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QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-1
Chapter 2
Graphs, Charts, and Tables –
Describing Your Data
Business Statistics:
Department of Quantitative Methods & Information Systems
Dr. Mohammad Zainal QMIS 120
Chapter Goals
After completing this chapter, you should be
able to:
Construct a frequency distribution both manually and with a computer
Construct and interpret a histogram
Create and interpret bar charts, pie charts, and
stem-and-leaf diagrams
Present and interpret data in line charts and
scatter diagrams
QMIS 120, by Dr. M. Zainal Chap 2-2
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-2
Raw data
Ages (in years) of 20 students selected from
CBA are reported in the way they are
collected. The data values are recorded in the
following table. Ages of 20 Students
23 24 22 30 19 20 18 24 19 21
25 24 18 20 19 23 25 22 21 20
QMIS 120, by Dr. M. Zainal Chap 2-3
Raw data
The same students were asked about their
status. The responses of the sample are
recorded in the following table
Status of 20 Students
S S J S F J F S F J
S S F J F J S J J F
QMIS 120, by Dr. M. Zainal Chap 2-4
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-3
Frequency Distributions
What is a Frequency Distribution?
A frequency distribution is a list or a table …
containing the values of a variable (or a set of
ranges within which the data fall) ...
and the corresponding frequencies with which
each value occurs (or frequencies with which
data fall within each range)
QMIS 120, by Dr. M. Zainal Chap 2-5
Frequency Distributions
Weekly Earnings of 100 Employees of a company
Number of employees
f
Weekly Earnings
(dollars)
9 401 to 600
22 601 to 800
39 801 to 1000
15 1001 to 1200
9 1201 to 1400
6 1401 to 1600
QMIS 120, by Dr. M. Zainal Chap 2-6
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-4
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
QMIS 120, by Dr. M. Zainal Chap 2-7
Frequency Distribution: Discrete Data
Discrete data: possible values are countable
Example: An
advertiser asks
200 customers
how many days
per week they
read the daily
newspaper.
Row Data
5,6,1,2,4,5,7,2,3,5,1,3,2,5,0,2,2,0,7,7,1,2,4
,3,5,6,7,1,1,1,1,2,5,0,0,0,1,20,7,5,3,6,2,1,6
,2,1,4,2,4,5,3,1,0,2,3,6,5,7,4,1,2,3,5,6,1,0,
0,0,0,0,1,1,1,2,3,5,1,4……………………..
QMIS 120, by Dr. M. Zainal Chap 2-8
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-5
Frequency Distribution: Discrete Data
Number of days
read Frequency
0 44
1 24
2 18
3 16
4 20
5 22
6 26
7 30
Total 200
It is called
“Single-Value”
approach
QMIS 120, by Dr. M. Zainal Chap 2-9
Relative Frequency
Relative Frequency: What proportion is in each category?
Number of days
read Frequency
Relative
Frequency
0 44 .22
1 24 .12
2 18 .09
3 16 .08
4 20 .10
5 22 .11
6 26 .13
7 30 .15
Total 200 1.00
.22200
44
22% of the
people in the
sample report
that they read
the newspaper
0 days per week
QMIS 120, by Dr. M. Zainal Chap 2-10
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-6
Frequency Distribution: Discrete Data
Example: Construct a frequency distribution table for
the following data
Home Runs Team Home Runs Team
139 Milwaukee 152 Anaheim
167 Minnesota 165 Arizona
162 Montreal 164 Atlanta
160 New York Mets 165 Baltimore
223 New York Yankees 177 Boston
205 Oakland 200 Chicago Cubs
165 Philadelphia 217 Chicago White Sox
142 Pittsburgh 169 Cincinnati
175 St. Louis 192 Cleveland
136 San Diego 152 Colorado
198 San Francisco 124 Detroit
152 Seattle 146 Florida
133 Tampa Bay 167 Houston
230 Texas 140 Kansas City
187 Toronto 155 Los Angeles
Home Runs Hit by Major League Baseball Teams During the 2002 Season
QMIS 120, by Dr. M. Zainal Chap 2-11
Frequency Distribution: Discrete Data
QMIS 120, by Dr. M. Zainal Chap 2-12
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-7
Frequency Distribution: Continuous Data
Continuous Data: may take on any value in some interval
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
Temperature is a continuous variable because it could
be measured to any degree of precision desired
QMIS 120, by Dr. M. Zainal Chap 2-13
Grouping Data by Classes
QMIS 120, by Dr. M. Zainal Chap 2-14
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-8
Frequency Distribution Example
Data from low to high:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
QMIS 120, by Dr. M. Zainal Chap 2-15
Frequency Histograms
The classes or intervals are shown on the
horizontal axis
frequency is measured on the vertical axis
Bars of the appropriate heights can be used
to represent the number of observations
within each class
Such a graph is called a histogram
QMIS 120, by Dr. M. Zainal Chap 2-16
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-9
Histogram
0
3
6
5
4
2
0
0
1
2
3
4
5
6
7
5 15 25 36 45 55 More
Fre
qu
en
cy
Class Midpoints
Histogram Example
Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
No gaps
between
bars, since
continuous
data
0 10 20 30 40 50 60
Class Endpoints Chap 2-17
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
QMIS 120, by Dr. M. Zainal Chap 2-18
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-10
How Many Class Intervals?
