Newbold_chap02.ppt

Chap 2-1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chapter 2
Describing Data: Graphical
Statistics for
Business and Economics
6th Edition

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 2-2
Chapter Goals
After completing this chapter, you should be able to:
 Identify types of data and levels of measurement
 Create and interpret graphs to describe categorical variables:
 frequency distribution, bar chart, pie chart, Pareto diagram
 Create a line chart to describe time-series data
 Create and interpret graphs to describe numerical variables:
 frequency distribution, histogram, ogive, stem-and-leaf display
 Construct and interpret graphs to describe relationships between
variables:
 Scatter plot, cross table
 Describe appropriate and inappropriate ways to display data
graphically

Types of Data
Data
Categorical Numerical
Discrete Continuous
Examples:
 Marital Status
 Are you registered to
vote?
 Eye Color
(Defined categories or
groups)
Examples:
 Number of Children
 Defects per hour
(Counted items)
Examples:
 Weight
 Voltage
(Measured characteristics)

Measurement Levels
Interval Data
Ordinal Data
Nominal Data
Quantitative Data
Qualitative Data
Categories (no
ordering or direction)
Ordered Categories
(rankings, order, or
scaling)
Differences between
measurements but no
true zero
Ratio Data
Differences between
measurements, true
zero exists

Graphical
Presentation of Data
 Data in raw form are usually not easy to use
for decision making
 Some type of organization is needed
 Table
 Graph
 The type of graph to use depends on the
variable being summarized

Graphical
Presentation of Data
 Techniques reviewed in this chapter:
Categorical
Variables
Numerical
Variables
• Frequency distribution
• Bar chart
• Pie chart
• Pareto diagram
• Line chart
• Frequency distribution
• Histogram and ogive
• Stem-and-leaf display
• Scatter plot
(continued)

Tables and Graphs for
Categorical Variables
Categorical
Data
Graphing Data
Pie
Chart
Pareto
Diagram
Bar
Chart
Frequency
Distribution
Table
Tabulating Data

The Frequency
Distribution Table
Example: Hospital Patients by Unit
Hospital Unit Number of Patients
Cardiac Care 1,052
Emergency 2,245
Intensive Care 340
Maternity 552
Surgery 4,630
(Variables are
categorical)
Summarize data by category

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
Hospital Patients by Unit
0
1000
2000
3000
4000
5000
Cardiac
Care
Emergency
Intensive
Care
Maternity
Surgery
Number
of
patients
per
year
Hospital Number
Unit of Patients
Cardiac Care 1,052
Emergency 2,245
Intensive Care 340
Maternity 552
Surgery 4,630

Hospital Patients by Unit
Emergency
25%
Maternity
6%
Surgery
53%
Cardiac Care
12%
Intensive Care
4%
Pie Chart Example
(Percentages
are rounded to
the nearest
percent)
Hospital Number % of
Unit of Patients Total
Cardiac Care 1,052 11.93
Emergency 2,245 25.46
Intensive Care 340 3.86
Maternity 552 6.26
Surgery 4,630 52.50

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”

Example: 400 defective items are examined
for cause of defect:
Source of
Manufacturing Error Number of defects
Bad Weld 34
Poor Alignment 223
Missing Part 25
Paint Flaw 78
Electrical Short 19
Cracked case 21
Total 400
Pareto Diagram Example

Step 1: Sort by defect cause, in descending order
Step 2: Determine % in each category
Source of
Manufacturing Error Number of defects % of Total Defects
Poor Alignment 223 55.75
Paint Flaw 78 19.50
Bad Weld 34 8.50
Missing Part 25 6.25
Cracked case 21 5.25
Electrical Short 19 4.75
Total 400 100%
(continued)

cumulative
%
(line
graph)
%
of
defects
in
each
category
(bar
graph)
Pareto Diagram: Cause of Manufacturing Defect
0%
10%
20%
30%
40%
50%
60%
Poor Alignment Paint Flaw Bad Weld Missing Part Cracked case Electrical Short
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Step 3: Show results graphically
(continued)

Graphs for Time-Series Data
 A line chart (time-series plot) is used to show
the values of a variable over time
 Time is measured on the horizontal axis
 The variable of interest is measured on the
vertical axis

Line Chart Example
Magazine Subscriptions by Year
0
50
100
150
200
250
300
350
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
Thousands
of
subscribers

Numerical Data
Stem-and-Leaf
Display
Histogram Ogive
Frequency Distributions
and
Cumulative Distributions
Graphs to Describe
Numerical Variables

What is a Frequency Distribution?
 A frequency distribution is a list or a table …
 containing class groupings (categories or
ranges within which the data fall) ...
 and the corresponding frequencies with which
data fall within each class or category
Frequency Distributions

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

Class Intervals
and Class Boundaries
 Each class grouping has the same width
 Determine the width of each interval by
 Use at least 5 but no more than 15-20 intervals
 Intervals never overlap
 Round up the interval width to get desirable
interval endpoints
intervals
desired
of
number
number
smallest
number
largest
width
interval
w




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

 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 interval width: 10 (46/5 then round up)
 Determine interval boundaries: 10 but less than 20, 20 but
less than 30, . . . , 60 but less than 70
 Count observations & assign to classes
(continued)

