In Fig. 6 curves A and B are Smedley's tests; curve C includes in addition Norsworthy's unrelated words, Pyle's memory for concrete and abstract terms, Anderson's letter-squares, Carpenter's memory for pictures, and Gilbert's for the time interval; curve E includes Pyle's two and Carpenter's two substitution tests; curve F includes Pyle's Marble Statue and Norsworthy's memory for related words and for sentences; curve S is Norsworthy's; curve D is the combination of these 17 tests.

In Fig. 7 curve H includes Gilbert's visual reaction time, Norsworthy's A and a-t tests, Carpenter's two A tests; curve I includes Gilbert's and Smedley's tapping tests; curve J is the median of the central tendencies of all 40 tests; curve K includes Norsworthy's two opposites and her part-whole and genus-species tests, the Pyle opposites, genus-species and part-whole tests; curve L is the same as D, curve M includes Smedley's strength of grip and ergograph tests and Gilbert's fatigue of tapping; curve N includes Pyle and Anderson's word building tests and Pyle's uncontrolled word association test.

In Fig. 8 curve P is Gilbert's visual reaction time test, curve S is Norsworthy's test for memory of unrelated words, the other curves are the median and quartiles for the central tendencies of all 40 tests after each was expressed at each age in terms of the gain from 8 to 9 years taken as a unit.

Several points are to be noted about the nature of the curves for different tests. In Fig. 6 showing the curves for different forms of memory tests, that for the memory of digits is very different in character from that for memory of related material. The most extreme differences in the time of maturity are shown by the test for memory for digits presented orally and the substitution of color in forms, the former continues to increase so rapidly relative to the absolute increase from 8 to 9 years that it cannot be represented in the graph reaching 539 units of the scale by 14 years of age, while improvement in ability in the latter is not measured after 9 years. We cannot take time to discuss how much of the differences between the various curves may be due to the nature of the tests themselves, the form of scoring the results, or the condition under which they were given, selection of subjects, etc. The conclusion is safe, however, that when groups of three or four tests of similar type show such marked differences as those for memory of digits and memory for related material we may expect similar differences in the rates of maturity of the corresponding processes.

From Fig. 7 we may learn that tests emphasizing functions such as speed of motor or perceptual motor reaction, curves H and I, are notably different in their form from curves for tests of imaginative processes, curve N. As we group tests together covering larger ranges of activity we approach the median curve for general ability. Note the median curve for 17 memory tests (curve L) compared with the median for the 40 tests (curve J). By empirical studies we might pick out types of tests which would most closely represent the maturity of average ability. For example, the median for the substitution tests, curve E, resembles the median for the memory tests, curve D, more closely than does that of the 4 digit tests, curve B. Curve K, for 7 association tests, resembles the median for the 40 tests, curve J, much more closely than the curve for the perceptual-motor speed tests, curve H. This difference can not be explained by the use of 7 instead of 5 tests in calculating the central tendency of the group. It probably means that the sort of psycho-physical processes usually tested more closely represent on the average the abilities shown in association tests than they do the abilities shown by speed of motor reaction. The significance of this sort of analysis for those constructing a scale for measuring intellectual ability is obvious.

Fig. 8 shows the median and quartile range for the central tendencies of the 40 tests and gives examples of two extremely different tests, visual reaction time and memory for unrelated words. How closely these particular tests represent fundamental differences in the maturity of different processes, we cannot, of course, be sure without prolonged research; but nobody would question that analogous differences would be found in different processes. When we think of curves of general ability we must, therefore, keep in mind the light which might be thrown on them by an analysis of the various processes tested in the particular scale used.

Another feature of all developmental curves which is apparent as soon as the causes of development are considered, is that growth in an individual is the result of several factors. These include the native capacity, the rate at which that capacity manifests itself instinctively, and the external stimuli which encourage or retard that manifestation. To some extent these factors vary independently. Our curves of development will never completely express all the facts until they analyse out all these factors for each of the processes. In the meantime we shall be able to think of general trends of development by considering average curves. The fact that they represent combinations of unanalyzed factors must, however, make us very cautious in interpreting our norms.

(b) Changes In The Rate Of Development.

