This document provides an overview of techniques for presenting numerical data in tables and charts. It discusses ordered arrays, stem-and-leaf displays, frequency distributions, histograms, polygons, ogives, bar charts, pie charts, and scatter diagrams. The chapter goals are to teach how to create and interpret these various data presentation methods using Microsoft Excel. Examples are provided for frequency distributions, histograms, polygons, and ogives to illustrate how to construct and make sense of these graphical representations of quantitative data.

1. Statistics for Managers
Using Microsoft® Excel
4th Edition
Chapter 2
Presenting Data in Tables and Charts
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-1

2. Chapter Goals
After completing this chapter, you should be able to:

Create an ordered array and a stem-and-leaf display

Construct and interpret a frequency distribution, polygon,
and ogive

Construct a histogram

Create and interpret bar charts, pie charts, and scatter
diagrams

Present and interpret category data in bar charts and pie
charts

Describe appropriate and inappropriate ways to display
data graphically
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-2

3. Organizing and Presenting
Data Graphically

Data in raw form are usually not easy to use
for decision making

Some type of organization is needed



Table
Graph
Techniques reviewed here:





Ordered Array
Stem-and-Leaf Display
Frequency Distributions and Histograms
Bar charts and pie charts
Contingency tables
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-3

4. Tables and Charts for
Numerical Data
Numerical Data
Frequency Distributions
and
Cumulative Distributions
Ordered Array
Stem-and-Leaf
Display
Histogram
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Polygon
Ogive
Chap 2-4

5. The Ordered Array
A sorted list of data:
 Shows range (min to max)
 Provides some signals about variability
within the range
 May help identify outliers (unusual observations)
 If the data set is large, the ordered array is
less useful
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-5

6. The Ordered Array
(continued)

Data in raw form (as collected):
24, 26, 24, 21, 27, 27, 30, 41, 32, 38

Data in ordered array from smallest to largest:
21, 24, 24, 26, 27, 27, 30, 32, 38, 41
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-6

7. Stem-and-Leaf Diagram

A simple way to see distribution details in a
data set
METHOD: Separate the sorted data series
into leading digits (the stem) and
the trailing digits (the leaves)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-7

8. Example
Data in ordered array:
21, 24, 24, 26, 27, 27, 30, 32, 38, 41

Here, use the 10’s digit for the stem unit:
Stem Leaf

21 is shown as
2
1

38 is shown as
3
8
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-8

9. Example
(continued)
Data in ordered array:
21, 24, 24, 26, 27, 27, 30, 32, 38, 41

Completed stem-and-leaf diagram:
Stem
Leaves
2
1 4 4 6 7 7
3
0 2 8
4
1
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-9

10. Using other stem units

Using the 100’s digit as the stem:

Round off the 10’s digit to form the leaves
Stem
Leaf

613 would become
6
1

776 would become
7
8
12
2


...
1224 becomes
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-10

11. Using other stem units
(continued)

Using the 100’s digit as the stem:

The completed stem-and-leaf display:
Data:
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Leaves
136
7
2258
8
346699
9
13368
10
356
11
47
12
613, 632, 658, 717,
722, 750, 776, 827,
841, 859, 863, 891,
894, 906, 928, 933,
955, 982, 1034,
1047,1056, 1140,
1169, 1224
Stem
6
2
Chap 2-11

12. Tabulating Numerical Data:
Frequency Distributions
What is a Frequency Distribution?

A frequency distribution is a list or a table …

containing class groupings (categories or
ranges within which the data falls) ...

and the corresponding frequencies with which
data falls within each grouping or category
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-12

13. Why Use Frequency Distributions?

A frequency distribution is a way to
summarize data

The distribution condenses the raw data
into a more useful form...

and allows for a quick visual interpretation
of the data
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-13

14. Class Intervals
and Class Boundaries


Each class grouping has the same width
Determine the width of each interval by
range
Width of int erval ≅
number of desired class groupings



Use at least 5 but no more than 15 groupings
Class boundaries never overlap
Round up the interval width to get desirable
endpoints
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-14

15. Frequency Distribution Example
Example: A manufacturer of insulation randomly
selects 20 winter days and records the daily
high temperature
24, 35, 17, 21, 24, 37, 26, 46, 58, 30,
32, 13, 12, 38, 41, 43, 44, 27, 53, 27
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-15

16. Frequency Distribution Example
(continued)

Sort raw data in ascending order:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

Find range: 58 - 12 = 46

Select number of classes: 5 (usually between 5 and 15)
Compute class interval (width): 10 (46/5 then round up)
Determine class boundaries (limits): 10, 20, 30, 40, 50, 60
Compute class midpoints: 15, 25, 35, 45, 55




Count observations & assign to classes
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-16

17. Frequency Distribution Example
(continued)
Data in ordered array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Frequency
Relative
Frequency
Percentage
10 but less than 20
20 but less than 30
30 but less than 40
3
6
5
.15
.30
.25
15
30
25
40 but less than 50
50 but less than 60
4
2
.20
.10
20
10
Class
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-17

