SECTION 2.1 GRAPHICAL SUMMARIES FOR QUALITATIVE DATA

Objectives 1. Construct frequency distributions for qualitative data 2. Construct bar graphs 3. Construct pie charts

Construct frequency distributions for qualitative data Objective 1

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.

Example 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. TabletLaptop NotebookDesktopLaptop NotebookDesktopLaptop Notebook DesktopLaptop DesktopLaptopTabletNotebookTablet Notebook TabletLaptopDesktop Laptop Desktop NotebookLaptopDesktopLaptop DesktopTabletDesktopLaptop DesktopTabletNotebookTabletLaptop

Solution Then we record the totals for each of the types of computers. Type of ComputerFrequency Desktop11 Laptop23 Notebook9 Tablet7

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

Computing Relative Frequencies The relative frequency of a category is the frequency of the category divided by the sum of all the frequencies.

Example Construct a relative frequency distribution for the computer sales data. Type of ComputerFrequency Desktop11 Laptop23 Notebook9 Tablet7

Solution We begin by finding the total number of observations by summing the frequencies: = 50 Next, we compute the relative frequency for each type of computer: Type of ComputerFrequencyRelative Frequency Desktop1111/50 = 0.22 Laptop2323/50 = 0.46 Notebook99/50 = 0.18 Tablet77/50 = 0.14

Construct bar graphs Objective 2

Bar Graphs A bar graph is a graphical representation of a frequency distribution. A bar graph 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.

Example Construct a frequency bar graph and the relative frequency bar graph for the computer sales data. Type of ComputerFrequencyRelative Frequency Desktop Laptop Notebook90.18 Tablet70.14

Solution Type of ComputerFrequencyRelative Frequency Desktop Laptop Notebook90.18 Tablet70.14

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.

Example Construct a relative frequency Pareto Chart for the computer sales data presented in the table below. Type of ComputerFrequencyRelative Frequency Desktop Laptop Notebook90.18 Tablet70.14

Solution Type of ComputerFrequencyRelative Frequency Desktop Laptop Notebook90.18 Tablet70.14

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.

Example Type of employmentRelative Frequency Farming, forestry, fishing Manufacturing, extraction, transportation and crafts Managerial, professional, technical Sales and Office Other Services Source: CIA – The World Factbook The following relative frequency distribution categorizes employed U.S. residents by type of employment in the year Construct a relative frequency bar graph.

Solution Type of employmentRelative Frequency Farming, forestry, fishing Manufacturing, extraction, transportation and crafts Managerial, professional, technical Sales and Office Other Services Source: CIA – The World Factbook

Side-by-Side Bar Graphs Sometimes we want to compare two bar graphs that have the same categories. The best way to do this is to construct both bar graphs 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.

Example – Side-by-Side Bar Graph Website February 2009 February 2010 Facebook Google YouTube 7093 Microsoft Network (MSN) 3738 Yahoo 66 Source: mostpopularwebsites.net

Construct pie charts Objective 3

Pie Chart A pie chart is an alternative to the bar graph 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.

Example Construct a pie chart for the computer sales data. Type of ComputerRelative Frequency Desktop0.22 Laptop0.46 Notebook0.18 Tablet0.14

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?