Percentiles of Normal distributions*
For Normal distributions, but not for other distributions, standard scores translate directly into percentiles.
Percentiles
The cth percentile of a distribution is a value such that percent of the observations lie below it and the rest lie above.
The median of any distribution is the 50th percentile, and the quartiles are the 25th and 75th percentiles. In any Normal distribution, the point one standard deviation above the mean (standard score 1) is the 84th percentile. Figure 13.10 shows why. Every standard score for a Normal distribution translates into a specific percentile, which is the same no matter what the mean and standard deviation of the original Normal distribution are. Table B at the back of this book gives the percentiles corresponding to various standard scores. This table enables us to do calculations in greater detail than does the 68–95–99.7 rule.
This material is not needed to read the rest of the book.
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EXAMPLE 4 Percentiles for college entrance exams
Madison’s score of 600 on the SAT translates into a standard score of 1.0. We saw that the 68–95–99.7 rule says that this is the 84th percentile. Table B is a bit more precise: it says that standard score 1 is the 84.13 percentile of a Normal distribution. Gabriel’s 21 on the ACT is a standard score of 0.5. Table B says that this is the 69.15 percentile. Gabriel did well, but not as well as Madison. The percentile is easier to understand than either the raw score or the standard score. That’s why reports of exams such as the SAT usually give both the score and the percentile.
EXAMPLE 5 Finding the observation that matches a percentile
How high must a student score on the SAT to fall in the top 10% of all scores? That requires a score at or above the 90th percentile. Look in the body of Table B for the percentiles closest to 90. You see that standard score 1.2 is the 88.49 percentile and standard score 1.3 is the 90.32 percentile. The percentile in the table closest to 90 is 90.32, so we conclude that a standard score of 1.3 is approximately the 90th percentile of any Normal distribution.
To go from the standard score back to the scale of SAT scores, “undo” the standard score calculation as follows:
observation = mean + standard score × standard deviation
= 500 + (1.3)(100) = 630
A score of 630 or higher will be in the top 10%. (More exactly, these scores are in the top 9.68% because 630 is exactly the 90.32 percentile.)
EXAMPLE 6 Finding the percentile below a particular value
Returning again to the distribution of SAT scores, if Ting obtains a score of 430 on this exam, what percentage of individuals score the same as or lower than Ting? To answer this question, we must first convert Ting’s score to a standard score. This standard score is
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From Table B, we can see that the standard score of −0.7 corresponds to a percentile of 24.20. Therefore, 24.20% of those individuals who took this exam score at or below Ting’s score of 430.
EXAMPLE 7 Finding the percentile above a particular value
If Jordan scores 725 on the SAT Mathematics college entrance exam, what percentage of individuals would score at least as high or higher than Jordan? To answer this question, we must again begin by converting Jordan’s score to a standard score. This standard score is
To use Table B, we must first round 2.25 to 2.3. From Table B, we can see that the standard score of 2.3 corresponds to a percentile of 98.93. We might be tempted to assume our final answer will be 98.93%. If we were attempting to determine the percentage of individuals who scored 725 or lower, our final answer would be 98.93%. We want to know, however, the percentage who score 725 or higher. To find this percentage, we need to remember that the total area below the Normal curve is 1. Expressed as a percentage, the total area is 100%. To find the percentage of values that fall at or above 725, we subtract the percentage of values that fall below 725 from 100%. This gives us 100% − 98.93% = 1.07%. Our final answer is that 1.07% of individuals who take the SAT Mathematics college entrance exam score at or above Jordan’s score of 725.
NOW IT’S YOUR TURN
Question 13.3
13.3 SAT scores. How high must a student score on the SAT Mathematics test to fall in the top 25% of all scores? What percentage of students obtain scores at or below 475? What percentage of students obtain scores at or above 580?