Interval Frequency
10 but less than 20 3 .15 15
20 but less than 30 6 .30 30
30 but less than 40 5 .25 25
40 but less than 50 4 .20 20
50 but less than 60 2 .10 10
Total 20 1.00 100
Relative
Frequency Percentage
Data in ordered array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
(continued)

Histogram
 A graph of the data in a frequency distribution
is called a histogram
 The interval endpoints 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

Histogram: Daily High Temperature
0
3
6
5
4
2
0
0
1
2
3
4
5
6
7
0 10 20 30 40 50 60
Frequency
Temperature in Degrees
Histogram Example
(No gaps
between
bars)
Interval
10 but less than 20 3
Frequency
0 10 20 30 40 50 60 70

Histograms in Excel
Select
Tools/Data Analysis
1

Choose Histogram
2
3
Input data range and bin
range (bin range is a cell
range containing the upper
interval endpoints for each class
grouping)
Select Chart Output
and click “OK”
Histograms in Excel
(continued)
(

Questions for Grouping Data
into Intervals
 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
Frequency
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
More
Temperature
Frequency
(X axis labels are upper class endpoints)

The Cumulative
Frequency Distribuiton
Class
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
Total 20 100
Percentage
Cumulative
Percentage
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Frequency
Cumulative
Frequency

The Ogive
Graphing Cumulative Frequencies
Ogive: Daily High Temperature
0
20
40
60
80
100
10 20 30 40 50 60
Cumulative
Percentage
Interval endpoints
Interval
Less than 10 10 0
10 but less than 20 20 15
Cumulative
Percentage
Upper
interval
endpoint

Distribution Shape
 The shape of the distribution is said to be
symmetric if the observations are balanced,
or evenly distributed, about the center.
Symmetric Distribution
0
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9
Frequency

Distribution Shape
 The shape of the distribution is said to be
skewed if the observations are not
symmetrically distributed around the center.
(continued)
Positively Skewed Distribution
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9
Frequency
Negatively Skewed Distribution
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9
Frequency
A positively skewed distribution
(skewed to the right) has a tail that
extends to the right in the direction of
positive values.
A negatively skewed distribution
(skewed to the left) has a tail that
extends to the left in the direction of
negative values.

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)

Example
 Here, use the 10’s digit for the stem unit:
21, 24, 24, 26, 27, 27, 30, 32, 38, 41
 21 is shown as
 38 is shown as
Stem Leaf
2 1
3 8

Example
 Completed stem-and-leaf diagram:
Stem Leaves
2 1 4 4 6 7 7
3 0 2 8
4 1
(continued)
21, 24, 24, 26, 27, 27, 30, 32, 38, 41

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

Using other stem units
 Using the 100’s digit as the stem:
 The completed stem-and-leaf display:
Stem Leaves
(continued)
6 1 3 6
7 2 2 5 8
8 3 4 6 6 9 9
9 1 3 3 6 8
10 3 5 6
11 4 7
12 2
Data:
613, 632, 658, 717,
722, 750, 776, 827,
841, 859, 863, 891,
894, 906, 928, 933,
955, 982, 1034,
1047,1056, 1140,
1169, 1224

Relationships Between Variables
 Graphs illustrated so far have involved only a
single variable
 When two variables exist other techniques are
used:
Categorical
(Qualitative)
Variables
Numerical
(Quantitative)
Variables
Cross tables Scatter plots

 Scatter Diagrams are used for 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
Scatter Diagrams

Scatter Diagram Example
Cost per Dayvs. Production Volume
0
50
100
150
200
250
0 10 20 30 40 50 60 70
Volume per Day
Cost
per
Day
Volume
per day
Cost per
day
23 125
26 140
29 146
33 160
38 167
42 170
50 188
55 195
60 200

Scatter Diagrams in Excel
Select the chart wizard
1
2
Select XY(Scatter) option,
then click “Next”
When prompted, enter the
data range, desired
legend, and desired
destination to complete
the scatter diagram
3

Cross Tables
 Cross Tables (or contingency tables) list the
number of observations for every combination
of values for two categorical or ordinal
variables
 If there are r categories for the first variable
(rows) and c categories for the second
variable (columns), the table is called an r x c
cross table

Cross Table Example
 4 x 3 Cross Table for Investment Choices by Investor
(values in $1000’s)
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

 Side by side bar charts
(continued)
Graphing
Multivariate Categorical Data
C omparing 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

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

Data Presentation Errors
Goals for effective data presentation:
 Present data to display essential information
 Communicate complex ideas clearly and
accurately
 Avoid distortion that might convey the wrong
message

 Unequal histogram interval widths
 Compressing or distorting the
vertical axis
 Providing no zero point on the
vertical axis
 Failing to provide a relative basis
in comparing data between
groups
Data Presentation Errors
(continued)

Chapter Summary
 Reviewed types of data and measurement levels
 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 distribution
 Bar chart
 Pie chart
 Pareto diagram
 Line chart
 Frequency distribution
 Histogram and ogive
 Stem-and-leaf display
 Scatter plot
 Cross tables and
side-by-side bar charts

Newbold_chap02.ppt

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