Making Raw Score Summaries Usefully Linear

Research is often reported as means and standard deviations of raw scores, percents or ratings. These scores compress towards the ends of their scale due to their finite range. Since log-odds extend to infinity in both directions, a simple log-odds transformation can remedy the compression problem.

Here is the transformation, when raw score means are available:

"Mean measure" is the linearized mean measure, relative to the center of the raw score scale. "Mean" is the reported mean raw score. "Bottom" is the lowest raw score possible. "Top" is the highest raw score possible.

Hagemann (1994) reports that teachers rate children's reading with an average raw score of 2.85. The raw score range was 1, "inadequate" at the bottom, to 4, "above average" at the top.


When a mean score, "m", and a standard deviation, "s", are reported, a better linearization is:

"b" is the lowest raw score possible. "t" is the highest raw score possible.

The linearized standard deviation around this mean measure is

In Hagemann, the raw score standard deviation is .80, so a better mean measure is:

with measure standard deviation

Stuart Luppescu 1994 RMT 8:1 p. 337

Hagemann M (1994) Interpreting teachers ratings requires caution. Paper presented at MOMS, May 1994.

Making raw score summaries usefully linear Luppescu S.. … Rasch Measurement Transactions, 1994, 8:1 p.337

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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
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