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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. 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