A casual observation of his standard percentile curves for the ergograph test at the different ages gives the impression that the distributions are decidedly skewed toward deficiency, but this impression is not justified by a careful analysis of his results ([51]). In the table which accompanies his standard percentile curves, giving his total results for the two years, we find that there is a sharp disagreement between the distributions of the boys and the girls. The distributions for the boys at each age between 6 and 13 years show a greater distance, measured in kilogram-centimeters, from the median to the 80-percentile than from the median to the 20-percentile, in 5 ages out of 8. The total difference is also slightly greater between the median and the upper 80-percentile. On the other hand, the table for the girls at these ages shows the 20-percentile farther from the median in 5 out of 8 ages, with a total difference considerably greater than that shown for the boys. Usually the differences were small compared with their errors. With the boys only at age 13 was the difference in favor of the 80-percentile three times its probable error, while with the girls the four oldest ages show the distance of the 20-percentile greater by three times its probable error.
A comparison with the reports of Smedley on this test for the previous year (Report No. 2), leaves his results still more uncertain. While he does not give the medians at each age, we may make less satisfactory comparisons between the distance of the 10-percentile from the 25-percentile and the distance of the 90-percentile from the 75-percentile. If we do this, we find the distance is uniformly greater at the upper end of the distributions for each age both for the boys and girls. The Smedley results are, therefore, decidedly contradictory. The first year shows distributions skewed toward excellence and total results for two years show distributions skewed mainly toward deficiency.
Broadly considered, the Binet records with school children point to a skewed distribution toward deficiency when large allowance is made for the difference in value of the year units. It is extremely rare to find a child testing 4 years in advance of his life-age, while 15-year-old idiots are presumed to test 12-year-units or more under a mature standard.
Pearson believes that “the Gaussian curve will be found to describe effectively the distribution of mental excess and defect” for intermediate ages as measured by Jaederholm's form of the Binet scale. The data on which Pearson places reliance are Jaederholm's results in testing 261 normal children 6-14 years of age in the Stockholm schools and 301 backward children in the special help classes of the same city. The best fit of a normal curve to the data was obtained with a group of 100 8-year-old children, in which case the chances were even that samples from a normal distribution would fit. With his larger normal and backward groups combined in proper proportions in one population the chances were 20 to 1 that such a distribution as was actually found would not fit into the Gaussian distribution. He admits that “this is not a very good result,” although it is better than when the Gaussian curve is fitted to either the normal or the backward group alone. In a subsequent paper he gives each child a score relative to the standard deviation of the normal child of his own age, a method comparable to his treatment of Norsworthy's data. He then finds that “10% to 20% or those from 4 to 4.5 years and beyond of mental defect could not be matched at all from 27,000 children” (164, p. 46). In each case the distributions actually found were skewed somewhat toward deficiency. Furthermore, when he suggests that -4 S. D. may be used as a borderline for tested deficiency, he recognized that the mental ability of children is skewed so far as the empirical data are concerned. With a normal distribution there would not be two children in 100,000 who would fall below this borderline. Nevertheless, the normal curve serves for most practical purposes to describe the middle ranges of ability.
Pearson thinks that the skewed distributions of his data may possibly be explained by the drawing off of older children of better ability to the “Vorgymnasium,” or to the higher-grade schools, by the incompleteness of the higher age testing, or by the “possibility of the existence of a really anomalous group of mental defectives, who, while continuously graded inter se, and continuously graded with the normal population as far as intelligence tests indicate, are really heterogeneous in origin, and differentiated from the remainder of the mentally defective population” (164, p. 34). The last hypothesis, of course, supposes that mental ability is skewed and suggests the cause. He supplements this explanation by stating that the heterogeneous cause of the “social inefficiency” of the deficients may not be connected directly with the intellect but affect rather the conative side of the mind. A skewed distribution under biological principles of interpretation supposes a single cause or group of causes especially affecting a portion of the population.
It is also to be noted that the apparent form of distribution may be the result of the nature of the test and the units in which it is scored. Some tests might not discriminate equally well a difference in ability at the lower and at the upper ranges of ability. If the test were too easy the group might bunch at the upper portion of the scale and the distribution appear to be skewed toward the lower extreme where there were only a few cases. If too difficult a test were used the form of distribution might shift in the opposite direction, most of the group ranking low. It is extremely difficult to formulate mental tests so that they will equally well measure differences at each degree of ability. This objection should not hold, however, if the scoring were in units of time for the same task, as with the form board test. The essential characteristics of a test in order that it may indicate the form of a distribution is that the units of scoring shall be objectively equal under some reasonable interpretation and that they shall be fine enough to discriminate ability at each position on the scale. Under such conditions the variations in the difficulty of tests should not obscure the form of the distribution of the ability tested.
Turning to the analogy of measurements of physical growth, a strong argument may be made for the hypothesis of shifting forms of distribution. As Boas points out regarding measurements of the body at adolescence, owing to the rapid increase of the rate of growth the distribution of the amounts of growth is asymmetrical, “the asymmetry of annual growth makes also all series of measurements of statures, weights, etc., asymmetrical.” Moreover, “acceleration and retardation of growth affects all the parts of the body at the same time, although not all to the same extent.... Rapid physical and rapid mental growth go hand in hand” ([80]). There is no reason to suppose that the brain is free from this phenomenon of asymmetrical distribution of annual increments of growth among children of the same age when the rate of growth is changing as at adolescence. It is therefore to be expected that the separate age distributions would be skewed at early ages and at adolescence even if the distribution should be normal with a static population. The presumption from physical measurements is that the form of distribution shifts with age.
Again we may note that if some of the idiots reach an arrest of development before any of the normal individuals, as several investigators contend, this would imply that the distributions must be skewed unless there is a curious corresponding acceleration of growth on the part of geniuses to balance this lagging by idiots.
In spite of these arguments and the evidence of asymmetry of measurements at least at some periods of life it is to be noted that current opinion is probably contrary to this hypothesis, although, as I believe, because it has been concerned mainly with those who are not of extreme ability. For all large medium ranges of ability slight skewness might well be negligible. It is interesting to note that Galton says that “eminently gifted men are raised as much above mediocrity as idiots are depressed below it” (159, p. 19). Measured by intelligence quotients with the Stanford scale, Terman finds among school children that deviations below normal are not more common than those above (197, p. 555). Burt, following a suggestion of Cattell as to college men, however, seems to incline to the opinion that the general distribution of ability, like wages, is skewed toward the upper end. He adds, “In crude language, dullards outnumber geniuses, just as paupers outnumber millionaires” ([85]).