Objective B : Construct a Frequency / Relative

Chapter 2
Descriptive Statistics
Copyright of the definitions and examples is reserved to Pearson Education,
Inc.. In order to use this PowerPoint presentation, the required textbook for
the class is the Fundamentals of Statistics, Informed Decisions Using Data,
Michael Sullivan, III, fourth edition.
Los Angeles Mission College
Prepared by DW
Ch 2.1 Organizing Qualitative Data
Objective A : Interpretation of a Basic Statistical Graph
Objective B : Construct a Frequency / Relative Frequency
Distribution, Bar Graph, Pareto Chart and Pie
Chart
Los Angeles Mission College
Prepared by DW
Ch 2.1 Organizing Qualitative Data
Objective A : Interpretation of a Basic Statistical Graph
Example 1 :
Identity Theft Identity fraud occurs someone else’s personal
information is used to open credit card accounts, apply for a job,
receive benefits, and so on. The following relative frequency bar
graph represents the various types of identity theft based on a study
conducted by the Federal Trade Commission.
Los Angeles Mission College
Prepared by DW
(a) Approximate what percentage of identity theft was loan fraud
(such as applying for a loan in someone else’s name)?
0.05
100%  5%
1
(b) If there were 10 million cases of identity fraud in 2008, how many
were credit card fraud (someone uses someone else’s credit card to
make a purchase) ?
0.26
10 million  2.6 million
1
Los Angeles Mission College
Prepared by DW
Ch 2.1 Organizing Qualitative Data
Objective A : Interpretation of a Basic Statistical Graph
Objective B : Construct a Frequency / Relative Frequency
Distribution, Bar Graph, Pareto Chart and Pie
Chart
Los Angeles Mission College
Prepared by DW
Ch 2.1 Organizing Qualitative Data
Objective B : Construct a Frequency / Relative Frequency
Distribution, Bar Graph, Pareto Chart and Pie
Chart
B1. Frequency / Relative Frequency Distribution
• A frequency distribution lists each category of data and the
frequency which is the number of occurrences for each category
data.
• A relative frequency distribution lists each category of data and the
relative frequency which is the proportion of observation within a
category.
Relative frequency 
Los Angeles Mission College
frequency
sum of all frequency
Prepared by DW
Example 1 : In a national survey conducted by the Centers of Disease
Control to determine health-risk behaviors among college
students, college students were asked, “How often do you
wear a seat beat when driving a car?” The frequencies were
as follows:
Response
I do not drive a car
Never
Rarely
Sometimes
Most of the time
Always
Los Angeles Mission College
Frequency
249
118
249
345
716
3093
Prepared by DW
(a) Construct a relative frequency distribution.
Response
I do not drive a car
Never
Rarely
Sometimes
Most of the time
Always
Los Angeles Mission College
Frequency (f ) Relative Frequency ( f / f )
249
118
249
345
716
3093
 f  4770
249 / 4770  0.052
118 / 4770  0.025
249 / 4770  0.052
345 / 4770  0.072
716 / 4770  0.150
3093/ 4770  0.648
Prepared by DW
(b) What percentage of respondents answered “Always”?
64.8%
(c) What percentage of respondents answered “Never” or “Rarely”?
2.5% + 5.2% = 7.7%
(d) Suppose that a representative from the Centers for Disease Control
says, “2.5% of the college students in this survey responded that
they never wear a seat belt.” Is this a descriptive or inferential
statement?
Descriptive statement.
Los Angeles Mission College
Prepared by DW
Ch 2.1 Organizing Qualitative Data
Objective B : Construct a Frequency / Relative Frequency
Distribution, Bar Graph, Pareto Chart and Pie
Chart
B2. Construct a Bar Graph, a Pareto Chart, or a Pie Chart
• A bar graph is constructed by labeling each category of data on
either the horizontal or vertical axis and the frequency or relative
frequency of the category on the other axis. Rectangles of equal
width are drawn for each category. The height of each rectangle
represents the category’s frequency or relative frequency.
• A Pareto chart is a bar graph whose bars are drawn in decreasing
order of frequency or relative frequency.
• A pie chart is a circle divided into sectors. Each sector represents a
category of data. The area of each sector is proportion to the
frequency of the category.
