OBJECTIVES By the end of this section, I will be able to …

The frequency, or count, of a category refers to the number of observations in each category. A frequency distribution for a qualitative variable is a listing of all the values (for example, categories) that the variable can take, together with the frequencies for each value.

EXAMPLE 1 Frequency distributions

Create a frequency distribution for the variable app type from Table 1.

Solution

For each app type, we compute the frequency; that is, we count (or tally) how many apps were of that particular app type. Table 2 shows the frequency distribution for the variable app type. For example, five of the apps were social networking apps. The frequency distribution summarizes the data set so that quick observations can be made, such as “The most popular app type in the Apple.com top 20 list of the most downloaded free apps is the Games app type.”

The relative frequency of a particular category of a qualitative variable is its frequency divided by the sample size. A relative frequency distribution for a qualitative variable is a listing of all values that the variable can take, together with the relative frequencies for each value.

EXAMPLE 2 Relative frequency distributions

Create a relative frequency distribution for the variable app type using Table 2.

Solution

The relative frequency of the games app type is the frequency 7 divided by the sample size 20:

The relative frequency of games apps is 0.35, or 35%. So, if someone told you that 35% of the apps were games, without telling you the sample size, you would have a better idea of the relative popularity of that app type. To construct the relative frequency distribution in Table 4, divide each frequency in the frequency distribution in Table 2 by the sample size 20.

A bar graph is used to represent the frequencies or relative frequencies for categorical data. It is constructed as follows:

EXAMPLE 3 Constructing bar graphs

Construct a frequency bar graph and a relative frequency bar graph for the distributions of app type in Tables 2 and 4.

Solution

The bar graphs are provided in Figure 1a and 1b. Across the horizontal axis are the five app type categories. Next, draw rectangles, the heights of which represent either the frequency or the relative frequency for that category represented on the vertical axis. For example, in Figure 1a, the first rectangle (Games) reaches a height of 7, while the second rectangle reaches only to 5. Note that the rectangles are of equal width, and none of them touch each other. Also notice that the two bar graphs are exactly alike, except for the scale indicated on the vertical axis. This is because we divide each frequency by the same number, the sample size, to get the relative frequency.

A Pareto chart is a bar graph in which the rectangles are presented in decreasing order from left to right.

A pie chart is a circle divided into sections (that is, slices or wedges), with each section representing a particular category. The size of the section is proportional to the relative frequency of the category.

EXAMPLE 4 Constructing a pie chart

Construct a pie chart for the app type data from Example 2.

Solution

The relative frequencies from Example 2 are shown in Table 5. We multiply each relative frequency by 360° to get the number of degrees for that section (slice) of the pie chart.

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Our pie chart will have five slices—one for each app type category. Use the compass to draw a circle. Then, use the protractor to construct the appropriate angles for each section. From the center of the circle, draw a line to the top of the circle. Measure your first angle using this line. For the Games app type, we need an angle of 126°. This angle is shown in Figure 3. Then, from there, measure your second angle—in this case, the 90° right angle for Social networking apps. Continue until your circle is complete.

Steps for Constructing a Crosstabulation

EXAMPLE 5 Constructing a crosstabulation

Table 6 contains information about the size (compact, midsize, or large) and the recommended gasoline (regular or premium) for a sample of ten 2014 automobiles.

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Solution

EXAMPLE 6 Working with tabular data

Table 8 is a crosstabulation showing the SAT Writing exam score for females and males in 2014. Note that the data are not presented in raw form, such as the automobiles in Table 6. Instead, the frequencies for each cell have been presented. Use Table 8 to construct the following:

Solution

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We work with the tabular data in Table 8 to construct further graphs and distributions in the exercises.

Use Table 8 to construct the following:

(The solutions are shown in Appendix A.)

EXAMPLE 7 Clustered bar graphs

Use the tabular data in Table 8 to construct a clustered bar chart of SAT Writing exam scores clustered by gender.

Solution

Gender is given as the clustering variable. Thus, for each category of the variable SAT Writing exam score, we will draw two bars: one representing females and the other representing males. For the first category, Low = SAT score between 200 and 390, we draw the female rectangle going up to 169.314 (because Figure 5 is given in 1000s), and we draw the male rectangle going up to 178.606. These two rectangles should touch each other but should not touch any other rectangles. Continue to draw two rectangles for each SAT score category: one for each of the females' and males' frequencies. The resulting clustered bar graph is shown here as Figure 5. We say that the Writing exam scores are clustered by gender. (Note: .)

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EXAMPLE 8 Comparison bar charts

Student-Run Café Business

Have you ever thought what it must be like to run a business? Business students at a Midwestern public university found out because they volunteered to manage a Student-Run café in the business school, which replaced a for-profit vendor that closed. In this chapter, we examine sales data from the Student-Run café, such as the number of coffees sold and the number of sodas sold, to help us learn about how to describe data using graphs and tables. The data set is called Café, a brief excerpt of which is shown in Table 10.

There are 47 days of sales data. The variable time period is a categorical variable that equals 1 for the first 16 days (late January to early February), 2 for the second 16 days (late February to early March), and 3 for the last 15 days (late March to early April). This variable allows us to examine changes in sales behavior over time. Construct a comparison bar chart of the number of sodas sold and the number of coffees sold, over the three time periods, and comment on the change in customer purchasing behavior as the winter months turned to spring.

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Solution

Figure 6 shows the comparison bar chart of the number of sodas sold and the number of coffees sold, over the three time periods. As winter turned to spring, soda sales increased from 332 to 576, while coffee sales decreased from 445 to 131.