There has been considerable discussion of the form of the curves of mental development. The logical aspects of the curves on the assumption of normal distribution of ability at each age and uniform age of maturity have been treated by Otis (163) and the bearing of these assumptions upon the Binet scale pointed out. Thorndike has plotted the developmental curves for a dozen tests on the basis of the variability at 12 years of age used as unit and gives a chapter in his Educational Psychology to the changes with maturity (198, Chap. XI). Bobertag suggests that the rates of development of normal and deficient children are analogous to the upward progress of two projectiles fired from such different heights that the force of gravity would retard the lower projectile more than the upper (81). This analogy supposes that the rate of maturity would continually decrease and that those who were feebler mentally would be arrested in their developmental earlier. Bobertag, Kuhlmann (137, 138) and Otis give evidence from the results of Binet testing that the rate of development decreases with age. The percentages of older children passing certain positions on the Binet scale or certain tests taken from it were found to change less at year intervals for the older ages. This evidence is not conclusive unless we know that the positions compared are at the same point in the distributions of ability at the beginning of the periods of growth. The same percentage change at a point farther away from the central tendency would mean a larger growth than at the middle of the distribution, when judged either in reference to a physical scale or to units of deviation.

While recognizing that the complete curve of mental development is logarithmic in form Pearson contends that, when measured by Jaederholm's adaptation of the Binet scale, development is adequately represented by a straight line from 6 to 15 years of age (164). As this conclusion is based upon the use, as equivalent units, of years of excess and deficiency at all these ages the data lacks the cogency of a scale of equal physical units.

With the Point Scale it is not known whether the units in different parts of the scale are equivalent. Without assuming that they are equal it is impossible to discover the form of curves of development from the records of children at a series of ages. Yerkes and Wood publish a curve of the increase of intellectual ability based upon point-scale measurements, which resembles in form the hypothetical curves. They say:

“The point-scale method has the merit of indicating directly the rate, or annual increments of intellectual growth. We do not claim for our measurements a high degree of accuracy, especially in the case of the early years of childhood. But even the roughly determined curve of intellectual growth from four to eighteen years, which we present below, has considerable interest for the genetic psychologist and for the psychological examiner. We have ascertained that whether measured by the ratio of the increment of increase, year by year, to the norm for the appropriate year or by the ratio of the extreme range of scores to appropriate year norms, intellectual development rapidly diminishes in rate, at least from the fifth year onward” (169, p. 603).

Waiving the question whether annual increases or the range of measurements relative to the age norms would be satisfactory indications of the change in the rate of growth, it seems to be fairly clear that neither of these criteria would be adequate unless we first knew that the units in which they were measured were equivalent at different portions of the scale. To show that the point scale units are even theoretically equivalent it would seem to be necessary to assume, on the basis of normal distribution of ability, that each unit of the deviation for each age distribution either equaled the same number of scale units or the same proportion of the total distance from lowest to highest ability at each age measured in the point-scale units. The originators of the scale do not seem to have planned it with this in view. Moreover, the difficulty of empirically demonstrating such equivalence of units on a point scale or any form of the Binet scale prevents its use for indicating curves of mental development, however serviceable it may be for other purposes.

The simplest demonstration of the form of the development curves is applying the same test, scored in equal physical units, to children of different ages. In Figs. 6, 7, and 8 the evidence from tests was assembled for ages 8 to 14 inclusive. It is probable, however, that the form of these development curves, when the unit of measurement was anything but time taken for the same task, has been affected by the difference in the real value of units called by the same name, e. g., giving the opposite of one word is not always equal to giving the opposite of another.

The best developmental curves empirically determined are probably those for the form board presented by Sylvester (191), Wallin (212) and Young (227) since in each of these cases the same test was presented at all ages and the scores were in equal physical units of seconds. It can hardly be supposed, however, that the form board curves alone would be typical of average mental development. To know something about the general curve of mental development we need a combination of a number of mental tests scored on scales of equal units. These may be either equal physical units or units on scales for mental development similar to those of Thorndike and others for measuring educational products, handwriting, arithmetic, spelling, etc.

