November 22, 2004

Prof targets student cheats
with statistical savvy


Final exams are just around the corner at universities nationwide, and with them the ever-present challenge of student cheating. But this year, a professor at California State University, Sacramento has prepared a new device to assist his colleagues.

Robert G. Mogull, who teaches business statistics, has shown that it takes only a modest amount of effort to thwart cheating students. He has created a simple statistical tool that any instructor can use to detect cheating on multiple-choice Scantron exams.

His technique was published in an article that appeared in the September 2004 issue of the Journal of College Teaching and Learning titled “A Device to Detect Student Cheating.” It involves computing the likelihood of more than one student missing the same test question – the easier the test question that students jointly miss, the more likely they cheated. (A brief description of the method appears below.)

“This is really a device to confirm an instructor’s suspicions. That’s what it’s best for,” Mogull says. “Of course, I’ve learned over the years that students are exceptionally clever at cheating. They can defeat any method of detection that we can come up with. If they are really determined to cheat, they are better at it than we are at stopping them.”

Mogull was inspired by a recent real-life classroom teaching experience, in which two of his students missed the same 26 questions out of a total of 100 questions over four different exams. That got the statistician’s heart pumping – but not because he was angry.

“It was clear in my mind that they had cheated. That wasn’t the question,” Mogull says. “I was mainly intrigued. When I saw their test scores and their identically missed questions, I wanted to calculate the probability of it happening at random. The ideas for the study just sort of exploded in my mind and it became one of the easiest papers that I’ve ever written.” He computed the chance of two students missing the same 26 questions to be roughly 0.000000000000000004 percent. As for the two students, he gave both of them a failing grade for the course.

Mogull, who believes cheating is much more common than most professors will acknowledge or even recognize, does not plan to employ his cheat-detecting device broadly. He says that even though it’s very easy to apply, it’s too time consuming to use all the time. He says that the best procedure to reduce student copying is to use different test versions simultaneously so that students can’t share answers with others sitting next to them.

Mogull’s method, in short: After grading the Scantron forms and conducting an Item Analysis, you have the probability that a student missed each test question. To calculate the likelihood that two students would jointly miss the same question, you square the probability of missing the particular question. Follow that procedure for each identically missed question. Then multiply all the probabilities, subtract from one, and you have the probability that the students collaborated. Stated another way, subtract the percentage chance that they didn’t cheat from 100 percent and you have the probability that they did cheat. The paper also offers an option for finding the chance that two (or more) students would miss the same question and also mark the same wrong answer.

Robert Mogull may be contacted at (916) 278-7142 or at For media assistance, contact the Sacramento State public affairs office at (916) 278-6156.


California State University, Sacramento • Public Affairs
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