18. Graphing Numerical Data:
The Histogram

A graph of the data in a frequency distribution
is called a histogram

The class boundaries (or class midpoints)
are shown on the horizontal axis

the vertical axis is either frequency, relative
frequency, or percentage

Bars of the appropriate heights are used to
represent the number of observations within
each class
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-18

19. Histogram Example
Class
Midpoint Frequency
10 but less than 20
20 but less than 30
30 but less than 40
40 but less than 50
50 but less than 60
(No gaps
between
bars)
15
25
35
45
55
3
6
5
4
2
Histogram : Daily High Tem perature
7
6
6
Frequency
Class
5
5
4
4
3
3
2
2
1
0
0
0
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
5
15
25
35
45
Class Midpoints
55
More
Chap 2-19

20. Histograms in Excel
1
Select
Tools/Data Analysis
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-20

21. Histograms in Excel
(continued)
2
Choose Histogram
(
3
Input data range and bin
range (bin range is a cell
range containing the upper class
boundaries for each class
grouping)
Select Chart Output
and click “OK”
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-21

22. Questions for Grouping Data
into Classes

1. How wide should each interval be?
(How many classes should be used?)

2. How should the endpoints of the
intervals be determined?



Often answered by trial and error, subject to
user judgment
The goal is to create a distribution that is
neither too "jagged" nor too "blocky”
Goal is to appropriately show the pattern of
variation in the data
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-22

23. How Many Class Intervals?
Many (Narrow class intervals)
3
2.5
2
1.5
1
0.5
More
60
56
52
48
44
40
36
32
28
24
20
16
8
0
4

may yield a very jagged distribution
with gaps from empty classes
Can give a poor indication of how
frequency varies across classes
12

3.5
Frequency

Temperature
Few (Wide class intervals)


may compress variation too much and
yield a blocky distribution
can obscure important patterns of
variation.
12
10
Frequency

8
6
4
2
0
0
30
60
More
Temperature
(X axis labels are upper class endpoints)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-23

24. Graphing Numerical Data:
The Frequency Polygon
Class
Midpoint Frequency
Class
10 but less than 20
20 but less than 30
30 but less than 40
40 but less than 50
50 but less than 60
15
25
35
45
55
3
6
5
4
2
Frequency Polygon: Daily High Temperature
7
6
(In a percentage
polygon the vertical axis
would be defined to
show the percentage of
observations per class)
Frequency
5
4
3
2
1
0
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
5
15
25
35
Class Midpoints
45
55
More
Chap 2-24

25. Tabulating Numerical Data:
Cumulative Frequency
Data in ordered array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Class
Frequency Percentage
Cumulative Cumulative
Frequency Percentage
10 but less than 20
3
15
3
15
20 but less than 30
6
30
9
45
30 but less than 40
5
25
14
70
40 but less than 50
4
20
18
90
50 but less than 60
2
10
20
100
20
100
Total
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-25

26. Graphing Cumulative Frequencies:
The Ogive (Cumulative % Polygon)
Less than 10
10 but less than 20
20 but less than 30
30 but less than 40
40 but less than 50
50 but less than 60
10
20
30
40
50
60
0
15
45
70
90
100
Ogive: Daily High Temperature
Cumulative Percentage
Class
Lower
Cumulative
class
boundary Percentage
100
80
60
40
20
0
10
20
30
40
50
60
Class Boundaries (Not Midpoints)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-26

27. Scatter Diagrams

Scatter Diagrams are used for
bivariate numerical data


Bivariate data consists of paired
observations taken from two numerical
variables
The Scatter Diagram:
 one variable is measured on the vertical
axis and the other variable is measured
on the horizontal axis
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-27

28. Scatter Diagram Example
Cost per Day vs. Production Volume
Cost per
day
23
125
250
26
140
200
29
146
33
160
38
167
42
170
50
188
55
195
60
200
Cost per Day
Volume
per day
150
100
50
0
0
10
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
20
30
40
50
60
70
Volume per Day
Chap 2-28

29. Scatter Diagrams in Excel
1
Select the chart wizard
2
Select XY(Scatter) option,
then click “Next”
3
When prompted, enter the
data range, desired
legend, and desired
destination to complete
the scatter diagram
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-29

30. Tables and Charts for
Categorical Data
Categorical
Data
Graphing Data
Tabulating Data
Summary
Table
Bar
Charts
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Pie
Charts
Pareto
Diagram
Chap 2-30

31. The Summary Table
Summarize data by category
Example: Current Investment Portfolio
Investment
Amount
Percentage
Type
(in thousands $)
(%)
(Variables are
Categorical)
Stocks
Bonds
CD
Savings
46.5
32.0
15.5
16.0
42.27
29.09
14.09
14.55
Total
110.0
100.0
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-31

