Goddard's Table I gives the data from which the following percentages of those who pass the norm are calculated, not counting those above 11 years, since the older groups are clearly affected by selection:—5 yrs., 88%; 6 yrs., 79%; 7 yrs., 81%; 8 yrs., 51%; 9 yrs., 60%; 10 yrs., 73%; 11 yrs., 44%. Kuhlmann's figures when using his own revised scale with public school children including the seventh grade, are:—6 yrs., 100%; 7 yrs., 95%; 8 yrs., 90%; 9 yrs., 87%; 10 yrs., 81%; 11 yrs., 80%; 12 yrs., 57%. It is clear that any change in the test norm from age to age must disturb the quotient which is based on these norms, although it would not affect the intelligence coefficient with the Point scale.
3. A third advantage of the percentage method arises from the fact that we cannot presume that the same ratio in terms of the scale units will exclude the same degrees of ability at different ages even when the norms for these ages are properly adjusted. The earlier results with the Stanford revision show a large variation as to the percentage excluded by the same I Q at different ages. For example, an I Q of .76 would have shut out 1% of 117 non-selected 6-year-olds, 2% of 113 9-year-olds and 7% of 98 13-year-olds. The lowest 1% of the last group was below a borderline of .66 ([197]).
With widely varying norms of the other scales, the I Q borderlines show much greater variation. In a recent review of the evidence, including Descoudres' report ([96]) on retesting the same children for several years Stern recognizes that an I Q index is not constant after 12 years ([187]). Doll records decided changes in quotients for the same individual at different ages ([99]). So far as the 1908 scale is concerned, using Goddard's data, our Table V shows that at five years of age the lowest 1.8% would fall at or below a quotient of .40, at eight years the lowest 1.9% would show a quotient of .62 or less, and at 15 years the lowest 2.8% fall below a quotient of .75. The rough tentative approximation of scale limits which I have suggested for the lowest 1.5% shows that a series of quotients for children from 5 to 15 years of age would be below .75 at every age and below .65 for half of these ages. For the presumably deficient group the quotients would be still lower in order to be as conservative as the borderlines that I have suggested with the Binet scale as at present standardized.
With the coefficient of intelligence and the Point scale, the Yerkes and Wood data show that their borderline of .70 excluded 13% of 196 children 8 and 9 years of age, while it excluded only 5% of each of the next two groups of double ages. With the group of 237 18-year-old Cincinnati working girls it excluded only 3% ([226]).
The data at present available thus indicate that we should not expect to find the same ratio at different ages excluding similar percentages. If the ratios have a value for comparing individuals of different ages, they seem to fluctuate so decidedly from age to age that they can hardly be trusted for stating the borderlines of deficiency without empirical confirmation for each age.
Pearson found that the children of the older ages in the special classes were more and more deficient, measured in terms of the standard deviation of the normal group. This shift on the average was four months of mental age downward for each year of life during the period 7-14 which he studied. It makes uncertain the definition of the borderline in terms of a constant multiple of the deviation or of a constant quotient, unless this shift is shown to be due to imperfections of the tests which can be corrected, or to changes in the selection of the tested groups at advanced ages.
Pearson's suggestion of -4 S. D. as a borderline with the Jaederholm data gives some very curious results with the group of children in the special schools at Stockholm. Under his interpretation at life-ages 8-11 from 0 to 5.2% of the pupils in these classes would be regarded as deficient, while for life-ages 12-14, 15.2% to 44.4% are beyond -4 S. D. In passing it is to be noted that if one accepted Pearson's suggestion that the borderline should be fixed at -4 S. D., in case the distribution of mental capacity were strictly normal, only four children in 100,000 would be found deficient, according to the probability tables.
With the method of the standard deviation it would be necessary either to show that the deviation was constant in terms of the year units or else to restate the borderline for different ages in terms of the scale units. The irregularity of the norms with the Binet scale could also be allowed for, of course, by stating different quotients for the different ages, but when this readjustment is required for either the ratio or the deviation in terms of the scale units, these methods lose all their advantage of simplicity. Instead of one ratio or one multiple of the years of deviation, we might have a different statement for each life-age. With the percentage method there would be only one statement of the borderline for all ages in terms of percentage, although the scale positions change which shut out the same lowest percentage.
4. All the quotient methods of defining the borderline encounter a serious practical difficulty in fixing the borderline for the mature, so that it will be equivalent to that for the immature. With the Stanford scale in calculating the quotient for adults, no divisor is used over 16 years. Yerkes and Bridges also think that this is about the time that the development of capacity ceases. Kuhlmann and others use 15 as the highest divisor. Wallin objects to either of these ages being used as the age of arrest of mental development (15, p. 67). Both the methods of the standard deviation and percentage have a similar difficulty, in that the borderline for the mature has to be empirically determined on a test scale. In this dilemma, however, the data collected with the random group of 15-year-olds in Minneapolis and published in the present study, places the borderline for the mature on either the 1908 or 1911 Binet scale in a much safer position, so far as empirical data is concerned, than the borderline for the mature for any other scale. This is true whether that borderline be then stated in terms of either the quotient or percentage methods. Translated into terms of the quotient, our percentage borderlines for the mature with these scales, below X for presumably deficient and below XI for the uncertain, would amount to quotients .60 and .66 on the basis of our findings with this random group of children who have presumably about reached adult development. Pearson does not attempt to define any borderline for the adults on the basis of the deviation, since Jaederholm tested only children. Moreover, this is not possible empirically with our group of 15-year-olds, since we tested only the lower extreme of this group.
Unfortunately, the borderlines of the mature for the Stanford and other scales depend upon empirical results obtained not with random groups, but upon a composite of selected groups of adults built up by the investigator on an estimate that this combined group represents a random selection among those with a typical advance in development, an almost superhuman task. Fortunately the empirical determination of this borderline for the mature might be improved later by obtaining data on less selected groups. The clearer significance of the empirical data for the borderline for the mature which I have presented for the Binet 1908 and 1911 scales from a random group of 15-year-olds seems to be an important practical advantage. It provides an empirical basis for judging the implication of test results with adults. It gives adults the benefit of the doubt if they improve after 15 years of age.