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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.

- Sample of Parts Cost for 50 Tune-ups

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.

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

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.

- 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.

- 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.

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

- 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.

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

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.

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