department of quantitative methods & information systems ... · qmis 120, by dr. m. zainal...

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



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


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




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)


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




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


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……………………..





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


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





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






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


Grouping Data by Classes




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


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




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




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)


(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




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 =


Histograms in Excel

Select “Data” Tab 1

Choose Histogram 3

2 Data Analysis




Histograms in Excel (continued)

4

Input data and bin ranges

Select Chart Output


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




Ogives

12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

(continued)


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



Excel will show the Ogive

graphically if the

“Cumulative Percentage”

option is selected in the

Histogram dialog box

Ogives in Excel


Other Graphical Presentation Tools

Categorical

Data

Bar

Chart

Stem and Leaf

Diagram

Pie

Charts

Quantitative

Data




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


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)




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


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)




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


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)




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


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.




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.


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.




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


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




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


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




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


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




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



Types of Relationships

Linear Relationships

X X

YY


Curvilinear Relationships

X X

YY

Types of Relationships (continued)




No Relationship

X X

YY

Types of Relationships (continued)


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




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?)


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

. . .





Quality rating is a qualitative variable with the rating categories of good, very good and excellent



Also, we can find the row percentage





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.



Try it for the meal price (column totals)





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








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




Copyright

The materials of this presentation were mostly

taken from the PowerPoint files accompanied

Business Statistics: A Decision-Making Approach,

7e © 2008 Prentice-Hall, Inc.


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