That either a straight line or a simple curve would represent the development of ability from birth to maturity is very doubtful. When we consider the entire developmental curve from birth nobody doubts that there is a change in the rate of development at the time of the arrest of instinctive changes at adolescence. There are probably fluctuations in the rate before this final arrest. Pintner and Paterson also assume a complex curve of development (44). Whether the fluctuations should be allowed for in the description of the borderline of deficiency is the important question in our study. With measurements of bodily growth we noted that changes in the rate of maturity are accompanied by a skewness of distribution of ability at the ages affected. The same effect may be expected with mental measurements. The percentage method of defining the borderline of deficiency has an advantage when the form of distribution at any age is uncertain (See Chap. XIV, d.). Since the changes in the rate of development are most likely to be important at the prepubertal and adolescent ages the description of the borderline in terms of deviation or quotient may be expected to be most uncertain at this period. Moreover, none of the quantitative definitions of the borderline, except the percentage method, remain equivalent if rates of development of normal and deficient children change relative to each other, a question we shall now consider.

(c) The Question Of Earlier Arrest Of Deficient Children.

It has been assumed by Bobertag (81), Stern (88), Goddard (117) and others that deficient children reach their maturity earlier than normal children. If this were true the curves of mental development for the average and for the deficient children should not be expected to retain their same relative positions after the idiots had begun to show arrested development. Moreover, unless this arrest were compensated by some peculiar form of accelerated growth among those above normal ability, we might expect that the distributions of ability would change in form at the various ages after arrest had begun. A relative increase in the distance of older deficients from the average as compared with younger deficients may be interpreted as meaning either the earlier cessation of growth of the deficients or a change in the relative rates of growth of individuals of different mental capacity. When fully considered the present evidence from the Binet tests fails, I believe, to demonstrate the earlier arrest of the deficients, although it is undoubtedly true that the Binet scale may not be fine enough to measure the improvement of idiots. We shall take up certain investigations that bear upon this point.

Goddard has reported tests upon the same group of 346 inmates in an institution for the feeble-minded who were tested three years in succession (117). The paper suggests that the idiots, as a group increased less in absolute ability than those of higher mental age. The average gain for 55 idiots who tested I or II mentally was about half a test in the two years. In order to reach our present problem, however, we must know that the idiots, for example, developed relatively less mentally than did those of the higher grades of ability in the imbecile and moron groups of the same life-ages. This question cannot be answered from the paper. It probably cannot be adequately answered from mental age results on account of the irregularity in the value of the year units at different points on the Binet scales.

Bobertag summarizes Chotzen's data obtained by the examination of the children in the Breslau Hilfsschulen with the Binet scale. He believes that the position on an objective scale attained by the average of these retarded children is progressively lower with advancing age relative to the average position attained by normal children, assuming that the quotient for normal children remained constant at each age. The average intelligence quotients of all the children in the special schools (exclusive of those testing III or less) was 0.79 for those 8 years of age, 0.72 for those 9 years, 0.70 at 10, and 0.67 at 11-12 (81, p. 534).

Stern also compiled a table from Chotzen's results which shows this decrease in intelligence quotients with life-age separately for each group of those whom Chotzen by his expert diagnosis regarded as imbeciles, morons, doubtful, and not feeble-minded although attending the special schools (188, p. 80). This table is reproduced here as Table XX. On the surface it suggests that the quotients of the extreme groups are nearer together at the older ages, instead of being farther apart. The objection to this evidence from the Binet scale is that the norms are not equivalent for different ages on the scale used. Since the objective norms on the Binet scale are more difficult to attain at the older ages this variation would tend to make older children show lower quotients than the same children would show at younger ages, so that such tables are quite uncertain in significance.

The Jaederholm data with his form of the Binet scale, as treated by Pearson, shows a straight regression line for the backward children which falls below the normal development line on the average four months of mental age for each additional year of life from 7-14 (167). Accepting Pearson's interpretation that a year of excess or deficiency and a year of growth is a constant unit, we find that the deficient group from special classes was falling continually behind the normals with increase of age a relatively greater distance from any rational reference point. Pearson accounts for this change in the distance between the two groups of normal and backward children, as I understand his paper, by supposing that with increase in age more and more normal children become deficient. It would seem that this data would be more easily explained by supposing that the distributions became skewed toward deficiency for the older ages, rather than that the distributions remained normal and became flatter.

The best evidence as to the relative positions of the curves for deficients and those for average ability would be provided by using psychological tests that could be adequately scored in terms of equal physical units for the same task. The position of various lower percentiles relative to the average or to an assumed reference point could then be compared on the same objective scale. I have reviewed studies of this type in discussing skewed distributions in Chap. XIII, A, c. I there reached the conclusion that the weight of the evidence was that the distributions were slightly skewed in the direction of deficiency, although the evidence was not conclusive. We are now raising the further question whether this skewness increases with age.

