Question

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?

Answer 1

`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.