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Chapter 2Descriptive Statistics:Tabular and Graphical Methods

- Summarizing Qualitative Data
- Summarizing Quantitative Data
- Exploratory Data Analysis
- Crosstabulations

and Scatter Diagrams

Summarizing Qualitative Data

- Frequency Distribution
- Relative Frequency
- Percent Frequency Distribution
- Bar Graphs and Pie Charts

Frequency Distribution

- A frequency distribution is a tabular summary of data showing the frequency (or number) of items in each of several nonoverlapping classes.
- The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data.

Example: Marada Inn

Guests staying at Marada Inn were asked to rate the

quality of their accommodations as being excellent,

above average, average, below average, or poor. The

ratings provided by a sample of 20 quests are shown

below.

Below Average Average Above Average

Above Average Above Average Above Average Above Average Below Average Below Average Average Poor Poor

Above Average Excellent Above Average

Average Above Average Average

Above Average Average

Example: Marada Inn

- Frequency Distribution

RatingFrequency

Poor 2

Below Average 3

Average 5

Above Average 9

Excellent 1

Total 20

Using Excel’s COUNTIF Functionto Construct a Frequency Distribution

- Formula Worksheet

Note: Rows 9-21 are not shown.

Using Excel’s COUNTIF Functionto Construct a Frequency Distribution

- Value Worksheet

Note: Rows 9-21 are not shown.

Relative Frequency Distribution

- The relative frequency of a class is the fraction or proportion of the total number of data items belonging to the class.
- A relative frequency distribution is a tabular summary of a set of data showing the relative frequency for each class.

Percent Frequency Distribution

- The percent frequency of a class is the relative frequency multiplied by 100.
- Apercent frequency distribution is a tabular summary of a set of data showing the percent frequency for each class.

Example: Marada Inn

- Relative Frequency and Percent Frequency Distributions

RelativePercent

RatingFrequencyFrequency

Poor .10 10

Below Average .15 15

Average .25 25

Above Average .45 45

Excellent .05 5

Total 1.00 100

Using Excel to Construct Relative Frequency and Percent Frequency Distributions

- Formula Worksheet

Note: Columns A-B and rows 9-21 and are not shown.

Using Excel to Construct Relative Frequency and Percent Frequency Distributions

- Value Worksheet

Note: Columns A-B and rows 9-21 and are not shown.

Bar Graph

- A bar graph is a graphical device for depicting qualitative data that have been summarized in a frequency, relative frequency, or percent frequency distribution.
- On the horizontal axis we specify the labels that are used for each of the classes.
- A frequency, relative frequency, or percent frequency scale can be used for the vertical axis.
- Using a bar of fixed width drawn above each class label, we extend the height appropriately.
- The bars are separated to emphasize the fact that each class is a separate category.

Using Excel’s Chart Wizardto Construct Bar Graphs

Step 1 Select cells C1:D6

Step 2 Select the Chart Wizard button

Step 3 When the Chart Type dialog box appears:

Choose Column in the Chart type list

Choose Clustered Column from the Chart

sub-type display

Select Next >

Step 4 When the Chart Source Data dialog box appears

Select Next >

… continued

Using Excel’s Chart Wizardto Construct Bar Graphs

Step 5 When the Chart Options dialog box appears:

Select the Titles tab and then

Type Customers’ Quality Ratings in the

Chart title box

Enter QualityRating in the Value (X) axis box

Enter Frequency in the Value (Y) axis box

Select the Legend tab and then

Remove the check in the Show Legend box

Select Next >

… continued

Using Excel’s Chart Wizardto Construct Bar Graphs

Step 6 When the Chart Location dialog box appears:

Specify the location for the new chart

Select Finish to display the bar graph

Pie Chart

- The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data.
- First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class.
- Since there are 360 degrees in a circle, a class with a relative frequency of .25 would consume .25(360) =

90 degrees of the circle.

