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Chapter

Displaying Descriptive Statistics

CHAPTER 2 MAP

2.1 The Role Technology Plays in Statistics

2.2 Displaying Quantitative Data

2.3 Displaying Qualitative Data

2.4 Contingency Tables

2.5 Stem and Leaf Display

2.6 Scatter Plots

2.1 The Role Technology Plays
in Statistics

Microsoft Excel has built-in options for data presentation and statistical analysis

You may need to activate Excel’s Analysis Tool Pak Add-in to see these options

Statistical Analysis Using Excel 2013

Open Excel 2013, then click on the File tab
Click Options shown in the drop down menu. This will open the Excel Options dialog box
Select Add-Ins in the left margin…

Statistical Analysis Using Excel 2010

Click on Go at the bottom of the screen

Select the check boxes for Analysis ToolPak and Analysis ToolPak - VBA in the popup menu and click OK

Statistical Analysis Using Excel 2013

Select the Data tab. Click on Data Analysis on the right side of the application bar

The Data Analysis pop-up menu should appear in the spreadsheet

Installing PHStat

PHStat is an Excel Add-in developed by Prentice Hall to provide students with additional features for statistical analysis

The software will be referred to throughout the book and is available from the book’s website: www.pearsonhighered.com/donnelly
To install PHStat on your Windows PC, follow the instructions on the book’s website
Mac users can also find instructions for PHStat on the book’s website

2.2 Displaying Quantitative Data

Recall the types of data from Chapter 1:

Quantitative

Qualitative

Types of Data

Displaying qualitative data is discussed in section 2.3

Displaying quantitative data is discussed in section 2.2

Constructing a Frequency Distribution

A frequency distribution shows the number of data observations that fall into specific intervals

Graphically summarize information not readily observable by merely looking at data in a table

Constructing a Frequency Distribution

Example: Number of iPads sold per day

Discrete vs. Continuous Data

Discrete data are values based on observations that can be counted and are typically represented by whole numbers

represent something that has been counted
take on whole numbers such as 0, 1, 2, 3

Continuous data are values that can take on any real numbers, including numbers that contain decimal points

usually measured rather than counted
Examples are weight, time, and distance

Discrete vs. Continuous Data

Examples of Discrete data

Number of children per family
Number of cars listed per insurance policy
Vacation days per month

Examples of Continuous data

Time required to read chapter 2
Thickness of paint applied to a car body
Voltage of batteries produced in August

Relative Frequency Distributions

Relative frequency distributions display the proportion of observations of each class relative to the total number of observations

shows the fraction of observations in each class
found by dividing each frequency by the total number of observations
the fractions in a relative frequency distribution add up to 1.00

Relative Frequency Distributions

Two iPads were sold on 28% of the days

Example:

Cumulative Relative Frequency Distributions

A cumulative relative frequency distribution totals the proportion of observations that are less than or equal to the class at which you are looking

Shows the accumulated proportion as values vary from low to high

Cumulative Relative Frequency Distributions

Example:

Three iPads or less were sold on 80% of the business days

Using a Histogram to Graph a Frequency Distribution

A histogram is a graph showing the number of observations in each class of a frequency distribution

Excel uses the term “bins” for the classes in the distribution

Constructing a Histogram in Excel

Select the Data tab, and click on Data Analysis in the upper right corner

In the pop-up menu, select Histogram and click OK…

Constructing a Histogram in Excel

In the Input Range text box, highlight the desired data

In the Bin Range text box, highlight the bin values (create bins if not already created before step 1)

For Output options, select New Worksheet Ply and Chart Output

Click OK

Histograms in Excel

7. Customize the Excel graph to make it more attractive

8. Stretch size to better proportion

9. Eliminate “more” bin
10. Modify the graph and axis labels

11. Remove the redundant “Frequency” legend

The Shape of Histograms

Symmetric

the right side is the mirror image of the left side of the distribution

Still symmetric, but wider spread

Not symmetric

Constructing a Frequency Distribution Using Grouped Quantitative Data

Ideally, the number of classes in a frequency distribution should be between 4 and 20