Many (Narrow class intervals)
may yield a very jagged
distribution with gaps from empty
classes
Can give a poor indication of how
frequency varies across classes
Few (Wide class intervals)
may compress variation too much
and yield a blocky distribution
can obscure important patterns of
variation.
0
2
4
6
8
10
12
0 30 60 More
Temperature
Fre
qu
en
cy
0
0.5
1
1.5
2
2.5
3
3.5
4 8
12
16
20
24
28
32
36
40
44
48
52
56
60
Mo
re
Temperature
Fre
qu
en
cy
(X axis labels are upper class endpoints)
QMIS 120, by Dr. M. Zainal Chap 2-19
(X axis labels are upper class endpoints)
General Guidelines
Number of Data Points Number of Classes
under 50 5 - 7
50 – 100 6 - 10
100 – 250 7 - 12
over 250 10 - 20
Class widths can typically be reduced as the
number of observations increases
Distributions with numerous observations are more
likely to be smooth and have gaps filled since data
are plentiful
QMIS 120, by Dr. M. Zainal Chap 2-20
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-11
Class Width
The class width is the distance between the
lowest possible value and the highest possible
value for a frequency class
The class width is
Largest Value - Smallest Value
Number of Classes W =
QMIS 120, by Dr. M. Zainal Chap 2-21
Histograms in Excel
Select “Data” Tab 1
Choose Histogram 3
2 Data Analysis
QMIS 120, by Dr. M. Zainal Chap 2-22
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-12
Histograms in Excel (continued)
4
Input data and bin ranges
Select Chart Output
QMIS 120, by Dr. M. Zainal Chap 2-23
Ogives
An Ogive is a graph of the cumulative relative
frequencies from a relative frequency
distribution
Ogives are sometime shown in the same
graph as a relative frequency histogram
QMIS 120, by Dr. M. Zainal Chap 2-24
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-13
Ogives
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
(continued)
QMIS 120, by Dr. M. Zainal Chap 2-25
Histogram
0
1
2
3
4
5
6
7
5 15 25 36 45 55 More
Fre
qu
en
cy
Class Midpoints
Ogive Example
100
80
60
40
20
0 Cum
ula
tive F
requency (
%)
0 10 20 30 40 50 60
Class Endpoints
Chap 2-26
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-14
Excel will show the Ogive
graphically if the
“Cumulative Percentage”
option is selected in the
Histogram dialog box
Ogives in Excel
QMIS 120, by Dr. M. Zainal Chap 2-27
Other Graphical Presentation Tools
Categorical
Data
Bar
Chart
Stem and Leaf
Diagram
Pie
Charts
Quantitative
Data
QMIS 120, by Dr. M. Zainal Chap 2-28
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-15
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
QMIS 120, by Dr. M. Zainal Chap 2-29
Bar Chart Example 1
Investor's Portfolio
0 10 20 30 40 50
Stocks
Bonds
CD
Savings
Amount in $1000's
(Note that bar charts can also be displayed with vertical bars)
QMIS 120, by Dr. M. Zainal Chap 2-30
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-16
Bar Chart Example 2
Newspaper readership per week
0
10
20
30
40
50
0 1 2 3 4 5 6 7
Number of days newspaper is read per week
Fre
uen
cy
Number of
days read
Frequency
0 44
1 24
2 18
3 16
4 20
5 22
6 26
7 30
Total 200
QMIS 120, by Dr. M. Zainal Chap 2-31
Pie Chart Example
Percentages
are rounded to
the nearest
percent
Current Investment Portfolio
Savings
15%
CD
14%
Bonds
29%
Stocks
42%
Investment Amount Percentage Type (in thousands $) Stocks 46.5 42.27 Bonds 32.0 29.09 CD 15.5 14.09 Savings 16.0 14.55 Total 110 100
(Variables are Qualitative)
QMIS 120, by Dr. M. Zainal Chap 2-32
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-17
