Interpreting Reliabilities

"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/(1-R)
Group size for
Sep.=3, R=0.9
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.

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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