THE NATURE OF STATISTICS

Copyright © Cengage Learning. All rights reserved.

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Copyright © Cengage Learning. All rights reserved.

14.1Frequency Distributions

and Graphs

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Frequency Distribution

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Frequency Distribution

The difference between the lower limit of one class and the lower limit of the next class is called the interval of the class. After determining the number of values within a class, termed the frequency.

The end result of this classification and tabulation is called a frequency distribution.

For example, suppose that you roll a pair of dice 50 times and obtain these outcomes:3, 2, 6, 5, 3, 8, 8, 7, 10, 9, 7, 5, 12, 9, 6, 11, 8, 11, 11, 8, 7,

7, 7, 10, 11, 6, 4, 8, 8, 7, 6, 4, 10, 7, 9, 7, 9, 6, 6, 9, 4, 4, 6,

3, 4, 10, 6, 9, 6, 11

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Frequency Distribution

We can organize these data in a convenient way by using a frequency distribution, as shown in Table 14.1.

Table 14.1

Frequency Distribution for 50 Rolls of a Pair of Dice

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Example 1 – Make a frequency diagram

Make a frequency distribution for the high temperatures (in °F) for October on Lake Erie as summarized in Table 14.2.

Table 14.2

High Temperatures on Lake Eriefor the Month of October

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Example 1 – Solution

First, notice that the included temperatures range from a low of 61 to a high of 86.

If we were to list each temperature and make tally marks for each, we would have a long table with few entries for each temperature.

Instead, we arbitrarily divide the temperatures into eight categories. Make the tallies and total the number in each group.

The result is called a grouped frequency distribution.

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Frequency Distribution

A graphical method useful for organizing large sets of data is a stem-and-leaf plot. Consider the data shown in Table 14.3.

Table 14.3

Best Actors, 1928–2009

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Frequency Distribution

Table 14.3

Best Actors, 1928–2009

cont’d

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Example 2 – Make a stem-and-leaf plot

Construct a stem-and-leaf plot for the best actor ages in Table 14.3.

Solution:

Stem-and-leaf plot of ages of best actor, 1928–2009.

This plot is useful because it is easy to see that most best actor winners received the award in their forties.

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Frequency Distribution

To help us understand the relationship between and among variables, we use a diagram called a graph.

In this section, we consider bar graphs, line graphs, circle graphs, and pictographs.

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Bar Graphs

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Bar Graphs

A bar graph compares several related pieces of data using horizontal or vertical bars of uniform width. There must be some sort of scale or measurement on both the horizontal and vertical axes.

An example of a bar graph is shown in Figure 14.1, which shows the data from Table 14.1.

Figure 14.1

Outcomes of experiment of rolling a pair of dice

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Example 3 – Make a bar graph

Construct a bar graph for the Lake Erie temperatures given in Example 1. Use the grouped categories.

Solution:To construct a bar graph, drawand label the horizontal andvertical axes, as shownin part a of Figure 14.2.

Draw and label axes and scales.

Bar graph for Lake Erie temperaturesFigure 14.2(a)

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Example 3 – Solution

It is helpful (although not necessary) to use graph paper.Next, draw marks indicating the frequency, as shown inpart b of Figure 14.2. Finally, complete the bars and shadethem as shown in part c.

cont’d

Bar graph for Lake Erie temperaturesFigure 14.2(b)

Mark the frequency levels.

Figure 14.2(c)

Bar graph for Lake Erie temperaturesComplete and shade the bars.

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Example 4 – Read a bar graph

Refer to Figure 14.3 to answer the following questions.

Figure 14.3

Share of U.S. employment in agriculture, manufacturing, and services from 1849 to 2049

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Example 4 – Read a bar graph

a. What was the share of U.S. employment in manufacturing in 1999?

b. In which year(s) was U.S. employment in services approximately equal to employment in manufacturing?

c. From the graph, form a conclusion about U.S. employment in agriculture for the period 1849–2049.

cont’d

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Example 4 – Solution

a. We see the bar representing manufacturing in 1999 is approximately 20%. (Use a straightedge for help.)

b. Employments in services and manufacturing were approximately equal in 1949.

c. Conclusions, of course, might vary, but one obvious conclusion is that there has been a dramatic decline of employment in agriculture in the United States over this period of time.

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Line Graphs

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Line Graphs

A graph that uses a broken line to illustrate how one quantity changes with respect to another is called aline graph.

A line graph is one of the most widely used kinds of graph.

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Example 5 – Make a line graph

Draw a line graph for the data given in Example 1. Use the previously grouped categories.

Solution

The line graph uses points instead of bars to designate the locations of the frequencies. These points are then connected by line segments.

To plot the points, use the frequency distribution to find the midpoint of each category (62 is the midpoint of the 60–64 category, for example); then plot a point showing the frequency (3, in this example).

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Example 5 – Solution

This step is shown in part a of Figure 14.4. The last step is to connect the dots with line segments, as shown in part b.

cont’d

Figure 14.4

Constructing a line graph for Lake Erie temperatures

a. Plot the points to represent frequency levels. b. Connect the dots with line segments.

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Circle Graphs

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Circle Graphs

Another type of commonly used graph is the circle graph, also known as a pie chart.

To create a circle graph, first express the number in each category as a percentage of the total. Then convert this percentage to an angle in a circle.

Remember that a circle is divided into 360°, so we multiply the percent by 360 to find the number of degrees for each category.

You can use a protractor to construct a circle graph, as shown in the next example.

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Example 7 – Make a circle graph

The 2010 expenses for Karlin Enterprises are shown in Figure 14.6. Construct a circle graph showing the expenses for Karlin Enterprises.

Figure 14.6

Karlin Enterprises expenses

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Example 7 – Solution

The first step in constructing a circle graph is to write the ratio of each entry to the total of the entries, as a percent.

This is done by finding the total ($120,000) and then dividing each entry by that total:

Salaries: Rents:

Utilities: Advertising:

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Example 7 – Solution

Shrinkage: Depreciation:

Materials/supplies:

cont’d

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Example 7 – Solution

A circle has 360°, so the next step is to multiply eachpercent by 360°:

Salaries: 360° 0.60 = 216°

Advertising: 360° 0.10 = 36°

Rents: 360° 0.20 = 72°

Shrinkage: 360° 0.01 = 3.6°

Materials/supplies: 360° 0.01 = 3.6°

Utilities: 360° 0.05 = 18°

Depreciation: 360° 0.03 = 10.8°

cont’d

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Example 7 – Solution

Finally, use a protractor to construct the circle graph as shown in Figure 14.7.

cont’d

Figure 14.7

Circle graph showing the expenses for Karlin Enterprises

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Pictographs

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Pictographs

A pictograph is a representation of data that uses pictures to show quantity. Consider the raw data shown in Table 14.4.

A pictograph uses a picture to illustrate data; it is normally used only in popular publications, rather than for scientific applications.

Table 14.4

Marital Status (for persons age 65 and older)

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Pictographs

For the data in Table 14.4, suppose that we draw pictures of a woman and a man so that each picture represents 1 million persons, as shown in Figure 14.8.

Figure 14.8

Pictograph showing the marital status of persons age 65 and older

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Misuses of Graphs

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Misuses of Graphs

The most misused type of graph is the pictograph. Consider the data from Table 14.4.

Such data can be used to determine the height of a three-dimensional object.

Table 14.4

Marital Status (for persons age 65 and older)

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Misuses of Graphs

When an object (such as a person) is viewed as three-dimensional, differences seem much larger than they actually are. Look at the Figure 14.9 and notice that, as the height and width are doubled, the volume is actually increased eightfold.

Doubling height and width increases volume 8-fold

Figure 14.9Examples of misuses in pictographs

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Misuses of Graphs

This pictograph fallacy carries over to graphs of all kinds,especially since software programs easily change twodimensional scales to three-dimensional scales.

For example, percentages are represented as heights on the scale shown in Figure 14.10a, but the software has incorrectly drawn the graph as three-dimensional bars.

Heights are used to graph three-dimensional objects.

Figure 14.10(a)

Misuse of graphs

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Misuses of Graphs

Another fallacy is to choose the scales to exaggerate or diminish real differences. Even worse, graphs are sometimes presented with no scale whatsoever, as illustrated in a graphical “comparison” between Anacin and “regular strength aspirin” shown in Figure 14.10b.

No scale is shown in this Anacin advertisement

Figure 14.10(b)

Misuse of graphs