"Are the sub-tests and the total test shown in the Table reliable
enough to support interpretations about the achievement of
individual students? And of groups of students at classroom,
school, district, and state levels? How many performance levels do
they support?"
Art Burke
| Sub-test | Reliability R |
Separation = R/(1-R) |
Group size for Sep.=3, R=0.9 |
| A B C D .... |
0.41 0.21 0.13 0.07 |
0.83 0.52 0.39 0.27 |
13 34 60 120 |
| Total | 0.88 | 2.71 | 2 |
Test "reliability", R, is a cryptic index because it amalgamates the distribution of the sample and the measurement characteristics of the test into one correlation reporting repeatability (not quality) of a local combination of the test and the sample. We are actually interested in two statistics: (i) the student standard deviation with measurement error removed; (ii) the average precisions of student measures. The ratio of the sample error-adjusted S.D. to the average measure S.E. is the "separation". It is easy to compute here because it is sqrt[R/(1-R)].
The total test has reliability 0.88, and so a student separation of 2.7. If the sample is normally distributed, there are about 3 measurably different levels of performance in this sample. But none of the sub-tests sustains even two measurably different levels.
If we are prepared to consider that all students within a group (classroom, school, district, ...) are randomly equivalent, then we can compute separations for group means. These are (student separation) * sqrt(number of students in group), i.e., for a separation of 3, group size = 3².(1-R)/R.
Of course, high separation of group-means does not stop group
distributions from overlapping considerably. Men are usually
taller than women, but not all men are taller than all women.
There is sufficient overlap that any serious study of men's and
women's heights (or any commercial clothing business) has to
consider more than just the group averages.
Benjamin D. Wright
Wright B.D. (1998) Interpreting Reliabilities. Rasch Measurement Transactions 11:4 p. 602.
Interpreting Reliabilities. Wright B.D. … Rasch Measurement Transactions, 1998, 11:4 p. 602.
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