A test consists of 910 true or false questions. If the student guesses on each question, what is the expected standard deviation of the number of correct answers?

1 Answer
Bill K.
Dec 10, 2015

sqrt{910 * 0.5 * 0.5}=sqrt{227.5} approx 15.08.

Explanation:

The number X of correct guesses in n=910 trials is a binomial random variable with probability p=0.5 of success. The standard deviation of such a variable is sqrt{np(1-p)}. In this case, that's sqrt{910 * 0.5 * 0.5}=sqrt{227.5} approx 15.08.

Such a variable would be well-modeled by a Normal distribution with mean np=910 * 0.5 = 455 and standard devation sqrt{227.5 approx 15. Using the 68-95-99.7 rule-of-thumb , about 68% of the people who guessed on such an exam would score between 440 and 470, about 95% of such people would score between 425 and 485, and about 99.7% of such people would score between 410 and 500.

These are approximations and we are not making a distinction between these ranges including the endpoints or not. The "half-unit (continuity) correction " could be used along with a table of Normal probabilities to try to make these estimates more precise.

Related questions
See all questions in Measures of Variability
Impact of this question
24353 views around the world
You can reuse this answer
Creative Commons License