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Monday, June 28, 2004

How to statistically detect cheating?

I blog while I am giving an exam. There is a reported incidence of cheating on the midterm exam, comprised of 75 multiple guess questions. I am amazed at the charge, not so much that I don't believe it can happen, but because it seems a very risky move in a class on sports economics!! Please, sports economics is too easy to cheat at. If the class had to do with nuclear physics, perhaps I can understand taking a peek at the neighbor's exam.

Any way, how to detect cheating? I have been mulling it over and here is an initial take.

Odds of missing a question upon guessing either 3 in 4 or 4 in 5. If each person involved in the cheating guesses on the same questions, then the odds that they would choose the same answer on the same question is .25x.25 = 1/16 or .20x.20 = 1/25. If they guess on two of the same questions independently, the odds of choosing the same answer on both questions is then 1/16x1/16 or 1/25x1/25 or 1/16x1/25.

We can continue the odds of independent draws to include all 75 questions.

However, this is probably not appropriate if the two colluded before the exam by studying together, which might lead them to know the correct answer simultaneously or to simultaneously think they know the right answer and choose the same wrong answer.

How to detect cheating in this case? Perhaps looking at the correlation between the wrong answers? Perhaps looking at the probability that when person 1 chooses option A, person 2 also chooses option A, perhaps in a multinomial logit framework? Does this approach run into the problem of Independence of Irrelevant Alternatives?

I will have to mull this over, but there should be a way to statistically reject or fail to reject the null hypothesis of no cheating. I haven't thought about this before because I have never had cheating reported to me.

Comments appreciated.

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