Using Excel’s Chart Wizardto Construct Pie Charts

Step 1 Select cells C2:C6 and F2:F6

Step 2 Select the Chart Wizard button

Step 3 When the Chart Type dialog box appears:

Choose Pie in the Chart type list

Choose Pie from the Chart sub-type display

Select Next >

Step 4 When the Chart Source Data dialog box appears

Select Next >

… continued

Using Excel’s Chart Wizardto Construct Pie Charts

Step 5 When the Chart Options dialog box appears:

Select the Titles tab and then

Type Customers’ Quality Ratingsat Marada in the Chart title box

Select the Legend tab and then

Remove the check in the Show Legend box

Select the Data Labels tab and then

Select Show Label and percent

Select Show leader lines

Select Next >

… continued

Using Excel’s Chart Wizardto Construct Pie Charts

Step 6 When the Chart Location dialog box appears:

Specify the location for the new chart

Select Finish to display the pie chart

Example: Marada Inn

- Insights Gained from the Preceding Pie Chart
- One-half of the customers surveyed gave Marada a quality rating of “above average” or “excellent” (looking at the left side of the pie). This might please the manager.
- For each customer who gave an “excellent” rating, there were two customers who gave a “poor” rating (looking at the top of the pie). This should displease the manager.

Summarizing Quantitative Data

- Frequency Distribution
- Relative Frequency and Percent Frequency Distributions
- Dot Plot
- Histogram
- Cumulative Distributions
- Ogive

Example: Hudson Auto Repair

The manager of Hudson Auto would like to get a

better picture of the distribution of costs for engine

tune-up parts. A sample of 50 customer invoices has

been taken and the costs of parts, rounded to the

nearest dollar, are listed below.

Frequency Distribution

- Guidelines for Selecting Number of Classes
- Use between 5 and 20 classes.
- Data sets with a larger number of elements usually require a larger number of classes.
- Smaller data sets usually require fewer classes.

Frequency Distribution

- Guidelines for Selecting Width of Classes
- Use classes of equal width.
- Approximate Class Width =

Example: Hudson Auto Repair

- Frequency Distribution

If we choose six classes:

Approximate Class Width = (109 - 52)/6 = 9.5 10

Cost ($)Frequency

50-59 2

60-69 13

70-79 16

80-89 7

90-99 7

100-109 5

Total 50

Using Excel’s FREQUENCY Functionto Construct a Frequency Distribution

- Formula Worksheet (showing data entered)

Note: Rows 9-51 are not shown.

Using Excel’s FREQUENCY Functionto Construct a Frequency Distribution

- The FREQUENCY function is not a “simple” Excel function.
- FREQUENCY is capable of providing multiple values.
- In Excel, a formula that can return multiple values is called an array formula.
- An array formula must be entered in a special way.

Using Excel’s FREQUENCY Functionto Construct a Frequency Distribution

- Entering the Necessary Array Formula

Step 1 Select D2:D7 (where the frequencies will

appear)

Step 2 Type the following formula:

=FREQUENCY(A2:A51,{59,69,79,89,99,109})

Step 3 Hold down CTRL and SHIFT keys while pressing ENTER key

(Array formula will be entered in D2:D7)

Using Excel’s FREQUENCY Functionto Construct a Frequency Distribution

- Value Worksheet

Note: Rows 9-51 are not shown.

Example: Hudson Auto Repair

- Relative Frequency and Percent Frequency Distributions

Relative Percent

Cost ($)FrequencyFrequency

50-59 .04 4

60-69 .26 26

70-79 .32 32

80-89 .14 14

90-99 .14 14

100-109 .1010

Total 1.00 100

Example: Hudson Auto Repair

- Insights Gained from the Percent Frequency Distribution
- Only 4% of the parts costs are in the $50-59 class.
- 30% of the parts costs are under $70.
- The greatest percentage (32% or almost one-third) of the parts costs are in the $70-79 class.
- 10% of the parts costs are $100 or more.

Histogram

- Another common graphical presentation of quantitative data is a histogram.
- The variable of interest is placed on the horizontal axis and the frequency, relative frequency, or percent frequency is placed on the vertical axis.
- A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percent frequency.
- Unlike a bar graph, a histogram has no natural separation between rectangles of adjacent classes.

Using Excel’s Chart Wizardto Construct a Histogram

Step 1 Select cells C1:D7

Step 2 Select the Chart Wizard button

Step 3 When the Chart Type dialog box appears:

Choose Column in the Chart type list

Choose Clustered Column from the Chart

sub-type display

Select Next >

Step 4 When the Chart Source Data dialog box appears

Select Next >

… continued

Using Excel’s Chart Wizardto Construct a Histogram

Step 5 When the Chart Options dialog box appears:

Select the Titles tab and then

Type Histogram for Parts Cost Data in the

Chart title box

Enter Parts Cost ($) in the Value (X) axis box

Enter Frequency in the Value (Y) axis box

Select the Legend tab and then

Remove the check in the Show Legend box

Select Next >

… continued

Using Excel’s Chart Wizardto Construct a Histogram

Step 6 When the Chart Location dialog box appears:

Specify the location for the new chart

Select Finish to display the histogram

Using Excel’s Chart Wizardto Construct a Histogram

- Eliminating Gaps Between Rectangles

Step 1 Right click on any rectangle in the column chart

Step 2 Select the Format Data Series option

Step 3 When the Format Data Series Option dialog box appears:

Select the Options tab and then

Enter 0 in the Gap width box

Click OK

Cumulative Distribution

- The cumulative frequency distribution shows the number of items with values less than or equal to the upper limit of each class.
- The cumulative relative frequency distribution shows the proportion of items with values less than or equal to the upper limit of each class.
- The cumulative percent frequency distribution shows the percentage of items with values less than or equal to the upper limit of each class.

Example: Hudson Auto Repair

- Cumulative Distributions

Cumulative Cumulative

Cumulative Relative Percent

Cost ($)FrequencyFrequencyFrequency

< 59 2 .04 4

< 69 15 .30 30

< 79 31 .62 62

< 89 38 .76 76

< 99 45 .90 90

< 109 50 1.00 100

Ogive

- An ogive is a graph of a cumulative distribution.
- The data values are shown on the horizontal axis.
- Shown on the vertical axis are the:
- cumulative frequencies, or
- cumulative relative frequencies, or
- cumulative percent frequencies
- The frequency (one of the above) of each class is plotted as a point.
- The plotted points are connected by straight lines.

Example: Hudson Auto Repair

- Ogive
- Because the class limits for the parts-cost data are 50-59, 60-69, and so on, there appear to be one-unit gaps from 59 to 60, 69 to 70, and so on.
- These gaps are eliminated by plotting points halfway between the class limits.
- Thus, 59.5 is used for the 50-59 class, 69.5 is used for the 60-69 class, and so on.

Example: Hudson Auto Repair

- Ogive with Cumulative Percent Frequencies

100

80

60

Cumulative Percent Frequency

40

20

Parts

Cost ($)

50 60 70 80 90 100 110

Exploratory Data Analysis

- The techniques of exploratory data analysis consist of simple arithmetic and easy-to-draw pictures that can be used to summarize data quickly.
- One such technique is the stem-and-leaf display.

Stem-and-Leaf Display

- A stem-and-leaf display shows both the rank order and shape of the distribution of the data.
- It is similar to a histogram on its side, but it has the advantage of showing the actual data values.
- The first digits of each data item are arranged to the left of a vertical line.
- To the right of the vertical line we record the last digit for each item in rank order.
- Each line in the display is referred to as a stem.
- Each digit on a stem is a leaf.

Example: Hudson Auto Repair

- Stem-and-Leaf Display

5 2 7

6 2 2 2 2 5 6 7 8 8 8 9 9 9

7 1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9

8 0 0 2 3 5 8 9

9 1 3 7 7 7 8 9

10 1 4 5 5 9

Stretched Stem-and-Leaf Display

- If we believe the original stem-and-leaf display has condensed the data too much, we can stretch the display by using two more stems for each leading digit(s).
- Whenever a stem value is stated twice, the first value corresponds to leaf values of 0-4, and the second values corresponds to values of 5-9.

Example: Hudson Auto Repair

- Stretched Stem-and-Leaf Display

5 2

5 7

6 2 2 2 2

6 5 6 7 8 8 8 9 9 9

7 1 1 2 2 3 4 4

7 5 5 5 6 7 8 9 9 9

8 0 0 2 3

8 5 8 9

9 1 3

9 7 7 7 8 9

10 1 4

10 5 5 9

Stem-and-Leaf Display

- Leaf Units
- A single digit is used to define each leaf.
- In the preceding example, the leaf unit was 1.
- Leaf units may be 100, 10, 1, 0.1, and so on.
- Where the leaf unit is not shown, it is assumed to equal 1.

Example: Leaf Unit = 0.1

If we have data with values such as

8.6 11.7 9.4 9.1 10.2 11.0 8.8

a stem-and-leaf display of these data will be

Leaf Unit = 0.1

8 6 8

9 1 4

10 2

11 0 7

Example: Leaf Unit = 10

If we have data with values such as

1806 1717 1974 1791 1682 1910 1838

a stem-and-leaf display of these data will be

Leaf Unit = 10

16 8

17 1 9

18 0 3

19 1 7

Crosstabulations and Scatter Diagrams

- Thus far we have focused on methods that are used to summarize the data for one variable at a time.
- Often a manager is interested in tabular and graphical methods that will help understand the relationship between two variables.
- Crosstabulation and a scatter diagram are two methods for summarizing the data for two (or more) variables simultaneously.

Crosstabulation

- Crosstabulation is a tabular method for summarizing the data for two variables simultaneously.
- Crosstabulation can be used when:
- One variable is qualitative and the other is quantitative
- Both variables are qualitative
- Both variables are quantitative
- The left and top margin labels define the classes for the two variables.

Example: Finger Lakes Homes

- Crosstabulation

The number of Finger Lakes homes sold for each style and price for the past two years is shown below.

PriceHome Style

RangeColonial Ranch Split A-Frame Total

< $99,000 18 6 19 12 55

> $99,000 12 14 16 3 45

Total30 20 35 15 100

Using Excel’s PivotTable Reportto Construct a Crosstabulation

- Formula Worksheet (showing data entered)

Note: Rows 10-101 are not shown.

Using Excel’s PivotTable Reportto Construct a Crosstabulation

- Changing the Default Order for the PivotTable Report

Step 1 Select the Tools pull-down menu

Step 2 Choose Options

Step 3 When the Options dialog box appears:

Select the Custom lists tab

In the List entries: box, type <= 99K and press Enter, and type > 99K Select Add

Click OK

Using Excel’s PivotTable Reportto Construct a Crosstabulation

- Using the PivotTable Report

Step 1 Select the Data pull-down menu

Step 2 Choose the PivotTable and PivotChart Report

Step 3 When the PivotTable and PivotChart Wizard Step 1 of 3 dialog box appears:

Choose Microsoft Excel list or database

Choose PivotTable

Select Next >

Step 4 When the PivotTable and PivotChart Wizard Step 2 of 3 dialog box appears:

Enter A1:C101 in the Range box

Select Next >

Using Excel’s PivotTable Reportto Construct a Crosstabulation

- Using the PivotTable Report

Step 5 When the PivotTable and PivotChart Wizard Step 3 of 3 dialog box appears:

Select New Worksheet

Click on the Layout button

When the PivotTable and PivotChart Wizard – Layout diagram appears:

Drag the Price ($) field button to the ROW section of the diagram

Drag the Style field button to the COLUMN section of the diagram

Using Excel’s PivotTable Reportto Construct a Crosstabulation

- Using the PivotTable Report (Step 5 continued)

Drag the Home field button to the DATA section of the diagram

Double click the Sum of Home field button in the data section

When the PivotTable Field dialog box appears:

Choose Count under Summarized by:

Click OK

Click OK

When the PivotTable and PivotChart Wizard Step 3 of 3 dialog box reappears:

Select Finish >

Using Excel’s PivotTable Reportto Construct a Crosstabulation

- Value Worksheet

Example: Finger Lakes Homes

- Insights Gained from the Preceding Crosstabulation
- The greatest number of homes in the sample (19) are a split-level style and priced at less than or equal to $99,000.
- Only three homes in the sample are an A-Frame style and priced at more than $99,000.

Crosstabulation: Row or Column Percentages

- Converting the entries in the table into row percentages or column percentages can provide additional insight about the relationship between the two variables.

Example: Finger Lakes Homes

- Row Percentages

PriceHome Style

RangeColonial Ranch Split A-Frame Total

< $99,000 32.73 10.91 34.55 21.82 100

> $99,000 26.67 31.11 35.56 6.67 100

Note: Row totals are actually 100.01 due to rounding.

Example: Finger Lakes Homes

- Column Percentages

PriceHome Style

RangeColonial Ranch Split A-Frame

< $99,000 60.00 30.00 54.29 80.00

> $99,000 40.00 70.00 45.71 20.00

Total 100 100 100 100

Scatter Diagram

- A scatter diagram is a graphical presentation of the relationship between two quantitative variables.
- One variable is shown on the horizontal axis and the other variable is shown on the vertical axis.
- The general pattern of the plotted points suggests the overall relationship between the variables.

Example: Panthers Football Team

- Scatter Diagram

The Panthers football team is interested in investigating the relationship, if any, between interceptions made and points scored.

x = Number of y = Number of

InterceptionsPoints Scored

1 14

3 24

2 18

1 17

3 27

Example: Panthers Football Team

- Scatter Diagram

y

30

25

20

Number of Points Scored

15

10

5

x

0

1

0

2

3

Number of Interceptions

Example: Panthers Football Team

- The preceding scatter diagram indicates a positive relationship between the number of interceptions and the number of points scored.
- Higher points scored are associated with a higher number of interceptions.
- The relationship is not perfect; all plotted points in the scatter diagram are not on a straight line.

Using Excel’s Chart Wizardto Construct a Scatter Diagram

- Formula Worksheet (showing data entered)

Using Excel’s Chart Wizardto Construct a Scatter Diagram

- Producing a Scatter Diagram

Step 1 Select cells A1:B6

Step 2 Select the Chart Wizard

Step 3 When the Chart Type dialog box appears:

Choose XY (Scatter) in the Chart type list

Choose Scatter from the Chart sub-type display

Select Next >

Step 4 When the Chart Source Data dialog box appears

Select Next >

… continued

Using Excel’s Chart Wizardto Construct a Scatter Diagram

- Producing a Scatter Diagram

Step 5 When the Chart Options dialog box appears:

Select the Titles tab and then

Delete Number of Points Scored in the Chart title box

Enter Number of Interceptions in the Value (X) axis box

Enter Number of Points Scored in the Value (Y) axis box

Select the Legend tab and then

Remove the check in the Show Legend box

Select Next >

… continued

Using Excel’s Chart Wizardto Construct a Scatter Diagram

- Producing a Scatter Diagram

Step 6 When the Chart Location dialog box appears:

Specify the location for the new chart

Select Finish to display the scatter diagram

Using Excel’s Chart Wizardto Construct a Scatter Diagram

- Value Worksheet

Tabular and Graphical Procedures

Data

Qualitative Data

Quantitative Data

Tabular

Methods

Graphical

Methods

Tabular

Methods

Graphical

Methods

- Frequency
- Distribution
- Rel. Freq. Dist.
- % Freq. Dist.
- Crosstabulation

- Bar Graph
- Pie Chart

- Dot Plot
- Histogram
- Ogive
- Scatter
- Diagram

- Frequency
- Distribution
- Rel. Freq. Dist.
- Cum. Freq. Dist.
- Cum. Rel. Freq.
- Distribution
- Stem-and-Leaf
- Display
- Crosstabulation

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