5
An estimate of the reliability or reproducibility of a test
can be extracted from the variation within the tabled right marks (Table 25).
The variance from within the item columns
is related to the variance from within the student score column.
The error within items variance (2.96) and total variance
(MSS) between student scores (4.08) are both obtained from columns in Table 25b (blue, Chart 68). The true variance
is then 4.08 – 2.96 = 1.12.
The ratio of true variance to the total variance between scores
(1.12/4.08) becomes an indicator of test reliability (0.28). This makes sense.
A test with perfect
reliability (4.08/4.08 = 1.0) would have no variation, error variance = 0,
within the item columns in Table 25. A test with no reliability (0.0/4.08) would show equal values (4.08) for within
item columns, and between test scores.
The KR20 formula
then adjusts the above value (0.28 x 21/20) to 0.29 [from a large population (n) to a
small sample value (n-1)]. The KR20 ratio has no unit labels (“var/var” = “”). All
of the above takes place on the upper (variance) level of the math model.
Doubling the number of students taking the test (Chart 69) has no effect on reliability. Doubling the number of items doubles the error variance but increases the total variance by the square. The test reliability
increases from 0.29 to 0.64.
The square root
of the total variance between scores (4.08) yields the standard deviation (SD) for the score distribution [(2.02 for (n)
and 2.07 for (n-1)] on the lower floor of the math model.
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