# 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
*M*_{0}will increase to the new median value of*M*_{1}.

These three requirements uniquely determine the curving
function. It maps the old score *S*_{0} to the new score
*S*_{1} by:

*S*_{1} = 100 - (100 - *S*_{0}) × (100 - *M*_{1}) / (100 - *M*_{0})

*M*

_{0}= 60 and the new median is

*M*

_{1}= 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
*M*_{1}≈ 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
*M*_{0}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 *Z*_{1}
(typically in the range 20 to 50), using a piecewise linear function
of the form:

*S*_{1} = 100 - (100 - *S*_{0}) × (100 - *M*_{1}) / (100 - *M*_{0}), if *S*_{0} > *M*_{0}

*S*_{1} = *Z*_{1} + *S*_{0} × (*M*_{1} - *Z*_{1}) / *M*_{0}, if *S*_{0} ≤ *M*_{0}

With the median values from the previous example and
*Z*_{1} = 20, this gives the function: