Perhaps the most interesting characteristic of the percentage method is that it automatically adjusts itself to any form of distribution. In case the distributions of ability turn out to be normal for each age and the arrests of development for different degrees of ability distribute alike, then the borderline fixed by the percentage method becomes identical with the corresponding borderlines by the quotient, deviation, or relative objective distance. It can be directly translated into a quotient or a multiple of the standard deviation. This fact affords a good check upon the empirical borderlines fixed by the percentage method for different ages. If the distribution is normal, the lowest 1.5% and 0.5% would be identical with -2.17 S. D. and -2.575 S. D. in samples of 10,000 cases. We may check these percentage borderlines by Goddard's results for ages 5-11 tested with the 1908 Binet scale. I have given the standard deviation for the ages 5-11 with this data in Chap. XIII a, 2. Applying the criterion of 2.575 S. D. to these deviations, we find that to be in the lowest 0.5%, if the distribution were normal, would be about a year less of deficiency than we have suggested, while Pearson's borderline of -4 S. D. would be close to that we suggest. The empirical data thus suggest that the assumption of a normal distribution is faulty at the borderline or else Goddard's data is incorrect for fixing the limits on the scales. I have already given the evidence for supposing that the distribution is skewed during the years of growth.
When approximately random samples are not available, a multiple of the deviation of an efficient group such as -4 S. D. at the particular age seems to afford a practical way of discovering a tentative borderline until a random sample can be measured. The serious theoretical objections to such a procedure as a regular method is that the efficient group would be selected by the subjective standard of somebody's opinion and that the form of distribution of ability may vary from age to age.
Recalling the practical advantages of the percentage method which we enumerated in the preceding section, we can now better understand the value of a method that is not disturbed by the form of distribution of mental capacity which may ultimately be found to prevail at different ages. It is safer at present to assume that the distributions do change enough in form at the lower end seriously to affect the borderlines of deficiency as defined by other methods. If, however, the form of distribution remains uniform, it would first be necessary for those advocating the use of any of the other quantitative definitions to show that the units of their scales are equal under some reasonable hypothesis. A ratio or a deviation statable only in scale units which are not demonstrably equal is a hazard, with the chances badly weighted against its reliability. So far as both the Binet and the Point scales are concerned we have found that the units are not equal. A quotient or coefficient arrived at by assuming their equality is sure to mean seriously erroneous fluctuations in the borderlines.
Referring to the percentage method, Yerkes and Wood say: “Frequency of occurrence is unquestionably a useful datum, which should be presented, if not instead of, then in addition to, certain other statistical indices which possess greater scientific value” ([226]). These other indices require both equal scale units and uniform distributions from age to age. The ratio and deviation methods fail at present in both of these particulars, so that it seems necessary to depend upon the percentage definition of tested deficiency, incomplete as that may be.
This leaves us in the unfortunate situation that the borderline positions on the scale will have to be stated separately for each age and will have to be found empirically. Moreover, we shall need to determine more accurately in what lowest percentage an individual must test in order reasonably to predict that he will require social care for the good of himself and society.
As soon as anybody can discover a means of defining the borderline, which is equally accurate and significant, and which, in addition to counting the proportion of better individuals to be met in the competition of life, will also evaluate the distance they are above the borderline, we all shall be eager to accept this better criterion of deficiency. A form which it might take is that of relative objective distance between zero and median ability. If measurable in equal objective units, this would be independent of the form of distribution and would improve the quantitative description of equivalent deficiency, provided that it also forecasted future social failure as well as the percentage method.
What form of stating the borderline of tested deficiency may ultimately meet with approval, a verbal definition of feeble-mindedness will never remain an ideal scientific statement until it finds expression in quantitative terms.
BIBLIOGRAPHY ON TESTED DELINQUENTS[[34]]
1. Baldwin, Bird T. The Learning of Delinquent Adolescent Girls as shown by a Substitution Test. J. of Educ. Psychol., 1913, 4, 317-332.
2. Blumel, C. S. Binet Tests on the Two Hundred Juvenile Delinquents. Training School Bull., 1915, 12, 187-193.