3.3 The Five-Number Summary; Boxplots
Resistant measures: are descriptive measures that are not influenced by
extreme measures.
Descriptive measures based on percentiles may be preferred over those
based on the mean and standard deviation.
Median: divides the data into two equal parts. It separates the bottom 50%
from the top 50%.
Percentiles: divides a data set into 100 equal parts.
Deciles: divides a data set into 10 equal parts.
Quintiles: divides a data set into 5 equal parts.
Quartiles: divides a data set into 4 equal parts and is the most commonly
used. A data set has three quartiles designated as Q1, Q2, and Q3.
Q1 is the first quartile. It divides the bottom 25% of the data from the top
75%.
Q2 is the second quartile. It divides the bottom 50% of the data from the
top 50%.
Q3 is the third quartile. It divides the bottom 75% of the data from the top
25%.
How to find the Quartiles
Arrange the data in increasing order and determine the median.
 The second quartile is the median of the entire data set
 The first quartile is the median of the part of the entire data set that
lies at or below the median of the entire data set.
 The third quartile is the median of the part of the entire data set that
lies at or above the median of the entire data set.
Example 1
Find the quartiles of the data set.
Example 2
Find the quartiles of the data set.
Interquartile Range
The interquartile range, or IQR, is the difference between the first and
third quartiles, that is,
IQR  Q3  Q1
Example 3
Find the interquartile range IQR for the data sets in Examples 1 and 2
Five-Number Summary
The five-number summary of a data set is Min, Q1, Q2, Q3, Max.
Example 4
Find the five-number summary for the data sets in Example 3.
Lower and Upper Limits
The lower limit and upper limit of a data set are
Lower limit = Q1 - 1.5  IQR
Upper limit = Q3  1.5  IQR
Example 5
Find the lower and upper limits for the data sets in Example 4.
Procedure to Construct a Boxplot
1. Determine the quartiles.
2. Determine potential outliers and the adjacent values.
3. Draw a horizontal axis on which the numbers obtained in Steps 1 and
2 can be located. Above the axis, mark the quartiles and the adjacent
values with vertical lines.
4. Connect the quartiles to make a box, and then connect the box to the
adjacent values with lines.
5. Plot each potential outlier with an asterisk.
Example 6
Construct boxplots for the data sets in Example 5.
Uses for Boxplots
1. Comparing two or more data sets. The same scale should be used.
2. Determining the shape of the data sets.
Example 7