. .| 21 | 17 | 11 | 5 | 3 . . .| 21 | 26 | 6 | 2 4 . . .| 28 | 31 | 16 | 3 | 4 . . .| 25 | 27 | 8 | 6 5 . . .| 26 | 34 | 12 | 13 | 5 . . .| 21 | 27 | 11 | 7 6 . . .| 24 | 27 | 13 | 13 | 6 . . .| 9 | 15 | 12 | 4 7 . . .| 9 | 20 | 16 | 10 | 7 . . .| 10 | 15 | 16 | 18 8 . . .| 5 | 14 | 21 | 19 | 8 . . .| 6 | 7 | 20 | 9 9 . . .| 3 | 9 | 11 | 13 | 9 . . .| 4 | 7 | 11 | 6
10 . . .| —­ | 4 | 6 | 10 |10 . . .| 4 | 9 | 7 | 13
11 . . .| 1 | —­ | 2 | 9 |11 . . .| 1 | 3 | 3 | 7
12 . . .| —­ | —­ | 2 | 6 |12 . . .| —­ | 2 | 10 | 10
13 . . .| —­ | —­ | 1 | 3 |13 . . .| —­ | 2 | —­ | 10
14 . . .| —­ | —­ | —­ | 3 |14 . . .| 1 | —­ | 1 | 4
15 . . .| —­ | —­ | —­ | —­ |15 . . .| —­ | 1 | 2 | 9
16 . . .| —­ | —­ | —­ | 1 |16 . . .| —­ | —­ | —­ | 2
17 . . .| —­ | —­ | —­ | —­ |17 . . .| —­ | —­ | —­ | 4
18 . . .| —­ | —­ | —­ | 1 |18 . . .| —­ | —­ | —­ | 2
19 . . .| —­ | —­ | —­ | 1 |19 . . .| —­ | —­ | —­ | 1
20 . . .| —­ | —­ | —­ | —­ |20 . . .| —­ | —­ | —­ | 1
21 . . .| —­ | —­ | —­ | —­ |21 . . .| —­ | —­ | —­ | 1
22 . . .| —­ | —­ | —­ | —­ |22 . . .| —­ | —­ | —­ | —­
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Total | | | | | | | | |
Papers | 156 | 184 | 119 | 111 | | 156 | 187 | 118 | 111
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If the tests had been given in the fourth or the third grade, it would have been found that there were children, even as low as the third grade, who could do as well or better than some of the children in the eighth grade.  Such comparisons of achievements among children in various subjects ought to lead at times to reorganizations of classes, to the grouping of children for special instruction, and to the rapid promotion of the more capable pupils.

In many of these measurements it will be found helpful to describe the group by naming the point above and below which half of the cases fall.  This is called the median.  Because of the very common use of this measure in the current literature of education, it may be worth while to discuss carefully the method of its derivation.[30]

[31]The median point of any distribution of measures is that point on the scale which divides the distribution into two exactly equal parts, one half of the measures being greater than this point on the scale, and the other half being smaller.  When the scales are very crude, or when small numbers of measurements are being considered, it is not worth while to locate this median point any more accurately than by indicating on what step of the scale it falls.