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5.2 Properties of the Normal
Distribution
LEARNING GOAL
Know how to interpret the normal distribution in terms of
the 68-95-99.7 rule, standard scores, and percentiles.
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A simple rule, called the 68-95-99.7 rule, gives
precise guidelines for the percentage of data values
that lie within 1, 2, and 3 standard deviations of the
mean for any normal distribution.
Figure 5.17 Normal distribution illustrating the 68-95-99.7 rule.
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The 68-95-99.7 Rule for a Normal Distribution
• About 68% (more precisely, 68.3%), or just over twothirds, of the data points fall within 1 standard deviation
of the mean.
• About 95% (more precisely, 95.4%) of the data points
fall within 2 standard deviations of the mean.
• About 99.7% of the data points fall within 3 standard
deviations of the mean.
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EXAMPLE 1 SAT Scores
The tests that make up the verbal (critical reading) and
mathematics SAT (and the GRE, LSAT, and GMAT) are
designed so that their scores are normally distributed with a
mean of m = 500 and a standard deviation of s = 100.
1. Estimate the percentage of students having test scores
between 400-500?
2. Estimate the percentage of students having test scores
between 300-700?
3.Estimate the percentage of students having test scores between
200-800?
4. Estimate the percentage of students having test scores above
500?
5. Estimate the percentage of students having test scores
between 300-800?
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EXAMPLE 4 Normal Heart Rate
You measure your resting heart rate at noon every day for a year
and record the data. You discover that the data have a normal
distribution with a mean of 66 and a standard deviation of 4. On
how many days was your heart rate below 58 beats per minute?
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EXAMPLE 5 Finding a Percentile
On a visit to the doctor’s office, your fourth-grade daughter is told
that her height is 1 standard deviation above the mean for her age
and sex. What is her percentile for height? Assume that heights of
fourth-grade girls are normally distributed.
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Standard Scores
Computing Standard Scores
The number of standard deviations a data value lies
above or below the mean is called its standard score (or
z-score), defined by
data value – mean
z = standard score =
standard deviation
The standard score is positive for data values above the
mean and negative for data values below the mean.
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EXAMPLE 6 Finding Standard Scores
The Stanford-Binet IQ test is scaled so that scores have a mean of
100 and a standard deviation of 16.
Find the standard scores for IQs of 85, 100, and 125.
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EXAMPLE 7 Cholesterol Levels
Cholesterol levels in men 18 to 24 years of age are normally
distributed with a mean of 178 and a standard deviation of 41.
a. What is the percentile for a 20-year-old man with a cholesterol
level of 190?
Solution:
a.The standard score for a cholesterol level of 190 is
data value – mean
190 – 178
z = standard score =
=
≈ 0.29
standard deviation
41
Table 5.1 shows that a standard score of 0.29 corresponds to about
the 61st percentile.
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EXAMPLE 7 Cholesterol Levels
Cholesterol levels in men 18 to 24 years of age are normally
distributed with a mean of 178 and a standard deviation of 41.
b. What cholesterol level corresponds to the 90th percentile?
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