Anticipated Date of Graduation

Summer 2022

Document Type


Degree Name

Master of Science in Mathematical Sciences


Mathematical Sciences

First Advisor

Douglas Darbro


Since the early 1980s, Xavier University (in Cincinnati, Ohio) has used its own placement exam as the sole criterion for placing incoming students into their first math courses. Even though the exam has been employed for over 40 years, there has never been a rigorous analysis of its design or of how appropriately it places students. So the purpose of this study was to provide such an analysis. The quality of the exam design was gauged in three ways. First, the study determined whether, for each of the 40 multiple choice questions that comprise the exam, the correct response was the most frequently chosen. Secondly, the study used a correlation matrix to determine whether there were any strongly correlated questions, which might therefore be redundant. Thirdly, an exploratory factor analysis was conducted in order to determine what mathematical competences the exam actually measures. The quality of placements made by the exam was gauged by a novel approach. The course performances of students taking each course as a second or third math class were used to establish baselines for performance in the course. The performances of exam-placed students could then be compared to these baselines in order to determine whether the exam-placed students outperformed, underperformed or performed equally to the other students in their classes. This was studied in two ways. First, a χ2 test of independence was conducted for each course in order to determine whether there was any significant correlation between how a student arrived in the course (exam placement or working up through prior courses) and whether the student succeeded in the course (earned a grade of C or better). Then an independent samples two-tailed t-test was conducted for each course in order to determine whether there was a statistically significant difference between the mean iv grades of exam-placed students and the other students in their courses. The analysis of the exam itself found that there were five questions for which at least one response was chosen more frequently than the correct choice. Though this finding does not directly threaten the validity of the exam, Xavier’s math department may wish to discuss whether or not questions with such attractive incorrect choices are appropriate. A correlation matrix of the exam’s 40 questions showed that there were no strongly correlated (redundant) questions. The exploratory factor analysis found that the exam questions have only two underlying factors: calculus and non-calculus, with nearly all questions loading cleanly to one factor or the other. Lastly, the χ2 tests of independence and t-tests both found that exam-placed students significantly outperform the other students in their classes, suggesting that the exam tends to place students into courses below what is appropriate for them. The results of the study may be used to reassess how math placement is conducted at Xavier University. However, beyond Xavier, the novel approach that was employed to gauge placement accuracy may be of use to any other institution seeking to evaluate the effectiveness of its math placement model.

Included in

Mathematics Commons