Five-Number Summary and Box-and-Whisker Plots

Five-Number Summary

Definitions:

• The minimum value of a data set is the least value in the set.
• The maximum value of a data set is the greatest value in the set.
• The range of a data set is the distance between the maximum and minimum value. To compute the range of a data set, we subtract the minimum from the maximum:
  range = maximum – minimum.
• The interquartile range of a data set is the distance between the two quartiles.
  Interquartile range = Q₃ – Q₁.

Example 1: Find the range and interquartile range of the set {3, 7, 8, 5, 12, 14, 21, 13, 18}.

First, we write the data in increasing order: 3, 5, 7, 8, 12, 13, 14, 18, 21.

range = max – min = 21 – 3 = 18.

Recall from the previous page that Q₁ = 6 and Q₃ = 16.

Therefore, the interquartile range = Q₃ – Q₁ = 16 – 6 = 10.

The range is 18 and the interquartile range is 10.

Example 2: Find the range and interquartile range of the set {3, 7, 8, 5, 12, 14, 21, 15, 18, 14}.

First, we write the data in increasing order: 3, 5, 7, 8, 12, 14, 14, 15, 18, 21.

range = max – min = 21 – 3 = 18.

Recall from the previous page that Q₁ = 7 and Q₃ = 15.

Therefore, the interquartile range = Q₃ – Q₁ = 15 – 7 = 8.

The range is 18 and the interquartile range is 8.

 Self Check Problem

The following dollar amounts were the hourly collections from a Salvation Army kettle at a local store one day in December: $19, $26, $25, $37, $32, $28, $22, $23, $29, $34, $39, and $31. Determine the range and interquartile range for the amount collected.

Solution

Definition: The five-number summary of a data set consists of the five numbers determined by computing the minimum, Q₁ , median, Q₃ , and maximum of the data set.

Example 1:  Find the five-number summary for the data set {3, 7, 8, 5, 12, 14, 21, 13, 18}.

From our Example 1's on the previous pages, we see that the five-number summary is:
Minimum: 3         Q₁ : 6            Median: 12            Q₃ : 16           Maximum: 21

Example 2: Find the five-number summary for the data set {3, 7, 8, 5, 12, 14, 21, 15, 18, 14}.

From our Example 2's on the previous pages, we see that the five-number summary is:

Minimum: 3         Q₁ : 7            Median: 13            Q₃ : 15           Maximum: 21  

Self Check Problem

The following dollar amounts were the hourly collections from a Salvation Army kettle at a local store one day in December: $19, $26, $25, $37, $32, $28, $22, $23, $29, $34, $39, and $31. Find the five-number summary for the amount collected.

Solution

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