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1
Question:

The mean and standard deviation (SD) of height in adult men and women in the United States are shown below.

Height (inches)

Mean

SD

Men

70

3

Women

64.5

2.5

Assuming that the heights of men and women are approximately normally distributed, which of the following approximates the percentile of heights for an adult woman who is 69.5 inches tall?

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Explanation:

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A normal distribution is a symmetrical, bell-shaped distribution with a fixed percentage of observations lying within a certain distance of the mean.  This distance is called the standard deviation (SD) and represents the degree of dispersion of the data from the mean.

The 68-95-99.7 rule for a normal distribution states that 68% of all observations lie within 1 SD of the mean, 95% of all observations lie within 2 SDs of the mean, and 99.7% of all observations lie within 3 SDs of the mean.  A percentile is the value in a normal distribution that has a specific percentage of observations below it (ie, area under the curve to the left of the specific value).  It is possible to use the 68-95-99.7 rule to identify the percentiles that correspond to values that are 1, 2, or 3 SDs from the mean.

The heights of adult women in the United States are approximately normally distributed with a mean of 64.5 inches and a SD of 2.5 inches.  Based on the 68-95-99.7 rule:

  • 68% of heights of adult women lie within 1 SD: 64.5 ± 2.5 = 62-67
  • 95% of heights of adult women lie within 2 SDs: 64.5 ± 2(2.5) = 59.5-69.5
  • 99.7% of heights of adult women lie within 3 SDs: 64.5 ± 3(2.5) = 57-72

The height of an adult woman who is 69.5 inches tall is exactly 2 SDs above the mean height.  This means that 2.5% of heights must lie above 69.5 inches and 100% − 2.5% = 97.5% must lie below 69.5 inches (Choice A).  Therefore, an adult woman who is 69.5 inches tall is at the 97.5 percentile of the distribution.

(Choice B)  In a normal distribution, the 50th percentile approximates the mean of the distribution.  In this case, the mean height of adult women is 64.5 inches, so a height of 69.5 inches is at a higher percentile.

(Choices C and E)  68.0 and 99.7 represent the percentage of heights that are within 1 SD and 3 SDs from the mean, respectively.

Educational objective:
A percentile is the value in a normal distribution that has a specified percentage of observations below it (ie, area under the curve to the left of the specific value).  It is possible to use the 68-95-99.7 rule to identify the percentiles that correspond to values that are 1, 2, or 3 standard deviations from the mean.