SAT Scores

Task

Suppose that SAT mathematics scores for a particular year are approximately normally distributed with a mean of 510 and a standard deviation of 100.

1. What is the probability that a randomly selected score is greater than 610?
2. Greater than 710?
3. Between 410 and 710?
4. If a student is known to score 750, what is the student’s percentile score (the proportion of scores below 750)?

IM Commentary

Students will want to go to their calculators immediately on questions like these. That is an option, but they should know the percentages given by the empirical rule for areas in intervals of one or two standard deviations to either side of the mean under a normal distribution. They should also note that the normal distribution is only an approximation to the true distribution of scores, so the answers are only approximate as well. (Thus, the empirical rule answers are fine, and an increase in the number of decimal places does not necessarily improve the answer!)

Solution

1. The score 610 is one standard deviation above the mean, so the tail area above that is about half of 0.32 or 0.16. The calculator gives 0.1586.
2. The score 710 is two standard deviations above the mean, so the tail area above that is about half of 0.05 or 0.025. The calculator gives 0.0227.
3. The area under a normal curve from one standard deviation below the mean to two standard deviations above is about 0.815. The calculator gives 0.8186.
4. Either using the normal distribution given or the standard normal (for which 750 translates to a z-score of 2.4) the calculator gives 0.9918.