Some data sets, particularly those with continuous data, require several values to be grouped together in a single class
This grouping prevents having too many classes in the frequency distribution, which can make it difficult to detect patterns

Number of Classes

One method to determine the number of classes in a frequency distribution is the rule

2k  n

where k = Number of classes

n = Number of data points

Find the lowest value of k that satisfies the rule

Suppose n = 50

25 = 32 < 50 (k = 5 is too small)

26 = 64 > 50 (k = 6 is a good choice)

Class Width

Once k is known, the width of each class can be found

The width is the range of numbers to put into each class

Round this estimate to a useful whole number that makes the frequency distribution more readable

Class Width

There is no one correct answer for the class width

The goal is to create a histogram to clearly and usefully show the pattern in the data
Often there is more than one acceptable way to accomplish this

Class Boundaries

Class boundaries represent the minimum and maximum values for each class

Choose class boundaries that are easy to read

 

3 to less than 6 minutes 3.21 to less than 6.21 minutes

6 to less than 9 minutes vs. 6.21 to less than 9.21 minutes

9 to less than 12 minutes 9.21 to less than 12.21 minutes

Class Frequencies

Find class frequencies by counting and recording the number of observations in each class

this is easier when the data are sorted

Example:

Rules for Classes for Grouped Data

Equal-size classes. All classes in the frequency distribution must be of equal width

Mutually exclusive classes. Class boundaries cannot overlap

Include all data values. Make sure all data values are accounted for in the total row of the frequency distribution

Avoid empty classes. It is undesirable for a histogram to display a class so narrow that there are no observations in it

Avoid open-ended classes (if possible). These violate the first rule of equal class sizes

Constructing a Histogram with
Grouped Quantitative Data

For grouped data, the bins in Excel are the upper boundary for each class

For continuous data, remove the gaps between the bars in the histogram:

Right-click on any histogram bar to get a pop-up menu

Left-click on Format Data Series

In the dialog box, move the Gap Width slide all the way to the left

Close the Format Data Series dialog box

Constructing a Histogram with
Grouped Quantitative Data

Additional formatting issues:

Use a descriptive title for the graph
Use descriptive labels for the axes
Remove the redundant “Frequency” legend
Remove gaps between bars

The Consequences of
Too Few or Too Many Classes

Wide classes results in few class intervals

Can obscure important patterns
Gives a “blocky” distribution graph
Summarizes the data too much
Tells us little about the true

distribution shape

Too many narrow classes in a histogram also

has consequences

Results in a “jagged” histogram
Some classes may be empty
Does not summarize the data enough

Are They Discrete or Continuous Data?

Some data are technically discrete (counted, not measured) but are displayed in a continuous format

Examples

Age
Income
Other discrete data sets containing a wide range of values

The Polygon

A percentage polygon graphs the midpoint of each class as a line rather than a column

The height of each midpoint represents the relative frequency of the corresponding class
Used to compare the shape of two or more distributions on one graph

The cumulative percentage polygon, or ogive, is a line graph that plots the cumulative relative frequency distribution

The Polygon

Percentage polygons and cumulative percentage polygons can be created using PHStat

The Polygon

2.3 Displaying Qualitative Data

Qualitative data are values that are categorical

Can be nominal or ordinal measurement level
Describe a characteristic, such as gender or level of education

Frequency distributions help display qualitative data by indicating the number of occurrences of various categories

Can use Excel’s COUNTIF function to count the number of values matching a category label

Displaying Qualitative Data

Figure 2.15 A-B |

Excel’s COUNTIF Function

Excel’s COUNTIF

Function Results

Bar Charts

Bar charts are a good tool for displaying qualitative data that have been organized in categories

