VALIDATION OF MEASURING INSTRUMENTS 167
selection. It will then be necessary in determining the
validity of the test to repeat the test and use an average of
the two scores.
The remedy for errors due to the second factor consists in
making the necessary improvements in the test and the procedure
 used in administering it.
The reliability coefficient of a test is raised by increasing
the range of ability measured. The relationship between
range of test scores and coefficient of reliability is given in
the following formula (86, p. 222):
¢ VI=R
(1) re ED e—
2 Vi—r
In this formula ¢ is the standard deviation of the more restricted
 group and 7 the reliability coefficient obtained with
this group, = the standard deviation of the other group and
R the reliability coefficient obtained with this group.
If a test has an observed reliability coefficient of . 50 when
used with a group having a standard deviation of 1, it should
yield a reliability coefficient of .87 with a group which has
twice as large a standard deviation.
The reliability of a test may also be raised by increasing
its length. The relationship between the length of a test and
its reliability is given by Spearman’s prediction formula,
often erroneously called Brown’s formula (86, p. 205):
Nrxx,
(2) Vipx, = 1+ (—1Dran,
Here r.., is the reliability coefficient of a test x, of given
length, = the number by which the length of the test is
multiplied, and r.,., the reliability of the lengthened test xy,
The assumption is made that all of the # parts have the same
reliability. Doubling the length of a test having a reliability
of .50 increases its reliability to .67.
The effect which a change in the length of a test will have
upon the correlation of the test with the criterion is shown in
the formula at the top of the following page (86, p. 200):