Los Angeles Mission College
Prepared by DW
Example 2 : A sample of 40 randomly selected registered voters in
Sylmar was asked their Political affiliation: Democrat (D),
Republican (R), Independent (I). The results of the survey
are as follows:
R D R D R R D R D D D D R R D R D D I D D R R D D D I D
R D D D I R D R D D D R
(a) Construct a frequency distribution of the data.
Political Party
Frequency ( f )
Republican
14
23
Democrat
Independen t
Los Angeles Mission College
3
Prepared by DW
(b) Construct a relative frequency distribution of the data.
Political Affilation
Frequency
Relative Frequency ( f / f )
Republican
14
Democrat
23
3
f  40
14 / 40  0.350
23 / 40  0.575
3 / 40  0.075
Independen t
(c) Construct a frequency bar graph.
25
Frequency
20
15
10
5
0
R
Los Angeles Mission College
D
Party
I
Prepared by DW
(d) Construct a relative frequency bar graph.
0.7
Relative Frequency
0.6
0.5
0.4
0.3
0.2
0.1
0
R
D
I
Party
(e) Construct a Pareto chart.
25
Frequency
20
15
10
5
0
D
R
I
Party
Los Angeles Mission College
Prepared by DW
(f) Construct a pie chart.
I
7%
R
35%
D
58%
Los Angeles Mission College
Prepared by DW
(g) Use StatCrunch to construct a pie chart.
Click StatCrunch navigation button of the Course Home page 
Click StatCrunch website  Click Open StatCrunch 
Input the raw data in Var 1 column  Click Graph  Pie chart 
with Data  Click Var 1 for Select column(s):
Los Angeles Mission College
Prepared by DW
Los Angeles Mission College
Prepared by DW
The pie chart is obtained from StatCrunch.
For more detailed instructions, please download “Q2.1.24 “ by
clicking the StatCrunch Handout navigation button of the course
homepage.
Los Angeles Mission College
Prepared by DW
Ch 2.2 Organizing Quantitative Data : The Popular
Displays
Objective A : Histogram
Objective B : Constructing a Stem-and-Leaf Plot
Objective C : Construct Frequency Distributions and
Histogram for Continuous Data
Objective D : Time Series Graphs
Los Angeles Mission College
Prepared by DW
Ch 2.2 Organizing Quantitative Data : The Popular
Displays
Objective A : Histogram
A histogram is constructed by drawing rectangles for each class of
data. If the discrete data set is small, each number is a class. If the
discrete data set is large or the data are continuous, the classes must
be created using interval of numbers. The height of each rectangle is
the frequency or relative frequency of the class. The width of each
rectangle is the same and the rectangles touch each other.
Los Angeles Mission College
Prepared by DW
Construct Frequency Distribution and Histogram for
Discrete Data
Example 1 : The following data represent the number of customers
waiting for a table at 6:00 p.m. for 40 consecutive
Saturdays at Bobak’s Restaurant:
11 5 11 3 6 8 6 7 4 5 13 9 6 4 14 11 13 10 9 6 8
10 9 5 10 8 7 3 8 8 7 8 7 9 10 4 8 6 11 8
(a) Are these data discrete or continuous? Explain
Discrete because the data are whole numbers.
Los Angeles Mission College
Prepared by DW
(b) Construct a frequency distribution of the data.
Number of customers' waiting
3
4
5
6
7
8
9
10
11
12
Los Angeles Mission College
Frequency ( f )
2
3
3
13
5
4
8
4
4
4
0
2
14
1
Prepared by DW
(c) Construct a relative frequency distribution of the data.
Number of customers' waiting
3
4
5
6
7
8
9
10
11
12
13
14
Los Angeles Mission College
Frequency
2
3
3
5
4
8
4
4
4
0
2
1
f  40
Relative Frequency ( f / f )
2 / 40  0.050
3/ 40  0.075
3 / 40  0.075
5 / 40  0.125
4 / 40  0.100
8 / 40  0.200
4 / 40  0.100
4 / 40  0.100
4 / 40  0.100
0 / 40  0.000
2 / 40  0.050
1/ 40  0.025
Prepared by DW
(d) What percentage of the Saturdays had 10 or more customers
waiting for a table at 6:00 p.m.?
0.100 + 0.100 + 0.000 + 0.050 + 0.025 = 0.275  0.3
(e) Construct a frequency histogram of the data.
