Curving Exam Scores
Linear curving function
We use a curving function that satisfies the following three properties:
- The curving function is linear.
- A perfect score of 100 will map to 100.
- The current median score M0 will increase to the new median value of M1.
These three requirements uniquely determine the curving function. It maps the old score S0 to the new score S1 by:
S1 = 100 - (100 - S0) × (100 - M1) / (100 - M0)As an example, if the old median was M0 = 60 and the new median is M1 = 80, then the function is:
The advantages of the above curving function are:
- Assuming that the new median is higher than the old median, it is guaranteed that every student's score increases, but cannot go over 100, and the ordering between scores is strictly maintained.
- Using the median as the specification point makes this function insensitive to outliers. We generally want M1 ≈ 80, putting half the class in the A/B range (more when high-scoring items like homeworks are included).
The disadvantages of the above curving function are:
- It compresses the range of student scores, which can be a problem if the old median M0 is very low. In such cases, it is better to also add a constant offset to all scores, and to then cap any scores that go over 100.
- Very poorly performing students can receive an enormous score increase (e.g., 0 mapping to 50 in the example above). If there are very-low-score students then it may be better to use a different curving function for low-scoring students.
Piecewise-linear curving function
One choice for a curving function that deals with low-scoring students is to map a score of 0 to the new score Z1 (typically in the range 20 to 50), using a piecewise linear function of the form:
S1 = 100 - (100 - S0) × (100 - M1) / (100 - M0), if S0 > M0
S1 = Z1 + S0 × (M1 - Z1) / M0, if S0 ≤ M0
With the median values from the previous example and Z1 = 20, this gives the function: