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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