9
8
7
Frequency
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Number of Customers' Waiting
Los Angeles Mission College
Prepared by DW
Identify the shape of each distribution.
Uniform distribution
Bell-shaped curve
Right-skewed distribution
Left-skewed distribution
Los Angeles Mission College
Prepared by DW
Ch 2.2 Organizing Quantitative Data : The Popular
Displays
Objective A : Histogram
Objective B : Constructing a Stem-and-Leaf Plot
Objective C : Construct Frequency Distributions and
Histogram for Continuous Data
Objective D : Time Series Graphs
Los Angeles Mission College
Prepared by DW
Objective B : Constructing a Stem-and-Leaf Plot
The stem of a data value will consist of the digits to the left of the
rightmost digit.
The leaf of a data value will be the rightmost digit.
Los Angeles Mission College
Prepared by DW
Example 1 : The following data represent the number of miles per gallon
achieved on the highway for small cars for the model year
2008.
27 31 28 30 52 25 33 33 29 23 27 37 30 45 24 32
34 35 31 44 42 26 43 35 36 36 54 33 32 35 34 37
Reorder: 23 24 25 26 27 27 28 29 30 30 31 31 32 32
33 33 33 34 34 35 35 35 36 36 37 37 42 43
44 45 52 54
(a) Construct a stem-and-leaf plot.
Stem Leaf
2
3 4 5 6 7 7 8 9
3
0 0 1 1 2 2 3 3 3 4 4 5 5 5 6 6 7 7
4
2 3 4 5
5
2 4
(b) Describe the shape of the distribution.
Slightly skewed to the right.
Los Angeles Mission College
Prepared by DW
Ch 2.2 Organizing Quantitative Data : The Popular
Displays
Objective A : Histogram
Objective B : Constructing a Stem-and-Leaf Plot
Objective C : Construct Frequency Distributions and
Histogram for Continuous Data
Objective D : Time Series Graphs
Los Angeles Mission College
Prepared by DW
Objective C : Construct Frequency Distributions and
Histogram for Continuous Data
• Classes are categories in which data are grouped.
• The lowest class limit is the smallest value within a class.
• The upper class limit is the largest value within a class.
• The class width is the difference between consecutive lower class
limits.
• The class width is computed by the following formula.
largest data value – smallest data value
Class width 
number of class
-------> Round this value up to the same decimal
place as the raw data.
Los Angeles Mission College
Prepared by DW
Example 1 : The following data represent the fall 2006 student
headcount enrollments for all public community colleges
in the state of Illinois.
(a) Find the number of class.
6
Los Angeles Mission College
Prepared by DW
(b) Find the class limits.
Lowest class limits:
0, 5,000, 10,000, 15,000, 20,000, 25,000
Upper class limits: 4,999, 9,999, 14,999, 19,999, 24,999, 29,999
(c) Find the class width.
Class width = 5000 – 0 = 5000
Los Angeles Mission College
Prepared by DW
Example 2 : Uninsured Rates The following data represent the
percentage of people without health insurance for the 50
states and the District of Columbia in 2009. (Ch 2.2 Q36
p. 94)
With the first class having a lower class limit of 4 and a class width of 2:
Los Angeles Mission College
Prepared by DW
(a) Construct a frequency distribution.
Put the data in ascending order first :
( 4.2) ( 8.6 9.2
9.6 9.6 9.7)(10.2 10.5 10.6 10.6 10.9 10.9 11.3 11.4 11.6)
( 12.3 12.6 13.0 13.3 13.4 13.9)(14.0 14.3 14.7 14.8 15.5 15.9 15.9)(16.1 16.1
16.1 16.2 17.8)(18.1 18.3 18.3 18.4 18.4 18.6 18.7 18.9 19.4 19.6 19.7)(20.6
21.1 21.2 21.3 21.4)(22.2) (25.0)
2
2
2
Los Angeles Mission College
Class Limit
4  5.9
6  7.9
8  9.9
10  11.9
12  13.9
14  15.9
16  17.9
18  19.9
20  21.9
22  23.9
24  25.9
Frequency( f )
1
0
5
9
6
7
5
11
5
1
1
Prepared by DW
(b) Construct a relative frequency distribution.
