# Ivars Peterson's MathTrek

June 12, 2006

## Dropping Lowest Scores

As a school year draws to a close, grades are inevitably on the minds of both students and teachers.

When computing final grades, teachers sometimes allow students to drop the lowest score—or even several of their lowest scores—from a sequence of quizzes, tests, homework assignments, or exams. Usually, it's the teacher who makes the decision on which scores to drop in any individual case. If the goal is to maximize a student's score, that's sometimes easy to do. But, in certain situations, it can be quite tricky to tell which score or scores to drop from the total.

In the June Mathematics Magazine, Daniel M. Kane of the Massachusetts Institute of Technology and Jonathan M. Kane of the University of Wisconsin-Whitewater describe some of the intricacies of this problem and present a simple, efficient algorithm for finding the best set of grades to drop. Jonathan is a math professor, and his son Daniel is a student at MIT. Daniel was a member of the U.S.A. International Mathematical Olympiad team in 2003 and 2004.

Suppose that Alan earned the quiz scores shown below and can drop two scores.

 Quiz 1 2 3 4 5 Score 2 6 24 3 6 Possible Score 8 12 40 4 24 Percentage 25 % 50 % 60 % 75 % 25 %

If his teacher computes Alan's final grade by simply adding up all the quiz scores, Alan does best if the teacher drops the scores for quizzes 1 and 4—even though quiz 4 has the highest percentage.

However, if a teacher bases a student's final grade on the ratio of total points earned to total points possible, the problem is a little more complicated.

"Intuitively, one might suspect that a way to obtain an optimal solution would be to drop those quiz grades where the student performed the worst either by obtaining the smallest number of points or by obtaining the smallest percentage grade," Daniel and Jonathan King note. "However, this is not always the case."

Consider Beth's quiz scores.

 Quiz 1 2 3 Score 80 20 1 Possible Score 100 100 20 Percentage 80 % 20 % 5 %

If Beth drops quiz 1, her final grade is 17.5 percent. If she drops quiz 2, it's 67.5 percent. If she drops quiz 3, it's 50 percent.

Note that dropping the quiz 2 score leaves quiz 3, which has both the lowest raw score and the lowest percentage. This result reflects the fact that quiz 3 is worth so few points that its impact is small compared with that of quiz 2.

Carl's quiz scores demonstrate another quirk.

 Quiz 1 2 3 4 Score 100 42 14 3 Possible Score 100 91 55 38 Percentage 100 % 46 % 25 % 8 %

If Carl is allowed to drop just one score, the best one to omit is quiz 4. That would give him a grade of 63.4 percent.

However, if Carl were allowed to drop two scores, it's best to omit quiz 2 and 3, and he would end up with a grade of 74.6 percent.

"Notice that the optimal deletion set of two grades does not include the best single grade to drop," Daniel and Jonathan Kane conclude. "Also surprising is how slight changes to a problem can result in radically different results."

One way to solve the problem of determining which grades to drop is by trying all the possibilities. But for large datasets, that's a time-consuming task. To drop 10 out of 100 grades, for example, you would have to consider 17,310,309,456,440 possibilities! At a rate of 1 million possibilities per second, it would take you about 200 days to go through them all.

To see the ingenious, efficient algorithm that Daniel and Jonathan Kane developed for solving the problem, check their paper (http://web.mit.edu/dankane/www/droplowest.pdf). The method appears to be particularly efficient when used in a computer gradebook program.

Of course, another answer for teachers is to give all quizzes the same possible value.