| Number of Children | ||
|---|---|---|
| November | February | |
| Rated at 0 | 0 | 0 |
| 1.83 | 1 | 1 |
| 2.60 | 6 | 4 |
| 3.69 | 12 | 6 |
| 4.74 | 8 | 11 |
| 5.85 | 3 | 4 |
| 6.75 | 1 | 3 |
| 7.72 | 1 | 2 |
| 8.38 | 0 | 1 |
| 9.37 | 0 | 0 |
A study of such a distribution would show not only that the average performance of the class has been raised, but also that those in the lower levels have, in considerable measure, been brought up; that is, that the teacher has been working with those who showed less ability, and not simply pushing ahead a few who had more than ordinary capacity. It would be possible to increase the average performance by working wholly with the upper half of the class while neglecting those who showed less ability. From a complete distribution, as has been given above, it has become evident that this has not been the method of the teacher. He has sought apparently to do everything that he could to improve the quality of work upon the part of all of the children in the class.
It is very interesting to note, when such complete distributions are given, how the achievement of children in various classes overlaps. For example, the distribution of the number of examples on the Courtis tests, correctly finished in a given time by pupils in the seventh grades, makes it clear that there are children in the fifth grade who do better than many in the eighth.
| Addition | Subtraction | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| No. of Examples Finished | Grades | No. of Examples Finished | Grades | ||||||
| 5 | 6 | 7 | 8 | 5 | 6 | 7 | 8 | ||
| 0 | 12 | 15 | 5 | 4 | 0 | 6 | 2 | 2 | -- |
| 1 | 26 | 23 | 14 | 9 | 1 | 5 | 6 | 2 | 1 |
| 2 | 27 | 31 | 8 | 6 | 2 | 7 | 8 | 1 | -- |
| 3 | 31 | 27 | 27 | 9 | 3 | 13 | 21 | 3 | 1 |
| 4 | 25 | 28 | 19 | 16 | 4 | 21 | 18 | 13 | 2 |
| 5 | 16 | 23 | 16 | 15 | 5 | 26 | 30 | 12 | 7 |
| 6 | 15 | 22 | 12 | 12 | 6 | 17 | 27 | 15 | 9 |
| 7 | 1 | 11 | 8 | 9 | 7 | 15 | 27 | 18 | 9 |
| 8 | 3 | 4 | 6 | 11 | 8 | 15 | 20 | 12 | 12 |
| 9 | 1 | 2 | 3 | 8 | 9 | 10 | 13 | 9 | 12 |
| 10 | -- | -- | -- | 6 | 10 | 8 | 6 | 13 | 11 |
| 11 | -- | -- | 1 | -- | 11 | 6 | 2 | 3 | 12 |
| 12 | -- | -- | 1 | 2 | 12 | 3 | 1 | 7 | 9 |
| 13 | -- | -- | -- | -- | 13 | 2 | 2 | 3 | 5 |
| 14 | -- | -- | -- | -- | 14 | 1 | 1 | 3 | 7 |
| 15 | -- | -- | -- | 2 | 15 | -- | -- | 2 | 3 |
| 16 | -- | -- | -- | 1 | 16 | -- | -- | 1 | 2 |
| 17 | -- | -- | -- | -- | 17 | -- | 1 | -- | 1 |
| 18 | -- | -- | -- | -- | 18 | -- | -- | -- | 1 |
| 19 | -- | -- | -- | -- | 19 | -- | -- | -- | 4 |
| 20 | -- | -- | -- | -- | 20 | -- | -- | -- | 2 |
| 21 | -- | -- | -- | -- | 21 | -- | -- | -- | 1 |
| 22 | -- | -- | -- | -- | 22 | -- | -- | -- | -- |
| Total papers | 157 | 86 | 119 | 111 | 155 | 185 | 119 | 111 | |
| Multiplication | Division | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| No. of Examples Finished | Grades | No. of Examples Finished | Grades | ||||||
| 5 | 6 | 7 | 8 | 5 | 6 | 7 | 8 | ||
| 0 . . . | 10 | 4 | -- | -- | 0 . . . | 17 | 7 | 1 | -- |
| 1 . . . | 10 | 4 | 3 | -- | 1 . . . | 19 | 17 | 2 | 1 |
| 2 . . . | 19 | 20 | 5 | 1 | 2 . . . | 18 | 22 | 8 | 4 |
| 3 . . . | 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 . . . | -- | -- | -- | -- |
| Total Papers | 156 | 184 | 119 | 111 | 156 | 187 | 118 | 111 | |
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. If the measuring instrument has been carefully derived and accurately scaled, however, it is often desirable, especially where the group being considered is reasonably large, to locate the exact point within the step on which the median falls. If the unit of the scale is some measure of the variability of a defined group, as it is in the majority of our present educational scales, this median point may well be calculated to the nearest tenth of a unit, or, if there are two hundred or more individual measurements in the distribution, it may be found interesting to calculate the median point to the nearest hundredth of a scale unit. Very seldom will anything be gained by carrying the calculation beyond the second decimal place.
The best rule for locating the median point of a distribution is to take as the median that point on the scale which is reached by counting out one half of the measures, the measures being taken in the order of their magnitude. If we let n stand for the number of measures in the distribution, we may express the rule as follows: Count into the distribution, from either end of the scale, a distance covered by *n/2 measures. For example, if the distribution contains 20 measures, the median is that point on the scale which marks the end of the 10th and the beginning of the 11th measure. If there are 39 measures in the distribution, the median point is reached by counting out 19-1/2 of the measures; in other words, the median of such a distribution is at the mid-point of that fraction of the scale assigned to the 20th measure.
The median step of a distribution is the step which contains within it the median point. Similarly, the median measure in any distribution is the measure which contains the median point. In a distribution containing 25 measures, the 13th measure is the median measure, because 12 measures are greater and 12 are less than the 13th, while the 13th measure is itself divided into halves by the median point. Where a distribution contains an even number of measures, there is in reality no median measure but only a median point between the two halves of the distribution. Where a distribution contains an uneven number of measures, the median measure is the (n+1)/2 measurement, at the mid-point of which measure is the median point of the distribution.