EXAMPLE 19 SAT Reasoning Test Scores
The distribution of scores on tests such as the SAT college entrance examination is close to normal. Scores on each of the three sections (math, critical reading, and writing) of the SAT are adjusted so that the mean score is about and the standard deviation is about . (The Greek letters and designate the population mean and standard deviation, respectively, as opposed to , which designate the mean and standard deviation of sample data). This information allows us to answer many questions about SAT scores.
- How high must a student score to fall in the top 25%? This score is the 75th percentile or the third quartile. The third quartile is points above the mean. So scores above 567 are in the top 25% (i.e., the top quarter).
- What percentage of scores falls between 200 and 800? Scores of 200 and 800 are 3 standard deviations on either side of the mean—for example, . The "99.7" part of the 68-95-99.7 rule says that 99.7% of all scores lie in this interval. (In practice, the SAT makes this 100% by reporting as 200 those rare scores below 200, or as 800 those rare scores above 800.)
- What percentage of scores is above 600? A score of 600 is 1 standard deviation above the mean. By the "68" part of the 68-95-99.7 rule, 68% of all scores fall between 400 and 600 and 32% fall below 400 or above 600. Because normal curves are symmetric, half of this 32% are above 600. So a score above 600 places a student in the top 16% of test-takers. We can also say that a score of 600 is at the 84th percentile of all test-takers.
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Sketching a normal curve and scaling the horizontal axis with the mean and points 1, 2, and 3 standard deviations from the mean can help you use the 68-95-99.7 rule. Figure 5.27 shows the distribution of SAT scores with the areas needed to find the percentage of scores above 600. Note that the tails of Figure 5.27, like those of any bell curve, technically stretch out forever in both directions (even as the amount of faraway area becomes vanishingly small). This is another reminder that the bell curve is a very good, but not perfect, model of reality. We know that real-life SAT subtest scores are scaled so that they do not go beyond 200 or 800.
