1. Fu Jen Catholic University
Slides Prepared by
Juei-Chao Chen
Fu Jen Catholic University

2. Chapter 2 Descriptive Statistics: Tabular and Graphical Presentations Part B
Exploratory Data Analysis
Crosstabulations and
Scatter Diagrams x y

3. 2.3 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.

4. 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.

5. Stem-and-Leaf Display
Example: Number of Questions Answered Correctly
on An Aptitude Test

6. Stem-and-Leaf Display
Stem: The numbers to the left of the vertical line
(6, 7, 8, 9, 10, 11, 12, 13, and 14).
Leaf: each digit to the right of the vertical line.

7. Stem-and-Leaf Display
Although the stem-and-leaf display may appear to offer the same information as a histogram, it has two primary advantages.
1. The stem-and-leaf display is easier to construct by
hand.
2. Within a class interval, the stem-and-leaf display
provides more information than the histogram
because the stem-and-leaf shows the actual data.

8. Example: Hudson Auto Repair
The manager of Hudson Auto
would like to have a better
understanding of the cost
of parts used in the engine
tune-ups performed in the
shop. She examines 50
customer invoices for tune-ups. The costs of parts,
rounded to the nearest dollar, are listed on the next
slide.

9. Example: Hudson Auto Repair
Sample of Parts Cost for 50 Tune-ups

10. Stem-and-Leaf Display5 6 7 8 9 10
2 7
a stem
a leaf

11. 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 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
value corresponds to leaf values of

12. Stretched Stem-and-Leaf Display5 6 7 8 9 10 2 7
1 3
1 4

13. 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.

14. a stem-and-leaf display of these data will be
Example: Leaf Unit = 0.1
If we have data with values such as
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

15. a stem-and-leaf display of these data will be
Example: Leaf Unit = 10
If we have data with values such as
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

16. 2.4 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.

17. 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.

18. Crosstabulation Example: Data from Zagat’s Restaurant Review.
Data on a restaurant’s quality rating and typical
meal price are reported.
Quality rating is a qualitative variable with rating
categories of good, very good, and excellent.
Meal price is a quantitative variable that ranges
from $10 to $49.

19. Crosstabulation

20. Crosstabulation Example: Crosstabulation of Quality Rating and
Meal Price for 300 Los Angeles Restaurants

21. Colonial Log Split A-Frame
Crosstabulation
Example: Finger Lakes Homes
The number of Finger Lakes homes sold for each style and price for the past two years is shown below.
quantitative
variable
qualitative
variable
Home Style
Price
Range
Colonial Log Split A-Frame
Total
< $99,000
> $99,000 55 45
Total
100

22. Crosstabulation Insights Gained from 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.

23. Crosstabulation Frequency distribution for the price variable
Home Style
Price
Range
Colonial Log Split A-Frame
Total
< $99,000
> $99,000 55 45
Total
100
Frequency distribution
for the home style variable

24. 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.

25. Crosstabulation: Row Percentages
Price
Range
Home Style
Colonial Log Split A-Frame
Total
100
< $99,000
> $99,000
Note: row totals are actually due to rounding.
(Colonial and > $99K)/(All >$99K) × 100 = (12/45) × 100

26. Crosstabulation: Column Percentages
Price
Range
Home Style
Colonial Log Split A-Frame
< $99,000
> $99,000
Total
(Colonial and > $99K)/(All Colonial) × 100 = (12/30) × 100

27. Crosstabulation: Simpson’s Paradox
Data in two or more crosstabulations are often
aggregated to produce a summary crosstabulation.
We must be careful in drawing conclusions about the
relationship between the two variables in the
aggregated crosstabulation.
Simpson’ Paradox: In some cases the conclusions
based upon an aggregated crosstabulation can be
completely reversed if we look at the unaggregated
data. suggests the overall relationship between the
variables.

28. Scatter Diagram and Trendline
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 trendline is an approximation of the relationship.

29. Scatter Diagram and Trendline
A Positive Relationship y x

30. Scatter Diagram and Trendline
A Negative Relationship y x

31. Scatter Diagram and Trendline
No Apparent Relationship y x

32. 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
Interceptions
y = Number of
Points Scored 1 3 2 14 24 18 17 30

33. Scatter Diagram y 5 10 15 20 25 30 35
Number of Points Scored x 1 2 3 4
Number of Interceptions

34. 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.

35. 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.
Crosstabulation
Dot Plot
Histogram
Ogive
Scatter
Diagram
Frequency
Distribution
Rel. Freq. Dist.
Cum. Freq. Dist.
Cum. Rel. Freq.
Stem-and-Leaf
Display
Crosstabulation

36. End of Chapter 2, Part B