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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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slide1

Chapter-2: Descriptive Statistics:

Tabular and Graphical PresentationsPart A

Exploratory Data Analysis

Cross-tabulations; Scatter Diagrams

tabular and graphical procedures
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
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.
slide4

Example: Hudson Auto Repair

  • Sample of Parts Cost for 50 Tune-ups
slide5

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
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.
slide7

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
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.
slide9

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

slide10

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
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

slide12

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 diagrams1
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.
slide15

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.
slide16

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
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
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.
slide19

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

slide20

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

slide21

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.

slide23

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
Scatter Diagram
  • A Positive Relationship

y

x

scatter diagram1
Scatter Diagram
  • A Negative Relationship

y

x

scatter diagram2
Scatter Diagram
  • No Apparent Relationship

y

x

example panthers football team
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

scatter diagram3

35

30

25

20

15

10

5

0

1

0

2

3

4

Scatter Diagram

y

Number of Points Scored

x

Number of Interceptions

slide29

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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