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I could not find any set of numbers that could explain the observed score tables under the assumption that weighted raw scores were summed, but I found solutions for every score table I tried under the second method.
That means that the composite writing score should be calculated in one of two ways: either by adding weighted raw scores to produce a composite raw score, which is then translated to a scaled score, or by assigning separate scales to the multiple-choice portion and the essay portion and adding the two to produce the final composite scale.
Based on the pattern of numbers in the scale tables, I strongly suspected that the second method was the one used, but I checked them both out to be sure.
The percentages of 11s and 12s seem to vary significantly more from test to test than any other score point.
If we look at the numbers in terms of incremental payoff, i.e., how many extra scaled-score points do you get, on average, for raising your essay score by one point, the biggest jump comes between the 8 and the 9 essay, which is worth on average 23.7 points to your score, followed next by the step between 7 and 8 (20.1 points).
In other words, $S_ = S_m S_e 200$, where $S_m$ is the scaled-score contribution for a multiple-choice raw score of $m$, and $S_e$ is the scaled-score contribution for an essay score of $e$.
Values of $S_m$ and $S_e$ need not be integers, or positive, but they must be monotonically increasing.
It's almost certainly based on the percentile ranks of essay scores for that test, but is the data smoothed?
Is it, for example, transformed to a normal distribution, and if so, what are the parameters of the target distribution?
The scores are rounded before they are reported so people are less likely to place unwarranted significance on small differences in scores.) One practical consequence of rounding is that, depending on your multiple-choice raw score, a single point difference in your essay score can mean a difference to your scaled score of anywhere from 0 to 30 points.
Such an answer is unsatisfying, however, so I set out to derive a more specific answer by inferring the unrounded contribution of particular essay scores based on the score scales released with publicly available tests.