Madison scored 600 on the SAT Mathematics college entrance exam. How good a score is this? That depends on where a score of 600 lies in the distribution of all scores. The SAT exams are scaled so that scores should roughly follow the Normal distribution with mean 500 and standard deviation 100. Madison’s 600 is one standard deviation above the mean. The 68–95–99.7 rule now tells us just where she stands (Figure 13.10). Half of all scores are below 500, and another 34% are between 500 and 600. So, Madison did better than 84% of the students who took the SAT. Her score report not only will say she scored 600 but will add that this is at the “84th percentile.” That’s statistics speak for “You did better than 84% of those who took the test.”

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Figure 13.10: Figure 13.10 The 68–95–99.7 rule shows that 84% of any Normal distribution lies to the left of the point one standard deviation above the mean. Here, this fact is applied to SAT scores.

Because the standard deviation is the natural unit of measurement for Normal distributions, we restated Madison’s score of 600 as “one standard deviation above the mean.” Observations expressed in standard deviations above or below the mean of a distribution are called *standard scores*. Standard scores are also sometimes referred to as z-scores.

Standard scores

The **standard score** for any observation is

A standard score of 1 says that the observation in question lies one standard deviation above the mean. An observation with standard score −2 is two standard deviations below the mean. Standard scores can be used to compare values in different distributions. Of course, you should not use standard scores unless you are willing to use the standard deviation to describe the variability of the distributions. That requires that the distributions be at least roughly symmetric.

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EXAMPLE 3 ACT versus SAT scores

Madison scored 600 on the SAT Mathematics exam. Her friend Gabriel took the American College Testing (ACT) test and scored 21 on the math part. ACT scores are Normally distributed with mean 18 and standard deviation 6. Assuming that both tests measure the same kind of ability, who has the higher score?

Madison’s standard score is

Compare this with Gabriel’s standard score, which is

Because Madison’s score is 1 standard deviation above the mean and Gabriel’s is only 0.5 standard deviation above the mean, Madison’s performance is better.

NOW IT’S YOUR TURN

13.2 Heights of young men. The distribution of heights of young men is approximately Normal with mean 70 inches and standard deviation 2.5 inches. What is the standard score of a height of 72 inches (6 feet)?