A similar method of calculation is employed in discovering the limits within which the middle fifty per cent of the cases fall.  It often seems fairer to ask, after the upper twenty-five per cent of the children who would probably do successful work even without very adequate teaching have been eliminated, and the lower twenty-five per cent who are possibly so lacking in capacity that teaching may not be thought to affect them very largely have been left out of consideration, what is the achievement of the middle fifty per cent.  To measure this achievement it is necessary to have the whole distribution and to count off twenty-five per cent, counting in from the upper end, and then twenty-five per cent, counting in from the lower end of the distribution.  The points found can then be used in a statement in which the limits within which the middle fifty per cent of the cases fall.  Using the same figures that are given above for scores in English composition, the lower limit is 2.64 and the limit which marks the point above which the upper twenty-five per cent of the cases are to be found is 5.08.  The limits, therefore, within which the middle fifty per cent of the cases fall are from 2.64 to 5.08.

It is desirable to measure the relationship existing between the achievements (or other traits) of groups.  In order to express such relationship in a single figure the coefficient or correlation is used.  This measure appears frequently in the literature of education and will be briefly explained.  The formula for finding the coefficient of correlation can be understood from examples of its application.

Let us suppose a group of seven individuals whose scores in terms of problems solved correctly and of words spelled correctly are as follows:[33]

======================================
INDIVIDUALS|No.  OF |No.  OF WORDS
MEASURED |PROBLEMS|SPELLED CORRECTLY
CORRECTLY | |
-----------+--------+-----------------
  A | 1 | 2
  B | 2 | 4
  C | 3 | 6
  D | 4 | 8
  E | 5 | 10
  F | 6 | 12
  G | 7 | 14
======================================

From such distributions it would appear that as individuals increase in achievement in one field they increase correspondingly in the other.  If one is below or above the average in achievement in one field, he is below or above and in the same degree in the other field.  This sort of positive relationship (going together) is expressed by a coefficient of +1.  The formula is expressed as follows: 

(Sum x . y)
r = ------------------------------
(sqrt(Sum x^2))(sqrt(Sum y^2))

Here r = coefficient of correlation.

x = deviations from average score in arithmetic (or difference between score made and average score).

y = deviations from average score in spelling.