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64 UNIT 1 Exploring One-Variable Data

HOW TO IDENTIFY OUTLIERS: THE 1.5 × IQR RULE

Call an observation an outlier if it falls more than 1.5 × IQR above the third

quartile or more than 1.5 × IQR below the first quartile. That is,

Lowoutliers < Q − 1.5× IQR Highoutliers > Q + 1.5× IQR
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EXAMPLE An NBA legend Skill 4.B
Identifying outliers

PROBLEM: Here are data on the average number of points per game that
LeBron James scored in each of his first 16 NBA seasons:
20.9 27.2 31.4 27.3 30.0 28.4 29.7 26.7
27.1 26.8 27.1 25.3 25.3 26.4 27.5 27.4
Use the 1.5 × IQR rule to identify any outliers in the distribution. Kevin C. Cox/Getty Images

SOLUTION:
20.9 25.3 25.3 26.4 26.7 26.8 27.1 27.1 27.2 27.3 27.4 27.5 28.4 29.7 30.0 31.4

Q = 26.55 Median = 27.15 Q = 27.95

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IQR = 27.95 − 26.55 = 1.40
• Find the interquartile range: IQR = Q 3 − .

Q 1
Low outliers < 26.55 − 1.5 × 1.40 = 24.45 • Then calculate the upper and lower cutoff values for outliers:
High outliers > 27.95 + 1.5 × 1.40 = 30.05 Low outliers Q 1 − 1.5× IQR
<
High outliers Q 3 + 1.5× IQR
>
LeBron James’s rookie-season average of 20.9
points per game is a low outlier because it is less than 24.45. The season when
LeBron averaged 31.4 points per game is a high outlier because it is greater than 30.55.
FoR PRAcTIce, TRY eXeRcISe 27
There are other methods for determining outliers, such as “any value that is
more than 2 standard deviations from the mean.” Let’s apply this 2 × SD rule to
the data from the preceding example. Here are summary statistics on the average
number of points scored by LeBron James in each of his first 16 NBA seasons:

n Mean SD Min Q 1 Med Q 3 Max
16 27.156 2.328 20.9 26.55 27.15 27.95 31.4

Low outliers < Mean − 2 × SD = 27.156 − 2 × 2.328 = 22.50

High outliers > Mean + 2 × SD = 27.156 + 2 × 2.328 = 31.812

By the 2× SD rule, the season in which LeBron James averaged 20.9 points per
game is an outlier because it is less than 22.50. However, the season in which
LeBron averaged 31.4 points per game is not an outlier by this rule because 31.4
is not greater than the upper cutoff of 31.812.
Unless otherwise indicated, we always use the 1.5 × IQR rule in this book to iden-

tify outliers because it is based on statistics that are resistant to extreme data values,
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