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.
Then
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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