Deficiency and Delinquency: An Interpretation of Mental Testing

by James Burt Miner

(c) Is Tested Capacity Distributed Normally?

Before leaving the question of the significance of units on a scale described in terms of the standard deviation we should ask whether tested mental abilities have been found to distribute normally, i. e., in the form of the symmetrical Gaussian curve with each extreme the same distance from the middle measurement. Contrary to the usual supposition in this matter, it seems as if the evidence was somewhat against this assumption, although neither position can be asserted at all dogmatically on the basis of our present data. A résumé of this evidence which I have given below makes it appear that the assumption of a normal distribution will not conflict with a practical use of normal probability tables for medium degrees of ability, but may quite seriously interfere with such use for the borderline of deficiency. There is little doubt, as Pearson believes, that the bulk of the children now in special classes for the retarded in the public schools would fall within the lower range of a normal distribution fitted to the general population. On the other hand, there is likely to be a respectable minority of the deficients which will be beyond such a normal curve. These facts are sufficiently evident, I believe, to make it impossible to base quantitative descriptions of borderline of deficiency on a hypothesis of normal distribution.

The best evidence on this point is probably the data of Norsworthy with eleven tests on groups of 100 to 150 feeble-minded children in institutions and special classes and 250 to 900 normal children. She expressed the position of each child in terms of the deviation of the group of normal children of his age for each test. Pearson has presented her data graphically on the assumption that her defective group represented 0.3% of a general population of 50,000 children, and then fitted a normal distribution curve to her data with her normal group. The result makes it evident, especially for the intelligence tests, that the defective group would better be described as part of a skewed distribution. To less extent this is also true for the maturity and memory tests (15, p. 30). Norsworthy's own table of data show that 43 of the 74 feeble-minded taking the intelligence tests were over -5 times the probable error of their ages below the averages of the normal children, a criterion which she proposes as indicating ability outside of that included in the normal species. Moreover, 9 children score between -22 P. E. and -32 P. E. which is far beyond any conceivable extension of the normal curve. Her figure for the composite results of all her mental tests is also manifestly skewed toward deficiency although she hesitates to adopt this conclusion, and was content with showing that they grade off into the distribution of normal children.

The other data, which I have found, that indicate that tested ability, when measured in equal physical units for the same task, is skewed toward deficiency, have to do with tests that are pre-eminently for psychomotor activities rather than intellectual. They consist of Sylvester's and Young's results with the form board test on Philadelphia school children, Stenquist's results with his construction test, and Smedley's results with the ergograph test on Chicago school children. Here we may apply the better criterion of the distance of the quartiles above and below the median of the group. These positions would be less likely, through extreme records, to be affected by chance conditions during the testing.

It is to be remembered that if the records of school pupils appear to be normally distributed this would not settle our problem, since it is apparent that idiots and many imbeciles are not sent to the public schools at all. The lowest children at any age would not be represented in the regular school groups. On the other hand, the brightest children are not generally drawn away from the public schools at least before 14 years of age in this country. We shall confine ourselves, therefore, to school-children 6-13 years of age. If we find that they show ability skewed toward deficiency the results will underestimate rather than over-estimate the skewness.

Sylvester ([191]) tested with the form board a group of 1537 children in the Philadelphia public schools, from 80 to 221 at each age from 5 to 14 inclusive. “Except that no especially backward or peculiar children were included there was no selection.” This study gives, with the complete distribution tables, the number of seconds required for the same task by the children at each age. If we find that the limit of the lower 25 percentile was farther from the median than the limit of the upper 25 percentile we can be reasonably sure that the difference would be still greater if the excluded deficient and backward children were also included. By calculating the quartiles and their differences from the medians at each age, I find that for only two of the eight ages is the upper quartile farther from the median than the lower quartile. The average excess of the distances of the lower quartile is .64 of a second. At only age 7 is the difference three times its probable error, 2.1 seconds, P. E. .67. The form board distributions thus tend to be slightly skewed toward deficiency. The errors of the quartiles were found by the method given in Yule's Introduction to the Theory of Statistics, Chap. XVII, which assumes normal distribution, so that they are too small. The skewness is more manifest when the extreme measurements are compared with medians at each age. It is not possible, unfortunately, to compare his group of normal children with those in the special classes since he did not use the same method of giving the test.

Since it was not important to compare the amounts of skewness in different data, I have not attempted the more elaborate calculations of coefficients of skewness. These would give the results a more elegant statistical expression. The simpler method I have here used affords more convincing evidence of asymmetry for the non-mathematical reader.

Young has published the results with Witmer's form board test on approximately two hundred Philadelphia children for each age, giving the results for the sexes separately for each half year of life-age ([227]). This affords 36 different groups in which he gives the median and upper and lower quintiles for the shortest time records. The lowest quintile is farther from the median in 25 cases, equal in 6 and less than the upper quintile in only 6 of the 36 comparisons. This skewness would have been even greater if children of the special classes had not been excluded from his groups.

Stenquist's results ([54]) with his construction test are scored in arbitrary units in which allowance is made for the quality of the score, but we should expect no constant effect on the form of the distribution from the character of these units of measurement. At ages 6 to 13 he tested from 27 to 74 pupils randomly selected from the public schools, a total of over 400. For six of these eight ages the lower quartile is farther from the median than the upper quartile, when calculated from his distribution table. The number of cases at each age, however, is so small that the largest difference, 15 units, is not three times its probable error, 6.

Smedley gave his ergograph test to about 700 school children of each of the ages we are considering. Since he tested so many more subjects than any other investigator this should provide the most valuable data on the question of distribution with a test recorded in the same physical units for the same task. Unfortunately, his results for two succeeding years are so directly contradictory to each other that they seem to have no significance for our problem. The simplest explanation of this contradiction is that the groups tested may have been selected on a different basis each year.