Tabulating and Graphing Multivariate Categorical Data
Investment in thousands of dollars
Investment Investor A Investor B Investor C Total Category
Stocks 46.5 55 27.5 129 Bonds 32.0 44 19.0 95 CD 15.5 20 13.5 49 Savings 16.0 28 7.0 51 Total 110.0 147 67.0 324
QMIS 120, by Dr. M. Zainal Chap 2-33
Tabulating and Graphing Multivariate Categorical Data
Side by side charts
Comparing Investors
0 10 20 30 40 50 60
S toc k s
B onds
CD
S avings
Inves tor A Inves tor B Inves tor C
(continued)
QMIS 120, by Dr. M. Zainal Chap 2-34
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-18
Side-by-Side Chart Example
Sales by quarter for three sales territories:
0
10
20
30
40
50
60
1st Qtr 2nd Qtr 3rd Qtr 4th Qtr
East
West
North
1st Qtr 2nd Qtr 3rd Qtr 4th Qtr
East 20.4 27.4 59 20.4
West 30.6 38.6 34.6 31.6
North 45.9 46.9 45 43.9
QMIS 120, by Dr. M. Zainal Chap 2-35
Dot Plot
A One of the simplest methods for graphing
and understanding quantitative data is to
create a dot plot.
A horizontal axis shows the range of values
for the observations.
Each data point is represented by a dot
placed above the axis.
QMIS 120, by Dr. M. Zainal Chap 2-36
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-19
Dot Plot
Dot plots can help us detect outliers (also
called extreme values) in a data set.
Outliers are the values that are extremely
large or extremely small with respect to the
rest of the data values.
QMIS 120, by Dr. M. Zainal Chap 2-37
Dot Plot
Example : The following table lists the number of runs
batted in (RBIs) during the 2004 Major League Baseball
playoffs by members of the Boston Red Sox team with
at least one at-bat. Create a dot plot for these data.
QMIS 120, by Dr. M. Zainal Chap 2-38
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-20
Dot Plot
Step 1. First we draw a horizontal line that includes the
minimum and the maximum values in this data set.
Step 2. Place a dot above the value on the numbers line
that represents each RBI listed in the table
QMIS 120, by Dr. M. Zainal Chap 2-39
Stem and Leaf Diagram
Another simple way to see distribution
details from qualitative data
METHOD
1. Separate the sorted data series into leading digits
(the stem) and the trailing digits (the leaves)
2. List all stems in a column from low to high
3. For each stem, list all associated leaves
QMIS 120, by Dr. M. Zainal Chap 2-40
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-21
Example:
Here, use the 10’s digit for the stem unit:
Data sorted from low to high: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
12 is shown as
35 is shown as
Stem Leaf
1 2
3 5
QMIS 120, by Dr. M. Zainal Chap 2-41
Example:
Completed Stem-and-leaf diagram:
Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 28, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Stem Leaves
1 2 3 7
2 1 4 4 6 7 8
3 0 2 5 7 8
4 1 3 4 6
5 3 8
QMIS 120, by Dr. M. Zainal Chap 2-42
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-22
Using other stem units
Using the 100’s digit as the stem:
Round off the 10’s digit to form the leaves
613 would become 6 1
776 would become 7 8
. . .
1224 becomes 12 2
Stem Leaf
QMIS 120, by Dr. M. Zainal Chap 2-43
Line charts show values of one variable vs.
time
Time is traditionally shown on the horizontal axis
Scatter Diagrams show points for bivariate
data
one variable is measured on the vertical axis and
the other variable is measured on the horizontal
axis
A trend line is a line that provides an approximation
of that relationship.
