[Illustration: FIG. 63.--The scores of pupils in grade 6 in city schools in the Woody Division Test A. The score is the number of correct answers obtained in 20 minutes. From Woody ['16, p. 61].] In spite of the effort by school officers to collect in any one school grade those somewhat equal in ability or in achievement or in a mixture of the two, the population of the same grades in the same school system shows a very wide range in any arithmetical ability. This is partly because promotion is on a more general basis than arithmetical ability so that some very able arithmeticians are deliberately held back on account of other deficiencies, and some very incompetent arithmeticians are advanced on account of other excellencies. It is partly because of general inaccuracy in classifying and promoting pupils. In a composite score made up of the sum of the scores in Woody tests,--Add. A, Subt. A, Mult. A, and Div. A, and two tests in problem-solving (ten and six graded problems, with maximum attainable credits of 30 and 18), Kruse ['18] found facts from which I compute those of Table 13, and Figs. 64 to 66, for pupils all having the training of the same city system, one which sought to grade its pupils very carefully. [Illustration: FIGS. 64, 65, and 66.--The scores of pupils in grade 6 (Fig. 64), grade 7 (Fig. 65), and grade 8 (Fig. 66) in a composite of tests in computation and problem-solving. The time was about 120 minutes. The maximum score attainable was 196.] The overlapping of grade upon grade should be noted. Of the pupils in grade 6 about 18 percent do better than the average pupil in grade 7, and about 7 percent do better than the average pupil in grade 8. Of the pupils in grade 8 about 33 percent do worse than the average pupil in grade 7 and about 12 percent do worse than the average pupil in grade 6. TABLE 13 RELATIVE FREQUENCIES OF SCORES IN AN EXTENSIVE TEAM OF ARITHMETICAL TESTS.[23] IN PERCENTS ============================================== SCORE | GRADE 6 | GRADE 7 | GRADE 8 ------------+-----------+-----------+--------- 70 to 79 | 1.3 | .9 | .4 80 " 89 | 5.5 | 2.3 | .4 90 " 99 | 10.6 | 4.3 | 2.9 100 " 109 | 19.4 | 5.2 | 4.4 110 " 119 | 19.8 | 18.5 | 5.8 120 " 129 | 23.5 | 16.2 | 16.8 130 " 139 | 12.6 | 17.5 | 16.8 140 " 149 | 4.6 | 13.9 | 22.9 150 " 159 | 1.7 | 13.6 | 17.1 160 " 169 | 1.2 | 4.8 | 9.4 170 " 179 | | 2.5 | 3.3 ============================================== [23] Compiled from data on p. 89 of Kruse ['18]. DIFFERENCES WITHIN ONE CLASS