Rasch Quick Norms

Norm-referenced interpretation compares an examinee's performance on a test with the distribution of performances of a reference population. Consequently, estimates of the mean and standard deviation of the reference population are the key ingredients to norming. Rasch theory provides an easy method to develop "quick norms" using only two pieces of information: a set of items with pre-calibrated difficulties, and the number of examinees in the reference group (representing the reference population) who succeeded on each item.

The quick norm procedure employs Cohen's PROX technique (Wright & Stone 1979 p. 126-128):
(1) Select a set of items whose pre-calibrated item difficulties span the expected range of abilities.
(2) Administer the set of items to a reference group.
(3) Count up the correct responses to each item.
(4) Calculate the log-odds = loge(number correct / number incorrect) for each item. (5) Plot the log-odds against the item difficulty for each item.
Verify they fall on a "statistical" line. Amend the items or reference sample, if necessary.
(6) Regress the log-odds on the item difficulties.
A = Intercept at 0 item difficulty
C = Slope of line.
(7) Reference population mean = A/C
(8) Reference population S.D. = 1.7*sqrt[(1-C^2)/C^2]
(9) Compute the measure for each possible raw score on the item set, or measure each new examinee using some or all of the item set.
(10) Compute norm-referenced indicators (percentiles, standard scores) using the computed measures and the reference population mean and S.D.

The Table summarizes the responses of 50 examinees to 15 pre- calibrated items. A plot of the log-odds correct answers against the item difficulties is shown in the Figure. The intercept of the least-squares regression line corresponds is "A" and the slope is "C".

               Rasch Quick Norms (N = 50 examinees)
Items    Examinees    Log-odds correct:  Pre-calibrated item
       succeeding: Si  loge(Si /(N-Si))    difficulty: Di
 1          42               1.66             -2.19
 2          42               1.66             -2.19
 3          37               1.05             -1.34
 4          34                .75              -.94
 5          33                .66              -.82
 6          29                .32              -.36
 7          29                .32              -.36
 8          24               -.08               .17
 9          24               -.08               .17
10          23               -.16               .27
11          20               -.41               .59
12          14               -.94              1.27
13          12              -1.15              1.53
14          11              -1.27              1.67
15           6              -1.99              2.54
                      Mean = 0.02       Mean = 0.00
                      S.D. = 1.03       S.D. = 1.34
Regression Intercept = A = 0.02
Regression Slope = C = 0.77
Reference (Norm) Mean = A/C = 0.02/0.77 = 0.03
Reference (Norm) S.D. = 1.7*sqrt[(1-C^2)/C^2] = 1.43

An examinee who scores 13 on this 15 item test has a measure of about 1.4 logits. The z-score is (1.4 - 0.03)/1.43 = 1.0. The T-score is 60, NCE is 70, percentile is 85%, and stanine is 7.

Quick norms efficiently replace cumbersome traditional test norming methods. Bush & Schumacker (1993) generated 100 replications of simulated data for all combinations of item difficulty distribution (normal and uniform), test length (10, 20, 30, 40, 50 items), and sample size (50, 100, 200, 300, 400 examinees). They discovered no statistically significant differences between the quick norm and conventional norming methods except for the very short 10-item tests.

Bush M.J., Schumacker, R.E. (1993). Quick norms with Rasch measurement. Paper presented at AERA Annual Meeting, Atlanta.

Schumacker RE, Bush MJ. (1993) Quick norms. Rasch Measurement Transactions, 7:1 p.270-1.

Wright B.D. Stone, M.H. (1979). Best Test Design. Chicago: MESA Press.

Quick norms. Schumacker RE, Bush MJ. … Rasch Measurement Transactions, 1993, 7:1 p.270

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 www.rasch.org
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 Rasch.org

www.rasch.org 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, www.rasch.org.

Coming Rasch-related Events
Aug. 11 - Sept. 8, 2023, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Aug. 29 - 30, 2023, Tue.-Wed. Pacific Rim Objective Measurement Society (PROMS), World Sports University, Macau, SAR, China https://thewsu.org/en/proms-2023
Oct. 6 - Nov. 3, 2023, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Facets), www.statistics.com
June 12 - 14, 2024, Wed.-Fri. 1st Scandinavian Applied Measurement Conference, Kristianstad University, Kristianstad, Sweden http://www.hkr.se/samc2024


The URL of this page is www.rasch.org/rmt/rmt71d.htm

Website: www.rasch.org/rmt/contents.htm