On account of the difficulty of determining the points for zero ability in terms of the physical scales used, let us see what conclusion might be reached if we calculated the relative distance of median and low ability of equivalent degree from the scores of the same higher degree of ability assumed as a reference point at the various ages. There seems to be no reason in the theory of measurement why the highest score instead of the lowest score in random samples might not be used for a reference point for comparing the distances between normal and deficient children at different ages. Instead of using the highest single score, it would be better to use the upper quartile or quintile since it would be less affected by a chance error in giving the test.

Applying this method to determining the relative position of median and retarded ability I have calculated the data for the form board test cited previously from Sylvester (191) and from Young (227). This affords the only adequate evidence of which I know, derived from tests scored in equal physical units given to sufficiently large groups to indicate whether or not the retarded group changes its relative position from the normal group at different ages. The comparison is shown in Fig. 9. With Sylvester's data the distance of the lower quartile in ability from the median is compared with the distance of the upper quartile from the median, the latter distance being taken as a unit. With Young's data for Witmer's form board the quintile is used instead of the quartile and each sex is given separately. Since Young's table shows the scores for half ages, it was necessary to take the average of the two scores, thus giving the approximate score for the middle of the complete age group. The graph discloses no pronounced tendency for the retarded group to fall relatively farther behind the median with increase in age. There are, however, notable fluctuations in the relative positions of the groups so that at 7 years with Young's data for boys and at 13 years for Sylvester's curve the retarded group is twice as far from the median relative to the distance between the median and the corresponding better group as it is at some other times. It is possible that the curves for the older groups of those of poorer ability are too high since it is likely that more of the actually deficient children tend to be dropped from the public school classes with increase in age. Nevertheless, so far as the evidence at present goes it is not sufficient to determine whether the backward and the corresponding better group show a general change in their relative distances from the median with approach to maturity.

On the other hand the curves indicate the tendency for the distributions to be skewed toward deficiency and for the relative distances to fluctuate as we should expect if the accelerations in growth occurred at different ages for those of different ability. The data of Young suggest that there may be sex differences in the age of acceleration, the backward girls showing accelerations, relative to the upper group at ages 7 and 12, a year or more before the boys. For Sylvester's data the ratio of the distance between the median and the lower quartile divided by the distance between the median and the upper quartile for each of the age groups is as follows: 5 yrs. 1.8, 6 yrs. 2.4, 7 yrs. 3.0, 8 yrs. 2.0, 9 yrs. 2.2, 10 yrs. 2.4, 11 yrs. 2.0, 12 yrs. 1.8, 13 yrs. 3.0, 14 yrs. 2.1. For Young's data the corresponding ratios are—Boys: 6 yrs. 1.5, 7 yrs. 1.9, 8 yrs. 1.5, 9 yrs. 0.8, 10 yrs. 1.6, 11 yrs. 1.2, 12 yrs. 1.4, 13 yrs. 1.0, 14 yrs. 1.3. Girls: 6 yrs. 1.7, 7 yrs. 1.0, 8 yrs. 1.5, 9 yrs. 0.9, 10 yrs. 1.0, 11 yrs. 1.3, 12 yrs. 0.9, 13 yrs. 1.5, 14 yrs. 1.4. Changes in the rate of growth causing asymmetrical distributions are to be expected throughout the periods of growth. A fundamental skewness toward deficient mental capacity, therefore, would be indicated only if it were found at maturity or at ages when the average rate is decreasing, when the more capable individuals would theoretically approach relatively nearer the deficients if the latter accelerated later.

So far as physical growth is concerned Baldwin (74, 75) has shown with repeated annual measurements on the same group of children that the period of adolescent acceleration shifts from 12½ years for the tallest boy to 16 years for the shortest boy. For the tallest girl the maximum height was attained at 14½, for the shortest at 17 years, 3 months. Maturity may be reached at 11 years by a tall well nourished girl, while with a short girl light in weight it may be delayed until 16. “Children above medium height between the chronological ages of 6-18 grow in stature and in physiological maturity in advance of those below the medium height, and they may be physiologically from one to four or five years older than those below the medium height. Those above the medium height have their characteristic pubescent changes and accelerations earlier than those below; there is a relative shifting of the accelerated period according to the individuals' relative heights” (74).