Can be arranged in a vertical or horizontal orientation

Bar Charts

Horizontal bar chart Vertical bar chart

Can display multiple series with clustered bar charts or stacked bar charts:

Displaying Qualitative Data: Example

Pareto Charts

Pareto charts are bar charts that show the frequency of the categories that cause quality control problems

Show quality problem categories in decreasing order

The most problematic categories are shown first

Pareto charts also plot the cumulative relative frequency as a line on the chart known as an ogive

Pareto Charts

Note: The categories are arranged from most frequent to least frequent

Follow the steps shown in the text, pages 49-50, to create a Pareto chart and ogive using Excel

Pie Charts

Pie charts are another excellent tool for comparing proportions for categorical data

Each segment of the pie represents the relative frequency of one category

All categories in the data set must be included in the pie
Use a pie chart to compare the relative sizes of all possible categories
Bar charts are more useful when you want to highlight the actual data values and when the classes combined don’t form a whole

Pie Charts

Constructing a Pie Chart in Excel

Figure 2.19A |

Pie Charts

Constructing a Pie Chart in Excel

(continued)

Figure 2.19B |

Pie Charts

Example:

2.4 Contingency Tables

Contingency tables provide a format to display observations that have more than one value associated with them

Use rows and columns for separate variables to summarize the data efficiently

Contingency Tables

7 females out of 20 customers paid using credit, 7/20 = 0.35

Contingency Table

Relative Contingency Table

Constructing a Contingency Table in Excel

Click on any cell within your data

Choose the Insert tab

Click on the Pivot Table icon

Click on Pivot Table in the drop-down menu

A Create Pivot Table dialog box will appear. Click OK…

Constructing a Contingency Table in Excel

A new worksheet will be created for your pivot table

From the Pivot Table Field List,

c. Drag the variable name to be summarized down into the Values box

a & b. Drag the desired variable names down into the Column or Row Labels boxes, as desired

Constructing a Contingency Table in Excel

Resulting pivot table:

Creating a Pivot Table in Excel (Final Result)

Figure 2.21C |

2.5 Stem and Leaf Display

A stem and leaf display splits the data values into stems (the larger place values) and leaves (the smaller place value)

By listing all of the leaves to the right of each stem, we can graphically describe how the data are distributed

All the original data points are visible on the display
Easy to construct by hand
Provides a histogram-like view of the distribution

Stem and Leaf Display

For this example, use the 10’s digit as the stem

Use the 1’s digit as the leaf

7 | 8

8 | 0

Stem and Leaf Display

Sort the data from lowest to highest

Determine the unique stem values

7, 8, 9 are the different stem values in this example

List the stems in a vertical column and then add the leaf values to the right of the appropriate stem, in ascending order

7 | 8 8 9 9 9

8 | 0 0 0 0 1 1 2 3 3 4 4 4 5 6 7 8

9 | 0 2 5

Stem and Leaf Display

To get more detail the stems can be split in half

7(5) | 8 8 9 9 9

8(0) | 0 0 0 0 1 1 2 3 3 4 4 4

8(5) | 5 6 7 8

9(0) | 0 2

9(5) | 5

The stem labeled 7(5) stores all the scores between 75 and 79
The stem 8(0) stores all the scores between 80 and 84

2.6 Scatter Plots

Scatter plots provide a picture of the relationship between two data points that are paired together

The dependent variable, which is placed on the vertical axis of the scatter plot, is influenced by changes in the independent variable, which is placed on the horizontal axis

Scatter Plots

Dependent variable

(y-axis)

Independent variable (x-axis)

Scatter Plots

Constructing a Scatter Plot in Excel

Figure 2.25A |

Line Charts

A line chart is a scatter plot in which the data points in the scatter plot are connected with line segments

Often used with time series data

When graphing a time series the convention is to place the time data on the horizontal axis

Line Charts

Constructing a Line Chart in Excel

Figure 2.26A |

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Printed in the United States of America.

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