# Thinking with Raw Scores

Why not use raw scores? Are they not immediate and real? Are they not the only indisputable "facts"?

But what have we counted? Real objects which, in this "reality", are manifestly not exchangeable, not equal in their contribution to the count. If we count four apples and each take two, it matters whether you get the two large ones or the two small ones. When we count, we assume exchangeability, i.e., equivalence. So even counting raw scores is not good enough. But it is worse because we expect "one more" always to be the same amount, no matter where we begin or end.

See what happens with raw scores, as shown in the "Equating" Figure overleaf. We expect someone who scores 100% on a hard test to score 100% on an easy test. We also expect someone who scores 0% on an easy test to score 0% on a hard test. What about someone who scores 50% on an easy test? That person must score less, say 20%, on a hard test, or else that test is not harder! Similarly someone who scores 50% on a hard test, must score more, say 80%, on an easy test. So the relationship between raw scores on an easy test and raw scores on a hard test is always the curved, "NOT-STRAIGHT equating line"!

This curvilinear relationship means that it is impossible for one more raw score point, or 1% more correct, to mean the same amount throughout the score range. What we want, of course, is the "STRAIGHT equating line" shown in the "Ability Measure" plot!

One more on a test is not always the same amount. What to do? The scores are the data, but not the desire. How can we make "one more apple" become "one more unit" that is always the same?

The solution is the Rasch model. This model straightens out the "NOT-STRAIGHT" line into the "STRAIGHT" line by means of a logistic ogive illustrated in the plot of "TEST SCORE" on "ABILITY MEASURE". Notice how a 10% raw score interval near 100% success covers 5 times the ability range of a 10% raw score interval around 50% success. One more score point near an extreme is worth much more than one more score point near the middle. This is why the raw score equating line is curved! Raw scores may be "facts", but facts are not enough, we want what they mean! We need the Rasch model to extract meaning from facts.

Benjamin D. Wright

# Equating a Hard and Easy Test

Test Score in Percent Correct
Ability Measure in Logits
{ Test curve is B = 1.5 loge [P/(100-P)] }

Thinking with raw scores. Wright BD. … 1993, 7:2 p.299-300

Thinking with raw scores. Wright BD. … Rasch Measurement Transactions, 1993, 1993, 7:2 p.299-300

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

 Forum Rasch 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
March 31, 2017, Fri. Conference: 11th UK Rasch Day, Warwick, UK, www.rasch.org.uk
April 2-3, 2017, Sun.-Mon. Conference: Validity Evidence for Measurement in Mathematics Education (V-M2Ed), San Antonio, TX, Information
April 26-30, 2017, Wed.-Sun. NCME, San Antonio, TX, www.ncme.org - April 29: Ben Wright book
April 27 - May 1, 2017, Thur.-Mon. AERA, San Antonio, TX, www.aera.net
May 26 - June 23, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 30 - July 29, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
July 31 - Aug. 3, 2017, Mon.-Thurs. Joint IMEKO TC1-TC7-TC13 Symposium 2017: Measurement Science challenges in Natural and Social Sciences, Rio de Janeiro, Brazil, imeko-tc7-rio.org.br
Aug. 7-9, 2017, Mon-Wed. In-person workshop and research coloquium: Effect size of family and school indexes in writing competence using TERCE data (C. Pardo, A. Atorressi, Winsteps), Bariloche Argentina. Carlos Pardo, Universidad Catòlica de Colombia
Aug. 7-9, 2017, Mon-Wed. PROMS 2017: Pacific Rim Objective Measurement Symposium, Sabah, Borneo, Malaysia, proms.promsociety.org/2017/
Aug. 10, 2017, Thurs. In-person Winsteps Training Workshop (M. Linacre, Winsteps), Sydney, Australia. www.winsteps.com/sydneyws.htm
Aug. 11 - Sept. 8, 2017, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Aug. 18-21, 2017, Fri.-Mon. IACAT 2017: International Association for Computerized Adaptive Testing, Niigata, Japan, iacat.org
Sept. 15-16, 2017, Fri.-Sat. IOMC 2017: International Outcome Measurement Conference, Chicago, jampress.org/iomc2017.htm
Oct. 13 - Nov. 10, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 5 - Feb. 2, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
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
May 25 - June 22, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 10 - Sept. 7, 2018, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com