Doll presents evidence from the physical measurements of a large feeble-minded group in institutions which he suggests shows that the shorter among them cease growing earlier. When the height of these feeble-minded is measured in relation to the Smedley percentiles of the height of normal children of their corresponding ages, he finds a correlation of -.20 between age and percentiles of height, the taller relative to normals being younger. He says: “This confirms Goddard's similar conclusion, but negatives for the feeble-minded at least, the theory affirmed by some writers, that children who grow at a retarded rate continue their growth to a later age” (98 p. 51). On the contrary this minus correlation is more likely to mean only that the Smedley norms on school children are too high for the older ages because of the excess of taller children who remain for the high school work. This would give the minus correlation without supposing that the taller individuals continue their growth to a later age, as he thinks.

Moreover, a total longer period of physical growth for smaller, less normal, children has been demonstrated. Boas (80) says: “Among the poor the period of diminishing growth which precedes adolescence is lengthened and the acceleration of adolescence sets in later; therefore, the whole period of growth is lengthened but the total amount of growth during the larger period is less than during the shorter period of the well-to-do” (80). A reversal in growth tendency between brain capacity and size of body, which is supposed when the mentally deficient are said to arrest earlier, would be one of the most puzzling paradoxes in the study of development. We should, therefore, be exceedingly cautious before accepting the hypothesis of the earlier maturity of deficient children.

A complicated situation is presented when we come to represent graphically the effect on the distributions of these differences in growth among those of different intellectual capacity. In the hypothetical diagrams, Fig. 5, it is shown how arrest of development might be presented graphically in relation to the distribution curves, ability being measured on the same physical scale. The earlier acceleration and earlier maturity of those of better ability are indicated. The distributions are shown as skewed at all ages after birth. Equivalent units of mental development at different ages can be found only in corresponding percentages of the groups, not in the units of the deviation or in development quotients relative to the averages at different ages. In other words the lowest 0.5% continues to be an equivalent unit while -3 S. D. measures different portions of the group and different portions of the distance from lowest to highest ability. Corresponding percentages retain one common significance, namely, that the same proportion of the group is ahead in the struggle for survival, regardless of the form of the distribution.

It is hoped that the discussion of the statistical problems connected with the quantitative study of mental development has given more meaning to the different attempts to devise scales for measuring mental ability. It should be noted that the same relative development at different ages, expressed relative to the distance from lowest to highest ability measured in equal objective units, does not correspond to the same relative development measured in percentages of the groups, as soon as the forms of the distributions change. The theoretical considerations show that we have available at once a perfectly definite and clear method of stating relative development in terms of corresponding percentages of corresponding groups. If the groups distribute normally these units are translatable into units of the standard deviation of the group. If the distributions change in symmetry the only equivalent units of deficiency available are in terms of corresponding percentages reading from either end of the group. On the other hand percentile units are not equivalent in amount of change for the same distribution, so they are of most importance for comparing different age distributions of uncertain forms.

Until we have a scale of equal objective units for mental ability, it is not possible to obtain a measure of relative development which shall take into account the amount of relative change. We must be content to measure the change in percentile rank (changes in serial position) of an individual relative to those of his own age.

Having clarified our conceptions of mental development and brought them into harmony with certain suppositions regarding the distribution of ability and its change from year to year, we are in a better position to evaluate in the following chapter the different objective methods of defining the borderline of feeble-mindedness.

On the basis of the detailed conception of the developmental curves and distributions of ability at different ages, which we have been considering, we can now compare the percentage method with other quantitative methods of describing the borderline on developmental test scales.

A. Different Forms of Quantitative Definitions

The earliest form of the quantitative description of the borderline on a scale of tests, was in terms of a fixed unit of years of retardation. This was taken over apparently from the rough method of selecting school children to be examined for segregation in special classes by choosing those who were two or three grades behind the common position for children of their ages. As this amount of school retardation was greater for older children, an additional year of retardation was required after the child had reached 9 years of age. I believe that nobody would seriously defend a practice of making an abrupt turning point of this kind, except on grounds of practical convenience. The theory of stating the borderline in terms of a fixed absolute unit of retardation is so crude that it has now been generally superseded by methods which make the amount of retardation a function of the age.

