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LESSON 2.4 • The Empirical Rule and Assessing Normality 121

(b) FYI

About 68% About 68% of ITBS scores fall between 5.29
and 8.39. We need to add the percent of scores Although the empirical rule applies
that are between 8.39 and 9.94. About 95%

of ITBS scores fall between 3.74 and 9.94. So only to normal distributions, there is a
=

−
About about 95% 68% 27% of scores are between similar rule that applies to even more
95%–68% 1 and 2 standard deviations from the mean. By
2 distributions: Chebyshev’s Inequality. It Lesson 2.4
=13.5% the symmetry of normal distributions, half of
these scores (27%/2 13.5%) are between 8.39 states that for any distribution with finite

=

=
and 9.94. So about 68% +13.5% 81.5% of ITBS mean µ and finite non-zero standard
scores are between 5.29 and 9.94. 1
deviation σ, at least −1 of the values
2.19 3.74 5.29 6.84 8.39 9.94 11.49 k 2
ITBS vocabulary score in a distribution are within k standard
=
About 68% +13.5% 81.5% , or 0.815 of Gary, Indiana, seventh-graders have deviations of the mean. Chebyshev’s

(C) 2021 BFW Publishers -- for review purposes only.
ITBS vocabulary scores between 5.29 and 9.94. FOR PRACTICE TRY EXERCISE 7.
Inequality applies to normal and many
non-normal distributions. Exercise 26
!
Note that the empirical rule applies only to normal distributions. There are other caution at the end of this lesson introduces

aution
rules for non-normal distributions, but they are beyond the scope of this book. students to Chebyshev’s Inequality.
You can also use the empirical rule to estimate the value that corresponds to a
given percentile in a normal distribution.
EXAMPLE AL TERNA TE EX AMPLE
What is Lexi Thompson’s score?
Stop the car!
Using the empirical rule in reverse
Using the empirical rule in reverse
PROBLEM: Golfer Lexi Thompson

PROBLEM: Many studies on automobile safety suggest that when automobile drivers must make emergency

stops, the stopping distances can be modeled by a normal distribution. Suppose that for one model of car is one of the top golfers on the LPGA
traveling at 60 mph under typical conditions on dry pavement, the distribution of stopping distances is tour. The distribution of scores for each
approximately normal with mean µ = 165 feet and standard deviation σ = 4 feet. What stopping distance is

at the 84th percentile of the distribution? Justify your answer. of the more than 700 rounds over her
LPGA career is approximately normal
SOLUTION:
=
About 68% About 16% +68% 84% of cars of this model would with a mean of about µ = 70.6 strokes

stop in less than 169 feet, so a stopping distance and a standard deviation of about
of 169 feet is at about the 84th percentile of the σ = 3.2 strokes. What score is at the 16th
32%
32% About = 16% distribution. percentile of the distribution? Justify
About = 16% 2
2
your answer.
SOLUTION:
153 157 161 165 169 173 177
Stopping distance (ft) FOR PRACTICE TRY EXERCISE 11. About 68% of all scores are between
67.4 and 73.8. That means about
100% –68% 32% are either less than
=
Assessing Normality 67.4 or greater than 73.8. Because
normal distributions are symmetric,
Normal distributions provide good models for some distributions of quantitative data.
=
Examples include SAT and IQ test scores, the highway gas mileage of 2020 Corvette about 32%/2 16% or 0.16 of Lexi
convertibles, weights of 9-ounce bags of potato chips, and heights of 3-year-old girls Thompson’s scores are less than 67.4.
(see Figure 2.11 ).
So, a score of 67.4 is at about the 16th
percentile of the distribution.
About 68%
03_StarnesSPA4e_24432_ch02_088_153.indd 121 07/09/20 1:56 PM About
About 100% – 68%
100% – 68% 2
2 = 16%
= 16%
61.0 64.2 67.4 70.6 73.8 77.0 80.2
Score
LESSON 2.4 • The Empirical Rule and Assessing Normality 121

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