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Variability and Box-and-Whisker Plots 10-4
While central tendency describes the middle of a data set, variability describes how spread out the data are. A box-and-whisker plot uses a number line to show how data are distributed and to illustrate the variability of a data set.
A box-and-whisker plot divides the data into four parts. The median, or second quartile, divides the data into a lower half and an upper half. The first quartile is the median of the lower half of the data, and the third quartile is the median of the upper half of the data.
Variability and Box-and-Whisker Plots 10-4
Variability and Box-and-Whisker Plots 10-4
Use the given data to make a box-and-whisker plot: 21, 25, 15, 13, 17, 19, 19, 21
Additional Example 1: Making a Box-and-Whisker Plot
Step 1: Order the data and find the least value, first quartile, median, third quartile, and greatest value.
13 15 17 19 19 21 21 25
least value: 13 greatest value: 25
first quartile: = 16 15 + 17
2 third quartile: = 21
21 + 21
2
median: = 19 19 + 19
2
Variability and Box-and-Whisker Plots 10-4
12 14 16 18 20 22 24 26 28
Step 2: Draw a number line and plot a point above each value from Step 1.
least value 13
13 15 17 19 19 21 21 25 first
quartile 16
median
19
third quartile
21
greatest value 25
Additional Example 1 Continued
Variability and Box-and-Whisker Plots 10-4
12 14 16 18 20 22 24 26 28
Step 3: Draw the box and whiskers.
13 15 17 19 19 21 21 25
Additional Example 1 Continued
Variability and Box-and-Whisker Plots 10-4
Check It Out: Example 1A
Use the given data to make a box-and-whisker plot.
31, 23, 33, 35, 26, 24, 31, 29
Variability and Box-and-Whisker Plots 10-4
Check It Out: Example 1B
Use the given data to make a box-and-whisker plot.
57, 53, 52, 31, 48, 58, 64, 86, 56, 54, 55
Variability and Box-and-Whisker Plots 10-4
The interquartile range of a data set is the difference between the third quartile and the first quartile. It represents the range of the middle half of the data.
Variability and Box-and-Whisker Plots 10-4
Additional Example 2: Using Interquartile Range
to Identify Outliers
Use interquartile range to identify any outliers.
75, 65, 78, 79, 76, 79, 72, 82
Step 1: Determine the first quartile, the third quartile, and the interquartile range.
65 72 75 76 78 79 79 82
Q1: 73.5 Q3: 79 IQR: 79 – 73.5 = 5.5
Variability and Box-and-Whisker Plots 10-4
Additional Example 2 Continued
Step 2: Determine whether there is an outlier less than the first quartile.
Q1 – (1.5 • IQR)
73.5 – (1.5 • 5.5)
73.5 – 8.25 = 65.25
The least value in the data set is 65. This value is less than 65.25.
Variability and Box-and-Whisker Plots 10-4
Additional Example 2 Continued
Step 3: Determine whether there is an outlier greater than the third quartile.
Q3 + (1.5 • IQR)
79 + (1.5 • 5.5)
79 + 8.25 = 87.25
The greatest value in the data set is 82. None of the values are greater than 87.25.
Variability and Box-and-Whisker Plots 10-4
Additional Example 2 Continued
Use interquartile range to identify any outliers.
75, 65, 78, 79, 76, 79, 72, 82
The data value 65 is an outlier.
Variability and Box-and-Whisker Plots 10-4
Check It Out: Example 2A
Use the interquartile range to identify any outliers.
25, 12, 31, 26, 27, 29, 32
Variability and Box-and-Whisker Plots 10-4
Check It Out: Example 2B
Use the interquartile range to identify any outliers.
35, 46, 50, 32, 54, 44, 40
