# Z-score equation

© Paul Cooijmans

## Explanation

Z-score equation is a norming method that is theoretically correct to use only when one may assume the (raw, scaled, or weighted) scores on the object test to form a linear (intervallic) scale. In practice, this is almost never the case with raw scores on high-range tests. Raw scores tend to behave in non-linear ways. If so desired, a more linear score pattern can be obtained with techniques like normalization or item weighting.

## How z-score equation is done

Like with rank equation, the raw scores of the object test must be paired to the normed scores of the tests selected for norming, such that there will typically be more pairs than there are initial raw scores (for instance, when one candidate has three scores on usable other tests, those form three pairs with that candidate's raw score).

Then, mean and standard deviation are computed for both the raw scores and the used scores on other tests. Do notice that the number of raw scores must be the same as the number scores on other tests. After all, it concerns paired data.

Then, the means of raw scores and other scores are equated, and the standard deviations of raw and other scores are equated. This results in a formula like the following:

Norm = (raw mean - (raw mean × (other sd / raw sd))) + raw score × (other sd / raw sd)

Notice that this is a specific case of the statistical technique called "regression", but differs from it in that the correlation between the two variables is not taken into account (or rather, is assumed to be perfect).