In order to relate the definition to the age of the child, at least during the period of growth, Stern suggested the “intelligence quotient,” consisting of the tested age divided by the life-age (188). This has been adopted by Kuhlmann with his revision of the Binet scale (139) and by Terman with the new Stanford scale (197). With the Point scale Yerkes utilized a similar ratio method for stating borderlines by what he calls a “coefficient of intelligence.” He defines it as “the ratio of an individual's point-scale score to the expected score, or norm” (226, p. 595). Haines also uses these coefficients, dividing the individual's score on the Point scale by the average number of points scored by those of his age (26). The difference between the “quotient” and the “coefficient” seems to be mainly empirical since they are theoretically alike in principle provided the scales by which they are determined are composed of equal units. Empirically, however, the units of the point scale would have to be compared with the 0.1 year units of the Binet scale to determine which showed the greater uniformity within its own scale. The coefficient has an advantage over the quotient in that the scale norms for the different ages would automatically become readjusted with additional data, and that physiological age norms could be more readily stated if they were ever available.

The suggestion of defining the borderline of tested deficiency in terms of a multiple of the standard deviation of ability of children who are efficient in school was made by Pearson in 1914. Tested inefficients did not with him include all inefficients, as he recognized other sources of deficiency. He had previously suggested a scale of mental ability in units called “mentaces”, 100 of which were equivalent to a unit of the standard deviation of all ability assumed to be normally distributed. On this scale of mentaces the imbeciles were 300 mentaces or more below average ability and would be expected to occur once among 1000 individuals chosen at random. Very dull, including some mentally defective individuals, were also to be found from 208 to 300 mentaces below the average (166, p. 109). Defining the borderline in terms of the deviation of a normal population was definitely forecasted by Norsworthy, although she did not specifically discuss the problem of the borderline. She indicated that if children tested below -5 P.E., they might be regarded as outside the normal group.

The following quotation from Pearson will make the method of stating the borderline in terms of a multiple of the deviation clearer:

“Now the question is, what we mean by a 'special or differentiated race': I should define it to mean that we could not obtain it by any selection from the large mass of the normal material. Now in the case of the mentally defective, we could easily obtain children of their height, weight, and temperature among the normals. We could, out of 50,000 normal children, obtain children practically with the same powers of perception and memory as the feeble-minded, as judged by Norsworthy's data. But not out of 50,000, nor out of 100,000 normal children, could we obtain children with the same defect of intelligence as some 50% of the feeble-minded children. In other words, when the deviation of a so-called feeble-minded child from the average intelligence of a normal-minded child is six times the quartile or probable deviation of the group of normal children of the same age, it falls practically outside the risk of being an extreme variation of the normal population. Now six times the quartile variation is almost exactly four times the standard deviation or the variability in intelligence of the normal child, and in the next material I am going to discuss [Jaederholm's], we have shown that the standard deviation in intelligence of the normal child is just about one year of mental growth” (164, p. 35).

With the Jaederholm data obtained in testing children in the regular and in the special classes in Stockholm by a modified form of the Binet scale, Pearson found that a year of excess or defect in intelligence was practically a uniform unit from 7 to 12 years of age and was about equivalent to the standard deviation of normal children measured in these year units. He, therefore, uses a year unit and the standard deviation as interchangeable for these data. He does not, however, always make it clear whether he means that the equivalence of the year units is determined by the standard deviation of the children of all these ages grouped together in one distribution, as it is in determining the regression lines, or by the equivalence of the standard deviations of the separate ages, especially when these two deviations are not equal in terms of the year units on the scale. I shall assume, however, that he would use the deviations of the separate years in case of such an inequality of the two concepts.

The quotation from Pearson, which we have given above, indicates that he would determine the borderline on the scale by the standard deviation of 'normal' children. In his case he actually used children who were efficient in school, as contrasted with those in special classes. On the other hand, he argues at length that all mental ability, including that of the social inefficients, is distributed in the form of the normal curve (167). Under this assumption it is, therefore, little theoretical change in his position to suppose that the borderline might be described in terms of the standard deviation of a random sample of the population. Defining the borderline in terms of a multiple of the deviation of a random sample at each age thus becomes directly comparable with the other forms of the quantitative definition, supposing that all refer to conditions to be found in a completely random sample. It is in this sense that I shall refer to the method of defining the borderline in terms of a multiple of the deviation.

