a scale perfected so far as its age norms are concerned, that the total distributions for each of the different years would be the same multiple of the year-units, the main significance of the age units is in permitting the statement that a child had reached the tested development normal for the children of a certain age.

It is also legitimate to use years of retardation as a short way of expressing rough borderlines when they happen thus to afford an easy method of empirically describing equivalent borderlines for a particular scale. This is what I have done for convenience in Part One of this book. I certainly do not mean to contend that four-years retardation has theoretically the same significance at different ages, in terms of the deviation of the separate ages. To me the Binet years are no more than names for certain positions on the scale.

To most psychologists who have been dealing with the measurement of mental development, I believe that the most significant concept of equivalent units would be in terms of the deviations for each age provided that the form of the distributions remained normal. But the deviations vary so much in the terms of the year units that it is not likely that they will be willing to accept a year of excess or deficiency as an equivalent unit for different ages with the common forms of the scale in use in English-speaking countries. Moreover, below the age of 6 and above 15, the limits which Pearson discusses, there is good reason to expect the year unit to vary still further. This Pearson recognizes for the complete developmental curve. It is only at the intermediate years, in which the average increases are most constant in relation to the deviations of the separate ages, that the year unit may be at all serviceable in measuring the deviation of a child from the norm of his age.

With the scales in use in this country the Binet year units are not equivalent in the sense in which they are usually spoken of as equivalent. We should recognize this and emphasize it. Even if the norms at each age marked off the same proportion of the individuals, as shown in A and B of Fig. 4, unless we knew that the forms of distribution were always alike, we should not know that the distance between successive age norms was the same on any sort of objective scale other than average age increments. Moreover, we would not have an objective scale of equal units applicable to measuring the deviation of children of any one age. The average annual increments would not necessarily represent the same proportion of the total distance from the lowest to the highest ability at different ages even if the distributions were all normal. With normal distributions it would also be necessary to demonstrate empirically that the annual average growth increment between successive ages always bore a constant relation to the deviations at these adjacent ages as shown in B of Fig. 4 where the increment is equal to 1 S. D. at each age. This could not possibly hold when the increment lessened near maturity.

Fig. 4. The Question of Equivalence of Year Units.

If the distributions of ability were variously skewed, the year units of excess or deficiency would not be shown to be equivalent at the different ages even if the proportion of individuals one year accelerated was equal to the number one year retarded, two years accelerated equal to those two years retarded, etc., at each age and the norm at each age shut out the same proportions of the age group. This is shown in C of Fig. 4 in which the year units are clearly not equal steps from lowest to highest ability even for the same age and yet the usual criteria which have been suggested for discovering the equivalence of the units are fulfilled. Whether the actual distribution of ability is skewed or normal cannot be determined by the Binet scale, of course, on account of the uncertain and probably varying size of its year units in measuring deviations at any age.

With the empirical evidence against the equivalence of the year units and the impossibility of determining their equivalence unless we first know that ability is distributed normally at each age, it is certainly hazardous to assume that individual deviations measured in terms of year units are equivalent at different ages.

It may be noted that it is quite as hazardous to suppose that the units of the Point scale are equivalent in any theoretical or practical sense. This question will be discussed later in Chap. XIII, B, (b).