32. Bar and Pie Charts

Bar charts and Pie charts are often used
for qualitative (category) data

Height of bar or size of pie slice shows
the frequency or percentage for each
category
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-32

33. Bar Chart Example
Current Investment Portfolio
Investment
Type
Amount
(in thousands $)
Percentage
(%)
Stocks
Bonds
CD
Savings
46.5
32.0
15.5
16.0
42.27
29.09
14.09
14.55
Total
110.0
100.0
Investor's Portfolio
Savings
CD
Bonds
Stocks
0
10
20
30
40
50
Amount in $1000's
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-33

34. Pie Chart Example
Current Investment Portfolio
Investment
Type
Amount
(in thousands $)
Percentage
(%)
Stocks
Bonds
CD
Savings
46.5
32.0
15.5
16.0
42.27
29.09
14.09
14.55
Total
110.0
100.0
Savings
15%
CD
14%
Bonds
29%
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Stocks
42%
Percentages
are rounded to
the nearest
percent
Chap 2-34

35. Pareto Diagram

Used to portray categorical data

A bar chart, where categories are shown in
descending order of frequency

A cumulative polygon is often shown in the
same graph

Used to separate the “vital few” from the “trivial
many”
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-35

36. Pareto Diagram Example
Current Investment Portfolio
100%
40%
90%
80%
35%
70%
30%
60%
25%
50%
20%
40%
15%
30%
10%
20%
5%
10%
0%
cumulative % invested
(line graph)
% invested in each category (bar
graph)
45%
0%
Stocks
Bonds
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Savings
CD
Chap 2-36

37. Tabulating and Graphing
Multivariate Categorical Data

Contingency Table for Investment Choices ($1000’s)
Investment
Category
Investor A
Investor B
Investor C
Total
Stocks
46.5
55
27.5
129
Bonds
CD
Savings
32.0
15.5
16.0
44
20
28
19.0
13.5
7.0
95
49
51
Total
110.0
147
67.0
324
(Individual values could also be expressed as percentages of the overall total,
percentages of the row totals, or percentages of the column totals)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-37

38. Tabulating and Graphing
Multivariate Categorical Data
(continued)

Side by side bar charts
Comparing Investors
S avings
CD
B onds
S toc k s
0
10
20
Inves tor A
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
30
Inves tor B
40
50
60
Inves tor C
Chap 2-38

39. Side-by-Side Chart Example

Sales by quarter for three sales territories:
East
West
North
1st Qtr
2nd Qtr
3rd Qtr
4th Qtr
20.4
27.4
59
20.4
30.6
38.6
34.6
31.6
45.9
46.9
45
43.9
60
50
40
East
West
Nor t h
30
20
10
0
1st Qt r
2nd Qt r
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
3rd Qt r
4t h Qt r
Chap 2-39

40. Principles of Graphical Excellence





Present data in a way that provides substance,
statistics and design
Communicate complex ideas with clarity,
precision and efficiency
Give the largest number of ideas in the most
efficient manner
Excellence almost always involves several
dimensions
Tell the truth about the data
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-40

41. Errors in Presenting Data

Using “chart junk”

Failing to provide a relative
basis in comparing data
between groups

Compressing or distorting the vertical axis

Providing no zero point on the vertical axis
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-41

42. Chart Junk
Bad Presentation
Good Presentation

Minimum Wage
1960: $1.00
1970: $1.60
1980: $3.10
$
Minimum Wage
4
2
0
1960
1970
1980
1990
1990: $3.80
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-42

43. No Relative Basis
listen
Bad Presentation
Freq.
300
200
100
0
A’s received by
students.
 Good Presentation
%
30%
A’s received by
students.
20%
10%
FR SO
JR SR
0%
FR SO JR SR
FR = Freshmen, SO = Sophomore, JR = Junior, SR = Senior
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-43

44. Compressing Vertical Axis
Bad Presentation
200
$
Good Presentation
Quarterly Sales
50
100
25
0
$
Quarterly Sales
0
Q1 Q2
Q3 Q4
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Q1
Q2
Q3 Q4
Chap 2-44

45. No Zero Point On Vertical Axis
Bad Presentation
$Good Presentations
Monthly Sales
45
45
$
Monthly Sales
39
36
42
0
39
36
42
J
or
J F M A M J
F
J
F
M
A
M
J
$
60
40
Graphing the first six months of sales
20
0
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
M
A
M
J
Chap 2-45

46. Chapter Summary

Data in raw form are usually not easy to use for
decision making -- Some type of organization is
needed:
♦ Table
♦ Graph

Techniques reviewed in this chapter:






Ordered array and stem-and-leaf display
Frequency distributions and histograms
Percentage polygons and ogives
Scatter diagrams for bivariate data
Bar charts, pie charts, and Pareto diagrams
Contingency tables and side-by-side bar charts
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Chap 2-46