======================================
| 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.
THE DISTRIBUTION OF THE NUMBER OF EXAMPLES CORRECTLY FINISHED
IN THE GIVEN TIME BY PUPILS IN THE SEVERAL GRADES
===================================================================
ADDITION | SUBTRACTION
No. OF |----------------------+ No. OF |------------------------
EXAMPLES| GRADES | EXAMPLES | GRADES
FINISHED| 5 | 6 | 7 | 8 | FINISHED | 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
===================================================================
THE DISTRIBUTION OF THE NUMBER OF EXAMPLES CORRECTLY FINISHED
IN THE GIVEN TIME BY PUPILS IN THE SEVERAL GRADES
=======================================================================
MULTIPLICATION | DIVISION
------------------------------------|----------------------------------
No. of | GRADES |No. of | GRADES
Examples|---------------------------|Examples|-------------------------
Finished| 5 | 6 | 7 | 8 |Finished| 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.
Much inaccurate calculation has resulted from misguided attempts to
secure a _median point_ with the formula just given, which is applicable
only to the location of the _median measure_. It will be found much more
advantageous in dealing with educational statistics to consider only the
median point, and to use only the _n_/2 formula given in a previous
paragraph, for practically all educational scales are or may be thought
of as continuous scales rather than scales composed of discrete steps.
The greatest danger to be guarded against in considering all scales as
continuous rather than discrete, is that careless thinkers may refine
their calculations far beyond the accuracy which their original
measurements would warrant. One should be very careful not to make such
unjustifiable refinements in his statement of results as are often made
by young pupils when they multiply the diameter of a circle, which has
been measured only to the nearest inch, by 3.1416 in order to find the
circumference. Even in the ordinary calculation of the average point of
a series of measures of length, the amateur is sometimes tempted, when
the number of measures in the series is not contained an even number of
times in the sum of their values, to carry the quotient out to a larger
number of decimal places than the original measures would justify. Final
results should usually not be refined far beyond the accuracy of the
original measures.
It is of utmost importance in calculating medians and other measures of
a distribution to keep constantly in mind the significance of each step
on the scale. If the scale consists of tasks to be done or problems to
be solved, then "doing 1 task correctly" means, when considered as part
of a continuous scale, anywhere from doing 1.0 up to doing 2.0 tasks. A
child receives credit for "2 problems correct" whether he has just
barely solved 2.0 problems or has just barely fallen short of solving
3.0 problems. If, however, the scale consists of a series of productions
graduated in quality from very poor to very good, with which series
other productions of the same sort are to be compared, then each sample
on the scale stands at the middle of its "step" rather than at the
beginning.
The second kind of scale described in the foregoing paragraph may be
designated as "scales for the _quality_ of products," while the other
variety may be called "scales for _magnitude_ of achievement." In the
one case, the child makes the best production he can and measures its
quality by comparing it with similar products of known quality on the
scale. Composition, handwriting, and drawing scales are good examples of
scales for quality of products. In the other case, the scales are placed
in the hands of the child at the very beginning, and the magnitude of
his achievement is measured by the difficulty or number of tasks
accomplished successfully in a given time. Spelling, arithmetic,
reading, language, geography, and history tests are examples of scales
for quantity of achievement.
Scores tend to be more accurate on the scales for magnitude of
achievement, because the judgment of the examiner is likely to be more
accurate in deciding whether a response is correct or incorrect than it
is in deciding how much quality a given product contains. This does not
furnish an excuse for failing to employ the quality-of-products scales,
however, for the qualities they measure are not measurable in terms of
the magnitude of tasks performed. The fact appears, however, that the
method of employing the quality-of-products scales is "by comparison"
(of child's production with samples reproduced on the scale), while the
method of employing the magnitude-of-achievement scales is "by
performance" (of child on tasks of known difficulty).
In this connection it may be well to take one of the scales for quality
of products and outline the steps to be followed in assigning scores,
making tabulations, and finding the medians of distributions of scores.
When the Hillegas scale is employed in measuring the quality of English
composition, it will be advisable to assign to each composition the
score of that sample on the scale to which it is nearest in merit or
quality. While some individuals may feel able to assign values
intermediate to those appearing on the Hillegas scale, the majority of
those persons who use this scale will not thereby obtain a more accurate
result, and the assignment of such intermediate values will make it
extremely difficult for any other person to make accurate use of the
results. To be exactly comparable, values should be assigned in exactly
the same manner.
The best result will probably be obtained by having each composition
rated several times, and if possible, by a number of different judges,
the paper being given each time that value on the Hillegas scale to
which it seems nearest in quality. The final mark for the paper should
be the median score or step (not the median point or the average point)
of all the scores assigned. For example, if a paper is rated five times,
once as in step number five (5.85), twice as in step number six (6.75),
and twice as in step number seven (7.72), it should be given a final
mark indicating that it is a number six (6.75) paper.
After each composition has been assigned a final mark indicating to what
sample on the Hillegas scale it is most nearly equal in quality, proceed
as follows:
Make a distribution of the final marks given to the individual papers,
showing how many papers were assigned to the zero step on the scale, how
many to step number one, how many to step number two, and so on for each
step of the scale. We may take as an example the distribution of scores
made by the pupils of the eighth grade at Butte, Montana, in May, 1914.
No. of papers 1 9 32 39 43 22 6 2
Rated at 0 1 2 3 4 5 6 7 8 9
All together there were 154 papers from the eighth grade, so that if
they were arranged in order according to their merit we might begin at
the poorest and count through 77 of them (n/2 = 154/2 = 77) to find the
median point, which would lie between the 77th and the 78th in quality.
If we begin with the 1 composition rated at 0 and count up through the 9
rated at 1 and the 32 rated at 2 in the above distribution, we shall
have counted 42. In order to count out 77 cases, then, it will be
necessary to count out 35 of the 39 cases rated at 3.
Now we know (if the instructions given above have been followed) that
the compositions rated at 3 were so rated by virtue of the fact that the
judges considered them nearer in quality to the sample valued at 3.69
than to any other sample on the scale. We should expect, then, to find
that some of those rated at 3 were only slightly nearer to the sample
valued at 3.69 than they were to the sample valued at 2.60, while others
were only slightly nearer to 3.69 than they were to 4.74. Just how the
39 compositions rated on 3 were distributed between these two extremes
we do not know, but the best single assumption to make is that they are
distributed at equal intervals on step 3. Assuming, then, that the
papers rated at 3 are distributed evenly over that step, we shall have
covered .90 (35/39 = .897 = .90) of the entire step 3 by the time we
have counted out 35 of the 39 papers falling on this step.
It now becomes necessary to examine more closely just what are the
limits of step 3. It is evident from what has been said above that 3.69
is the middle step 3 and that step 3 extends downward from 3.69 halfway
to 2.60, and upward from 3.69 halfway to 4.74. The table given below
shows the range and the length of each step in the Hillegas Scale for
English Composition.