The percentage method of defining the borderline seems to have been the spontaneous natural working out of the problem in the minds of several investigators. At the same time that I suggested this method in a paper before the American Psychological Association (151) Pintner and Paterson had prepared a paper suggesting a percentage definition of feeble-mindedness (44) and Terman had worked out his use of the quotient so that the borderline in terms of the quotient was given equivalent form in terms of percentage. Nobody, however, seems to have attempted to work out the details of the method as in the present monograph.

As a point of detail it is to be remembered that in translating percentages into terms of the deviation, the size of the group for which the percentages are determined is important if the groups are small, since the same percentage lies above slightly different multiples of the standard deviation with different sized groups. On this point the reader may see a paper by Cajori and the references cited there (86).

B. Common Characteristics of Quantitative Definitions

In distinction from qualitative methods of describing the mentally deficient, all quantitative definitions assume that those of deficient mentality do not represent a different species of mind; but that they are only the extreme representatives of a condition of mental ability which grades up gradually to medium ability. The deficient are not an anomalous group such as we find with some mental diseases. Except for the comparatively rare cases of traumatic or febrile origin, the deficient individual is a healthy individual so far as his nervous system is concerned, even though his capacity for brain activity is below that of those who socially survive. They are not as a group abnormal in the sense of diseased, but only unusual in the sense of being extreme variations from medium ability in a distribution which is uninterrupted in continuity. This distinction has been fully discussed by Goring in his work on The English Convict, which those who are interested in a full mathematical discussion of the significance of mental deficiency are urged to read.

Schmidt urges that the deficients are qualitatively different in being “unable to plan”, and then suggests tests which most markedly bring out this distinction between deficient and normal children (178). As I have said before, however, this seems rather to be a failure to recognize that such an attempt to find tests which “qualitatively” distinguish the two groups is only an effort to pick those tests which best make measurable the differences between individuals at the extreme of mental ability. As such it is a valuable contribution to this problem. If it is intended as an attempt to set up a qualitative distinction in a mathematical or biological sense, between deficient and passable ability, it seems to me wholly to fail. As I take it, a “qualitative” distinction with Schmidt is only a bigger quantitative distinction and is intended only to mean this.

None of those who advocate quantitative definitions would contend, I believe, as some of their opponents seem to think, that such definitions afford a final diagnosis for particular cases. In attempting to place the borderlines on a scale of tests, this is always done with the clear recognition that such borders are only symptomatic of deficiency. The diagnosis of “social inefficiency,” to use Pearson's term, rests upon many facts among which the test result is only one, albeit the most important.

Other characteristics which each of the above quantitative definitions, except that of a constant absolute amount of deficiency, have in common, or might easily have if they were stated in their best forms, include the possibility of adaptation to any developmental scale, the suggestion of borderlines for both the mature and immature, the distinction of a group which might be regarded as presumably deficient from one that was of better but doubtful ability and of this from a still better group which was presumably socially efficient.

Perhaps the most curious and important thing about these definitions is that they are all substantially identical, except in their terminology so long as general mental capacity is found to distribute in the form of the normal probability curve and to extend to absolute zero ability at each age. This can easily be seen by comparing the distribution curves in Fig. 3. The position of the percentage borderline would always represent the same distance from the average in terms of the standard deviation of each age and the same ratio when the life-age of arrest of development had been determined as the largest divisor. Under these conditions, therefore, these main statements of the quantitative definition agree in supposing that the same proportion of the individuals of each life-age would test deficient. Those who advocate any of these quantitative definitions logically commit themselves to assuming that the percentage of deficients at each age is practically constant, unless they suppose the symmetry of distribution varies or does not extend to the same zero point.

If the distributions do not extend to the same zero points of lowest ability on an objective scale (see Fig. 5), the ratio is clearly at a disadvantage compared with either of the other methods, since it assumes that the same percentage of average ability is an equivalent measure. This does not hold when the lowest ability at different ages is not at the same point on the scale of objective units. For example, .7 of an average 100 units above 0 is not equivalent to .7 of an average 150 points above a zero ability of 30 points on the objective scale. The idea of regarding percentages of averages as equivalent is therefore generally avoided in mental measurement. In case the position of the absolute zero points of ability may be different, the distance from the average should be stated in terms of the deviation. In this respect the method of the deviation or the lowest percentage are equally good so long as the form of distribution does not change.