Tuesday, March 4, 2008

Tuesday Statitsticz: Rethinking Multiple Choice

As I was taking my psychology midterm today (it was a doozie), I faced a quandary on a multiple choice test. Through process of elimination, I parsed the possible answers down to two, but I could not choose decide which was the best option between c and d. (Obviously, I chose c).

But I got to thinking about how we could improve this situation. I'm not the kid in the class that always moans about multiple choice, as I'd much rather circle some letters than write an essay, but I do think that multiple choice could be improved. From my perspective, it will be essentially luck as to whether c or d was the right answer, and luck should play as little a role as possible on tests.

My solution is for students to be allowed to assign a numerical probability to each of the possible answers. Then, they would be awarded the total worth of the question divided by the probability that they assigned to that answer. Now, before you say "this isn't fair for the students that actually know the correct answer!", let me tell you that you are wrong. This where statistics enter the picture.

The expected value of each answer will remain exactly the same. For example, if there are four options, and you have no idea what the answer is, you might answer "b" on a whim. Assuming that you receive one point for a correct answer, the expected value of this choice is:

1 point if answer "b" is correct, times a .25% chance it will be correct = .25 points

If you follow my strategy, then you'd receive the same expected value:

.25 points if answer "a" is correct times a .25% chance = .0625 points
.25 points if answer "b" is correct times a .25% chance = .0625 points
.25 points if answer "c" is correct times a .25% chance = .0625 points
.25 points if answer "d" is correct times a .25% chance = .0625 points
Total = .25 points

If this was my math homework, I'd box it and move on the next question. But it's not homework.

Why is this practice not commonplace? It certainly would have been nice today, when I couldn't decide between c and d. I would have assigned a 50 percent probability to both option c and d, and been on with it. I can't see too many cons to this approach, except that it would take a little bit more time to answer questions.

Of course, a real probability theorist might scoff at this post and claim that over time your guesses will eventually equal out to the same probabilities as you'd assign anyway. But tell that to the student that guesses 50/50 on 5 multiple choice questions in a final and gets them all wrong. Or the student who knew the answers, but for whom the curve was lowered because other students blindly guessed and got lucky. On a test to test basis, this strategy would make multiple choice more fair. Isn't that what it's supposed to be all about?