The National Center for Educational Statistics report that in the year ending 20

Question

The National Center for Educational Statistics report that in the year ending 2009 400,000 students took the Graduate Record Examination (GRE). For this set of examinees, the observed mean Verbal score was 560 with a standard deviation (SD) of 90, and the observed mean Quantitative score was 490 with a standard deviation (SD) of 100. Assuming that the distribution of scores on the 2009 GRE was normal, answer the following questions regarding both tests (Verbal and Quantitative): (Show all your computations and use statistical evidence to interpret your answer).

1. What percent of student scores were below the mean?

2. How many student scores were above the mean?

3. How many student scores fall between 1 SD below the mean (-1SD) and 1 SD above the mean (+1SD).

4.How many students scores were higher than 2 SD above the mean.

Explanation / Answer

As all the questions are in terms of standard deviations (SDs) from the mean, the actual scores to not matter here. We directly get the percentage from the information regarding number of SDs from the mean. To get the actual count of student, we multiply by the total number of students.

1. As we are using the normal distribution, we know that this is a symmetric distribution about the mean. So percentage of students below the mean = 50%.

2. Again, because this is a symmetric distribution, percentage of students above the mean = 50%. So, number of students above the mean = 400,000 * 50% = 200,000.

3. For number of students between +/- 1 SD from the mean, we use the 68-95-99.7 rule. This gives us 68.27% for +/- 1 SD from the mean. So, number of students within 1 SD from the mean = 400,000 * 68.27% = 273,080.

4. For number of students above 2 SD from the mean, we again use the 68-95-99.7 rule. This gives us 95.45% for +/- 2 SD from the mean. This will come to 50% - half of 95.45% = 50%-47.725% = 2.275% of students 2 SD above the mean. So, number of students 2 SD above the mean = 400,000 * 2.275% = 9,100.

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