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SECTION 2.1GRAPHICAL SUMMARIES FOR QUALITATIVE DATA
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Section 2.1 - Objectives
1. Construct frequency distributions for qualitative data
2. Construct bar charts
3. Construct pie charts
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Construct frequency distributions for qualitative data
Objective 1
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Frequency Distribution
The frequency of a category is the number of times it occurs in the data set.
A frequency distribution is a table that presents the frequency for each category.
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Example 2.1
A computer retailer compiles a list of the types of computers sold to the last 50 customers.
To construct a frequency distribution, we begin by tallying the number of observations in each category.Tablet Laptop Laptop Laptop Laptop
Laptop Laptop Notebook Desktop LaptopNotebook Desktop Laptop Laptop Laptop
Laptop Notebook Notebook Desktop LaptopDesktop Laptop Tablet Notebook Tablet
Notebook Notebook Tablet Laptop DesktopLaptop Laptop Laptop Laptop Desktop
Desktop Notebook Laptop Desktop LaptopDesktop Tablet Desktop Laptop LaptopDesktop Tablet Notebook Tablet Laptop
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Solution
Then we record the totals for each of the types of computers.
Type of Computer Frequency
Desktop 11Laptop 23
Notebook 9Tablet 7
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Relative Frequency Distribution
A frequency distribution makes it easy to see exactly how many observations are in each category. Sometimes we are interested in the proportion of observations in each category.
The proportion of observations in a category is called the relative frequency of the category.
A relative frequency distribution is a table that presents the relative frequency for each category
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Computing Relative Frequencies
The relative frequency of a category is the frequency of the category divided by the sum of all the frequencies.
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Example 2.2
Construct a relative frequency distribution for the computer sales data.
Type of Computer Frequency
Desktop 11Laptop 23
Notebook 9Tablet 7
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Solution
We begin by finding the total number of observations by summing the frequencies:
11 + 23 + 9 + 7 = 50
Next, we compute the relative frequency for each type of computer:
Type of Computer Frequency
Relative Frequency
Desktop 11 11/50 = 0.22Laptop 23 23/50 = 0.46
Notebook 9 9/50 = 0.18Tablet 7 7/50 = 0.14
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Construct bar graphs
Objective 2
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Bar Graphs
A bar graph is a graphical representation of a frequency distribution. A bar chart consists of rectangles of equal width, with one rectangle for each category. The heights of the rectangles represent the frequencies or relative frequencies of the categories.
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Example 2.3
Construct a frequency bar graph and the relative frequency bar graph for the computer sales data.
Type of Computer FrequencyRelative
FrequencyDesktop 11 0.22Laptop 23 0.46
Notebook 9 0.18Tablet 7 0.14
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Solution
Type of Computer FrequencyRelative
FrequencyDesktop 11 0.22Laptop 23 0.46
Notebook 9 0.18Tablet 7 0.14
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Pareto Chart
Sometimes it is desirable to construct a bar graph in which the categories are presented in order of frequency, with the largest frequency on the left and the smallest frequency on the right. This type of graph is called a Pareto chart.
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Example 2.4
Construct a relative frequency Pareto Chart for the computer sales data presented in the table below.
Type of Computer FrequencyRelative
FrequencyDesktop 11 0.22Laptop 23 0.46
Notebook 9 0.18Tablet 7 0.14
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Solution
Type of Computer FrequencyRelative
FrequencyDesktop 11 0.22Laptop 23 0.46
Notebook 9 0.18Tablet 7 0.14
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Horizontal Bars
The bars in a bar graph may be either horizontal or vertical. Horizontal bars are sometimes more convenient when the categories have long names.
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Example 2.5
Type of employmentRelative
FrequencyFarming, forestry, fishing 0.007Manufacturing, extraction, transportation and crafts 0.203Managerial, professional, technical 0.373Sales and Office 0.242Other Services 0.176
Source: CIA – The World Factbook
Following is the relative frequency distribution categorizes employed U.S. residents by type of employment in the year 2009. Construct a relative frequency bar graph.
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Solution
Type of employmentRelative
FrequencyFarming, forestry, fishing 0.007Manufacturing, extraction, transportation and crafts 0.203Managerial, professional, technical 0.373Sales and Office 0.242Other Services 0.176
Source: CIA – The World Factbook
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Side-by-Side Bar Graphs
Sometimes we want to compare two bar charts that have the same categories. The best way to do this is to construct both bar charts on the same axes, putting bars that correspond to the same category next to each other. This is called a side-by-side bar graph.
The following example presents the number of visitors, in millions, to several popular websites in February 2009 and in February 2010.
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Example – Side-by-Side Bar Graph
WebsiteFebruary 2009
February 2010
Facebook 65 125Google 126 137YouTube 70 93Microsoft Network (MSN) 37 38Yahoo 66 66
Source: mostpopularwebsites.net
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Construct pie charts
Objective 3
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Pie Chart
A pie chart is an alternative to the bar chart for displaying relative frequency information. A pie chart is a circle. The circle is divided into sectors, one for each category. The relative sizes of the sectors match the relative frequencies of the categories.
For example, if a category has a relative frequency of 0.25, then its sector takes up 25% of the circle.
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Example 2.6
Construct a pie chart for the computer sales data.
Type of ComputerRelative
FrequencyDesktop 0.22Laptop 0.46
Notebook 0.18Tablet 0.14
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Do You Know…
• How to construct a frequency and relative frequency distribution for qualitative data?
• How to construct the various kinds of bar graphs?
• How to construct a pie chart?
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