# Let's Do the Math, Shall We?

Everybody "knows" that since standardized testing companies calculate your score on multiple-choice items by subtracting a fraction of the number of wrong answers from the number of correct answers, you shouldn't guess. Well, by now you should be rather skeptical about what everybody "knows," and particularly skeptical about what large corporations tell you about their products. Think about it - these companies collect a LOT of money from people who are told that the tests they are spending big bucks for are absolutely perfect in every way. Of course these companies will say that students will be penalized for guessing - but what's the truth? Since we're mathematicians, let's "do the math."

## The Hypothetical Test:

Suppose we are giving a 60-question multiple-choice test, and each question has 5 possible answers. The score is calculated for this test as follows:

Score = (# correct) - (1/4)(# wrong)

## Abe's Score:

Abe writes his name on his test, but does not attempt to answer any questions. Therefore, he has 0 correct answers and 0 wrong answers.

Abe's Score = (# correct) - (1/4)(# wrong) = 0 - (1/4)(0) = 0

No big surprise here (I hope), except that people often suspect that Abe would get a negative score.

## Betty's Score:

Betty writes her name on her test, and randomly marks each answer without looking at the test. Since there is a 20% probability that Betty will get a particular question correct (since there are 5 possible answers), on average Betty will have 60/5 = 12 correct answers and 60 - 12 = 48 incorrect answers.

Betty's Score = (# correct) - (1/4)(# wrong) = 12 - (1/4)(48) = 12 - 12 = 0

This is the "guessing penalty" in action. Betty didn't know anything, and her test score is 0 - there's nothing wrong with that. But Betty wasn't penalized for guessing! The so-called "guessing penalty" is not a penalty - it simply removes the advantage of random guessing. Without the score correction, Betty would have a test score of 12 - but she didn't earn it - she deserves a zero, and she got a zero.

## Charlie's Score:

Charlie believes he knows the answers to 50 questions (out of 60) on the test. He does not mark the 10 answers he is not sure of - after all, he will be "penalized" for wrong answers, but blanks don't "count against" him! Of the 50 questions he answers, he gets 38 correct and 12 wrong.

Charlie's Score = (# correct) - (1/4)(# wrong) = 38 - (1/4)(12) = 38 - 3 = 35

No problem.

## Denise's Score:

Denise believes she knows the answers to 50 questions (out of 60) on the test. She picks a random answer (guesses) the other 10 answers because she really isn't sure about their answer. Of the 50 answers she thinks she knows, she gets 38 correct and 12 wrong (same as Charlie above). Of the 10 answers she guesses, probability says that she will get 10/5 = 2 correct and 10 - 2 = 8 wrong. That makes a total of 38 + 2 = 40 correct answers and 12 + 8 = 20 wrong answers.

Denise's Score = (# correct) - (1/4)(# wrong) = 40 - (1/4)(20) = 40 - 5 = 35

Was Denise penalized for guessing? No. She wasn't rewarded, but she wasn't penalized either. There is no statistical advantage (or disadvantage) to leaving the answers blank as opposed to random guessing from all possible answers.

## Edgar's Score:

Edgar believes he knows the answers to 50 questions (out of 60) on the test. For the remaining 10, he can eliminate all but 2 of the possible answers for each question, and he randomly chooses an answer from these 2 choices for these 10 questions. Of the 50 answers he thinks he knows, he gets 38 correct and 12 wrong (same as Charlie and Denise above). Of the 10 answers he guesses, probability says that he will get 10/2 = 5 correct and 10 - 5 = 5 wrong. That makes a total of 38 + 5 = 43 correct answers and 12 + 5 = 17 wrong answers.

Edgar's Score = (# correct) - (1/4)(# wrong) = 43 - (1/4)(17) = 43 - 4.25 = 38.75 = 39

## Summary:

So what's the best strategy?

• If you have no idea whatsoever, it makes no difference whether you guess or leave the answer blank. You can take the attitude, "If you swing the bat, you are likely to miss the ball, but if you don't swing the bat you are certain to miss the ball," or you can be cautious - whatever you are comfortable with.
• If you can eliminate even one answer from the possibilities, you absolutely must "take a swing" and answer the question. Probability works. Have you ever heard of a casino that went bankrupt?

What about "hunches?" In other words, if you eliminate one or more of the answer choices, and you just have a "gut feeling" that one of the remaining choices is correct, should you go with it or choose randomly?

• Well, this is a "touchy" one. You see, testing companies aren't just evil - they are evil and smart. They have generally done their homework, and there is likely to be an incorrect answer choice (a "distracter") that will appeal to a student's naive intuition. In other words, your hunch may be a trap. The only thing that really helps here is that you need to do your homework. You can find practice exams for many standardized tests - try a few and see what works for you.

last update November 20, 2009 by JL Stanbrough