Line Charts and Scatter Diagrams
QMIS 120, by Dr. M. Zainal Chap 2-44
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-23
Line Chart Example
U.S. Inflation Rate
0
1
2
3
4
5
6
1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006
Year
Infl
ati
on
Rate
(%
)
Year Inflation Rate 1985
3.56 1986 1.86
1987 3.65
1988 4.14
1989 4.82
1990 5.40
1991 4.21
1992 3.01
1993 2.99
1994 2.56
1995 2.83
1996 2.95
1997 2.29
1998 1.56
1999 2.21
2000 3.36
2001 2.85
2002 1.59
2003 2.27
2004 2.68
2005 3.39
2006 3.24 QMIS 120, by Dr. M. Zainal Chap 2-45
Scatter Diagram Example
Volume
per day
Cost per
day
23 125
26 140
29 146
33 160
38 167
42 170
50 188
55 195
60 200
Chap 2-46
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-24
Types of Relationships
Linear Relationships
X X
YY
QMIS 120, by Dr. M. Zainal Chap 2-47
Curvilinear Relationships
X X
YY
Types of Relationships (continued)
QMIS 120, by Dr. M. Zainal Chap 2-48
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-25
No Relationship
X X
YY
Types of Relationships (continued)
QMIS 120, by Dr. M. Zainal Chap 2-49
Cross-tabulation
Example: Draw a scatter diagram for the following data which lists the total amount spent in KD by costumers in a restaurant.
x (person) 1 1 2 2 3 3 4 4 5 5
y (KD) 8 7 14 18 20 22 21 26 29 33
QMIS 120, by Dr. M. Zainal Chap 2-50
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-26
Cross-tabulation
A cross tab is a tabular summary of data of two variables.
They are usually presented in a matrix format. Not like a frequency distribution (one variable).
A contingency table describes the distribution of two or more variables simultaneously.
Each cell shows the number of respondents that gave a specific combination of responses
It can be used with any level of data (What are they?)
QMIS 120, by Dr. M. Zainal Chap 2-51
Cross-tabulation example
Example: In a survey of the quality rating and the meal price conducted by a consumer restaurant review agency, the following table was produced:
Restaurant Quality rating Meal Price
1 Good 18
2 Very Good 22
3 Good 28
4 Excellent 38
5 Very Good 33
6 Good 28
. . .
QMIS 120, by Dr. M. Zainal Chap 2-52
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-27
Cross-tabulation example
Quality rating is a qualitative variable with the rating categories of good, very good and excellent
QMIS 120, by Dr. M. Zainal Chap 2-53
Cross-tabulation example
Also, we can find the row percentage
QMIS 120, by Dr. M. Zainal Chap 2-54
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-28
Cross-tabulation example
Dividing the totals in the right margin of the cross tab by the grand total provides relative and percentage frequency distribution for the quality rating variable.
QMIS 120, by Dr. M. Zainal Chap 2-55
Cross-tabulation example
Try it for the meal price (column totals)
QMIS 120, by Dr. M. Zainal Chap 2-56
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-29
Cross-tabulation example
Example: The following data are for 30 observations involving two qualitative variables x (A, B and C) and y (1 and 2).
Obs. x y Obs. x y Obs. x y
1 A 1 11 A 1 21 C 2
2 B 1 12 B 1 22 B 1
3 B 1 13 C 2 23 C 2
4 C 2 14 C 2 24 A 1
5 B 1 15 C 2 25 B 1
6 C 2 16 B 2 26 C 2
7 B 1 17 C 1 27 C 2
8 C 2 18 B 1 28 A 1
9 A 1 19 C 1 29 B 1
10 B 1 20 B 1 30 B 2
1- Construct a cross tabulation for the data
2- Calculate the row percentages
QMIS 120, by Dr. M. Zainal Chap 2-57
Cross-tabulation example
QMIS 120, by Dr. M. Zainal Chap 2-58
QMIS 120, by Dr. M. Zainal
Chapter 2 Student Lecture Notes 2-30
Cross-tabulation example
QMIS 120, by Dr. M. Zainal Chap 2-59
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:
Frequency Distributions, Histograms, and Ogives
Bar Charts and Pie Charts
Stem and Leaf Diagrams
Line Charts and Scatter Diagrams
QMIS 120, by Dr. M. Zainal Chap 2-60