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.

Please help with Standard Dataset 4: Andrich Rating Scale Model

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

To be emailed about new material on
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from welcomes your comments:

Your email address (if you want us to reply):


ForumRasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website,

Coming Rasch-related Events
Sept. 15-16, 2017, Fri.-Sat. IOMC 2017: International Outcome Measurement Conference, Chicago,
Oct. 13 - Nov. 10, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps),
Oct. 25-27, 2017, Wed.-Fri. In-person workshop: Applying the Rasch Model hands-on introductory workshop, Melbourne, Australia (T. Bond, B&FSteps), Announcement
Jan. 5 - Feb. 2, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps),
Jan. 10-16, 2018, Wed.-Tues. In-person workshop: Advanced Course in Rasch Measurement Theory and the application of RUMM2030, Perth, Australia (D. Andrich), Announcement
Jan. 17-19, 2018, Wed.-Fri. Rasch Conference: Seventh International Conference on Probabilistic Models for Measurement, Matilda Bay Club, Perth, Australia, Website
April 13-17, 2018, Fri.-Tues. AERA, New York, NY,
May 25 - June 22, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps),
June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps),
Aug. 10 - Sept. 7, 2018, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets),
Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps),
The HTML to add "Coming Rasch-related Events" to your webpage is:
<script type="text/javascript" src=""></script>


The URL of this page is