Z-Transformation of Raw Test Scores

When the instructor decides that a test is unusually difficult, or if there is some other classwide reason for adjusting the raw scores, a z-transformation is a fair way to do it. It is a more realistic way of “curving” the grades than just adding points to everyone’s score, since it adjusts the spread of the scores in addition to adjusting the class average. It is fair to the students because it keeps their scores in the same relative position to each other, so that no student jumps over another who had a higher raw score.

A z-transformation adjusts the raw scores of a test to fit a predetermined range of scores having a specific mean (arithmetic average) and standard deviation (a dispersion measure of the scores - how close together or spread apart they are). For example, suppose the class average on a given test was a 70, with a standard deviation of 20; but the usual test scores for this class have an average of 85 with a standard deviation of 10. The instructor realizes that there was not the usual amount of time for review (perhaps the class lost a day due to an assembly, etc.), and that this fact contributed to the unusually low scores on this particular test. Therefore, the instructor decides to perform a z-transformation on the scores to make them more like the usual test performance of the class (or perhaps their peers in previous years, if it is early in the school year).

Each raw test score is transformed by the following four steps:

1) Subtract the raw test mean (this may result in a negative amount).

2) Divide by the raw test standard deviation (this produces a z-score).

3) Multiply the z-score by the usual test standard deviation.

4) Add the usual test mean.

It is usually a good idea to limit the adjusted scores to a specific range, such as from 50 to 99. The minimum score of 50 protects against a student having such a low test score that it makes it nearly impossible to raise his/her average. It also takes into account that a student may have had an abnormally bad day taking the test. The maximum score of 99 accounts for the idea that a curve of any kind should not bring a student’s grade above the “perfect” score of 100. Of course, if the 99 maximum is used, the instructor must allow students who score 100 (or better, due to extra credit) to keep their superior raw score instead of the z-transformation score. In no case should a transformed score ever lower a student’s raw score – if that rare event occurs, the student should be allowed to keep the higher score.

The assumption behind this procedure is that a relatively large number of students perform approximately the same across test situations, and that the difficulty level of a given test is more likely to fluctuate than the average student effort across a class. If the instructor has evidence that the students have dropped their usual level of effort (a low homework class average, for example), it may justify keeping relatively low raw test scores rather than applying a z-transformation.