# Rank equation

## Explanation

The principle norming method used is rank equation. A group of candidates provide raw scores on the test to be normed, paired to protonorms (or whatever norms, as long as they are all on the same scale) on one or (mostly) multiple other tests. Both the raw scores and the norms are ranked from highest to lowest; this implies that all norms are detached from the raw scores they were originally paired to (beginners tend to overlook this fact, hence its mention here). Each raw score gets a rank and each norm gets a rank, and each raw score receives the norm of the corresponding (that is, same) rank. For instance, the highest score has rank 1, the next has rank 2.

Do note that missing scores have rank too; the possible missing scores between the highest and the next-highest score all have rank 1.5, for example. Missing scores are normed just like existing scores. Scores with tied ranks (that is, scores occurring more than once) receive the rank number that is the median of the tied ranks. E.g., if there are four scores of value x that have the ranks 3, 4, 5, and 6, those four scores all receive rank 4.5

Norming the missing scores rigidly according to this method may result in irregular norm tables when there is relatively little data; a solution to that is to norm only the actually occurring scores, and interpolate the norms for the missing scores linearly. But whichever method of equation is used, a certain amount of data is always required for acceptable results.

An advantage of rank equation over z-score (mean and standard deviation) equation is that it produces correct norms in all cases - both with linear and non-linear scales - while z-score equation only produces correct norms when both variables (raw scores and norms) are linear and have a close to normal distribution. In reality, raw scores are almost never linear, so that rank equation must be the principle method.

A special case is the top score on a low-ceiling test, which will receive a far too generous norm with this method when the tests against which it is ranked have higher ceilings; in that case, the top score must be normed at the lowest of the ranks corresponding to it, instead of at the median thereof. In extreme cases this ceiling effect expands over the top several scores.

Weighting of score pairs is possible with rank equation; simply use each score pair from a test that deserves a particular weight more than once, depending on the weight. If the weight is 3, use the pairs from that test 3 times. Easiest to use as weights are whole numbers; fractions are possible too but awkward. 