Should you ever guess on the SAT® or PSAT standardized tests?

My son is getting ready to take the preliminary SAT (PSAT), which is a practice test for the SAT. A teacher gave his class this advice regarding guessing: For a multiple-choice questions, if you can eliminate one or two choices, then it is okay to guess from the remaining possible answers.

By using probability and statistics, you can understand why the teacher's advice is correct. It turns out that, on average, your score will be higher if you eliminate choices that you know are wrong, then guess an answer from the remaining choices. That is, there is no penalty for guessing on the SAT or PSAT. To learn why, read on.

The PSAT web site describes how the multiple-choice portion of the test is scored:

• The multiple-choice questions have five possible answers.
• Students receive one point for each correct answer, regardless of difficulty.
• For incorrect answers to multiple-choice questions, a quarter (1/4) of a point is deducted.
• Nothing is deducted for unanswered questions.

The multiple-choice questions on the SAT test are scored similarly.

It is clear that you get no points for leaving a question unanswered. But how many points would you expect to get by guessing?

In high school, you might have learned about a statistical concept called the expected gain. If you have N items and the reward (or penalty) for choosing the ith item is R_i, then the expected gain is (R₁ + R₂ +...+ R_N)/N, assuming that each choice is equally likely to be chosen.

That means that you can calculate the expected points that you will receive for each guess. It pays to guess when the expected number of points is greater than zero. Let's see how it works:

• If you know the correct answer, your expected gain is 1 point. Obviously, this is the best strategy!
• If you can eliminate three of the five options, then the correct answer is one of the remaining two choices. On average, guessing will add (1 - 1/4)/2 = 3/8 points to your score.
• If you can eliminate two options, then the correct answer is one of the remaining three choices. On average, guessing will add (1 - 1/4 - 1/4)/3 = 1/6 points to your score.
• If you can eliminate one option, then guessing will add (1 - 1/4 - 1/4 - 1/4)/4 = 1/16 points to your score, on average. That's a tiny expected gain, so if you are pressed for time, don't bother filling in the oval.
• If you have no idea which of the five choices are correct, your expected gain is zero. You might as well save time by not filling in the oval. However, even in this case, guessing does not penalize you.

The following table summarizes the results of guessing on the SAT or PSAT:

To maximize your score on the SAT, review the basic concepts and know the correct answers. However, if you are not certain which answer is correct during the multiple choice portion of the test, you might be able to eliminate the wrong answers and guess from among the choices that remain. On average, guessing adds a small amount to your score, and, statistically speaking, guessing is never a bad decision.

SAS is proud to celebrate 2013 as the International Year of Statistics. Not only can statistics make our world safer and healthier, but it can actually increase your SAT scores!