### Odds ratio method and grading on a curve

The odds ratio method, applied to a binary outcome (e.g, http://www.insidethebook.com/ee/index.php/site/comments/the_odds_ratio_method/), says

$x = \mathrm(odds) = \frac{p}{1-p} \frac{q}{1-q} \frac{1-a}{a}$

that is, the odds are the odds(p) times the odds(q), divided by the odds(a). p here represents, say a batters OBP, q a pitchers OBP-against, and a the league average OBP. Applied to grades, one can think of the material as the pitcher and the students grades as the batting performance. In other words, I know how my students did (p) in a course with a certain true talent (a), and can solve for the expected performance in a “league” with a different mean (q). Of course then you get back to a percentage by inverting the computed odds, p’ = x/(1+x). Two possibilities for choosing the value of q are requiring that the highest grade achieves some value, or that the overall mean after adjusting achieves some value. In either case you end up multiplying the odds for each students observed performance by a constant, c, which is

$c = \frac{q}{1-q} \frac{1-a}{a}$.

Notice that a and q don’t matter by themselves, only the ratio of the odds for q to the odds for a. You can then invert that to get back to the adjusted percentages, which end up being,

$p' = \frac{c p}{1 - p(1-c)}$.

When c approaches infinity, i.e. the material in the hypothetical course is really really easy, p’ approaches 1 for all values of p. When c approaches 0, i.e. the material in the hypothetical course is really really hard, p’ approaches 0. When c is 1, of course you get back p.