z-scores

While students occasionally approach me with concerns that z-scores are somehow treating them unfairly, in fact z-scores are one of the fairest approaches to grading.

The idea is that if you get, say, a 75 on an exam, just a good a score that really is depends on how the rest of the class did. If the class average was 78, clearly 75 is not a great score. Hence, the mean matters. Likewise, suppose the mean was 60. While 75 is a better-than-average score, just how much better depends on the spread of class scores (formally, the standard deviation, or SD). The larger the SD, the more spread out the scores. The idea behind z-scores is to express a score in terms of how many SD above (or below) the mean a score is. Hence, if the SD is 10, then with a mean of 60, a score of 75 is (75-60)/10 = 1.5 SD above the mean. Likewise, a 55 is (55-60)/10 = -0.5 SB below the mean.

Formally, the z-score is computed as the

Assuming the test scores are normally distributed, z-scores translate into the precentages. For example, a z-score of 1.5 (a score of 1.5 SD above the mean) corresponds to a score in the upper 7 percent (i.e. 93 percent of all scores are expected to be lower).

z-score translate into grades as follows: