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Chapter-2: Descriptive Statistics:. Tabular and Graphical Presentations Part A. Exploratory Data Analysis Cross-tabulations; Scatter Diagrams. Tabular and Graphical Procedures. Data. Qualitative Data. Quantitative Data. Tabular Methods. Graphical Methods. Tabular Methods.

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Chapter-2: Descriptive Statistics:

Tabular and Graphical PresentationsPart A

Exploratory Data Analysis

Cross-tabulations; Scatter Diagrams

Tabular and Graphical Procedures

Data

Qualitative Data

Quantitative Data

Tabular

Methods

Graphical

Methods

Tabular

Methods

Graphical

Methods

• Bar Graph
• Pie Chart
• Frequency
• Distribution
• Rel. Freq. Dist.
• Percent Freq.
• Distribution
• Crosstabulation
• Dot Plot
• Histogram
• Ogive
• Scatter
• Diagram
• Frequency
• Distribution
• Rel. Freq. Dist.
• Cum. Freq. Dist.
• Cum. Rel. Freq.
• Distribution
• Stem-and-Leaf
• Display
• Crosstabulation
Stem-and-Leaf Display
• A stem-and-leaf display is an exploratory data

analysis method that shows both the rank order

• and shape of the distribution of the data.
• It is similar to a histogram, but it has the
• advantage of showing the actual data values.

Example: Hudson Auto Repair

• Sample of Parts Cost for 50 Tune-ups

Stem-and-Leaf Display

5

6

7

8

9

10

2 7

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

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

0 0 2 3 5 8 9

1 3 7 7 7 8 9

1 4 5 5 9

Steps to Construct a Stem-and-Leaf Display
• Arrange the first digits of each data to the
• left of a vertical line.
• To the right of the vertical line, 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.

Stem-and-Leaf Display

5

6

7

8

9

10

2 7

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

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

0 0 2 3 5 8 9

1 3 7 7 7 8 9

1 4 5 5 9

a stem

a leaf

Stretched Stem-and-Leaf Display
• If we believe the display has condensed the data too

much, we can stretch thedisplay by using two stems

for each leading digit(s).

• Whenever stretched, the first value of a stem

corresponds to leaf values of 0 - 4, and the second

• value of a stem corresponds to leaf values of 5 - 9.

Stretched Stem-and-Leaf Display

5

5

6

6

7

7

8

8

9

9

10

10

2

7

2 2 2 2

5 6 7 8 8 8 9 9 9

1 1 2 2 3 4 4

5 5 5 6 7 8 9 9 9

0 0 2 3

5 8 9

1 3

7 7 7 8 9

1 4

5 5 9

Stem-and-Leaf Display—Some notes

• 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

9

10

11

6 8

1 4

2

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

17

18

19

8

The 82 in 1682

is rounded down

to 80 and is

represented as an 8.

1 9

0 3

1 7

Crosstabulations and Scatter Diagrams
• Thus far we have focused on methods that are used
• to summarize the data on one variable at a time.
• Cross tabulationand a scatter diagram are methods for
• summarizing the data on two (or more) variables
• simultaneously.
• Help us understand the relationship between two variables.

Crosstabulation

• A crosstabulation is a tabular summary of data for

two variables.

• Crosstabulation can be used when:
• one variable is qualitative and the other is
• quantitative,
• both variables are qualitative, or
• both variables are quantitative.
• The left and top margin labels define the classes for
• the two variables.

Cross-tabulation

• Example:

The number of Homes sold by style and price for the past two years is shown below.

qualitative

variable

quantitative

variable

Home Style

Price

Range

Colonial Log Split A-Frame

Total

18 6 19 12

55

45

< \$99,000

> \$99,000

12 14 16 3

30 20 35 15

Total

100

Cross-tabulation
• Insights Gained from Preceding Cross-tabulation
• 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.
Cross-tabulation: 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.

Cross-tabulation

Frequency distribution

for the price variable

Home Style

Price

Range

Colonial Log Split A-Frame

Total

18 6 19 12

55

45

< \$99,000

> \$99,000

12 14 16 3

30 20 35 15

Total

100

Frequency distribution

for the home style variable

Cross-tabulation: Row Percentages

Home Style

Price

Range

Colonial Log Split A-Frame

Total

32.73 10.91 34.55 21.82

100

100

< \$99,000

> \$99,000

26.67 31.11 35.56 6.67

(Colonial and > \$99K)/(All >\$99K) x 100 = (12/45) x 100

Simpson’s Paradox

• Often Times data in two or more cross-tabulations are

aggregated to produce a summary cross-tabulation.

• We must be careful in drawing conclusions about the
• relationship between the two variables in the
• aggregated cross-tabulation.
• Simpson’ Paradox: In some cases the conclusions

based upon an aggregated cross-tabulation can be

completely different from the un-aggregated

data.

Scatter Diagram and Trend line

• 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.
• A trend line is an approximation of the relationship.
Scatter Diagram
• A Positive Relationship

y

x

Scatter Diagram
• A Negative Relationship

y

x

Scatter Diagram
• No Apparent Relationship

y

x

Example: Panthers Football Team
• The Panthers football team is interested

in investigating the relationship, if any,

between interceptions made and points scored.

x = Number of

Interceptions

y = Number of

Points Scored

14

24

18

17

30

1

3

2

1

3

35

30

25

20

15

10

5

0

1

0

2

3

4

Scatter Diagram

y

Number of Points Scored

x

Number of Interceptions

Example: Panthers Football Team

• Insights Gained from the Preceding Scatter Diagram
• The 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.