Class Limit
4  5.9
6
8
10
12
14
16
18
20
22
24










Los Angeles Mission College
7.9
9.9
11.9
13.9
15.9
17.9
19.9
21.9
23.9
25.9
Frequency ( f )
1
0
5
9
6
7
5
11
5
1
1
f  51
Relative Frequency ( f )
1/ 51  0.0196
0 / 51  0
5 / 51  0.0980
9 / 51  0.1765
6 / 51  0.1176
7 / 51  0.1373
5 / 51  0.0980
11/ 51  0.2157
5 / 51  0.0980
1/ 51  0.0196
1/ 51  0.0196
Prepared by DW
(c) Construct a frequency histogram of the data.
Los Angeles Mission College
Prepared by DW
(d) Construct a relative frequency histogram of the data.
(e) Describe the shape of the distribution.
Skewed to the right.
Los Angeles Mission College
Prepared by DW
(f) Use StatCrunch to repeat parts (a) to (e) with the first class
having a lower class of 4 and a class width of 4.
Note: For qualitative data, we can use StatCrunch to construct a
frequency histogram first, then from the histogram we obtain
the frequency distribution.
Construct a frequency histogram first  part (c)
Step 1: 1) Log in StatCunch  Data Sets from your textbook
 Click Chapter 2  Click 2.2.36
2) Click Graph → Histogram.
Los Angeles Mission College
Prepared by DW
Step 2: 1) Click Uninsured Rates under Select Column(s):
2) Choose Frequency under Type:
3) Under Bins: --> enter 4 for Start at: --> enter 4 for Width:
4) Under Display option: --> check √ Value above bar.
5) Under Graph Properties: --> enter Uninsured Rates for X-axis label,
Frequency for Y-axis label, and Frequency Histogram for Uninsured
Rates for Title.
6) Click Compute!
Los Angeles Mission College
Prepared by DW
The frequency histogram is obtained from StatCrunch.
Los Angeles Mission College
Prepared by DW
Construct a frequency distribution.
From the frequency histogram, the classes of uninsured rates can
be determined.
Class Limit
Frequency( f )
4  5.9
1
6  7.9
14
8  9.9
13
10  11.9
16
12  13.9
6
14  15.9
1
For more detailed instructions, please download “Q2.R.6 “ by
clicking the StatCrunch Handout navigation button of the course
homepage.
Los Angeles Mission College
Prepared by DW
Example 3 : The largest value of a data set is 125 and the smallest
value of the data set is 27. If six classes are to be
formed, calculate an appropriate class width.
Class width = Roundup (
= Roundup (
largest data value – smallest data value
number of class
125 – 27
6
)
)
= Roundup( 16.3333)
 17
Los Angeles Mission College
Prepared by DW
Ch 2.2 Organizing Quantitative Data : The Popular
Displays
Objective A : Histogram
Objective B : Constructing a Stem-and-Leaf Plot
Objective C : Construct Frequency Distributions and
Histogram for Continuous Data
Objective D : Time Series Graphs
Los Angeles Mission College
Prepared by DW
Objective D : Time Series Graphs
A time series graph represents the values of a variable that have
been collected over a specified period of time. The horizontal axis is
the time and the vertical axis is the value of the variable. Line
segments are drawn by connective consecutive points of time and
corresponding value of the variable.
Los Angeles Mission College
Prepared by DW
Example 1: The following time-series graph shows the annual U.S.
motor vehicle production from 1990 through 2008.
Los Angeles Mission College
Prepared by DW
(a) Estimate the number of motor vehicles produced in the United
States in 1991.
8900 thousands vehicles.
(b) Estimate the number of motor vehicles produced in the United
States in 1999.
13000 thousands vehicles.
(c) Use the results from (a) and (b) to estimate the percent increase in
the number of motor vehicles produced from 1991 to 1999.
Amount Increase
100  13000  8900 100  46.1%
Original
8900
(d) Estimate the percent decrease in the number of motor vehicles
produced from 1999 to 2008.
Amount Decrease
100  8800  13000 100  32.3%
Original
13000
Los Angeles Mission College
Prepared by DW
Ch 2.3 Graphical Misrepresentations of Data
Los Angeles Mission College
Prepared by DW
Ch 2.3 Graphical Misrepresentations of Data
The most common graphical misinterpretation of data is accomplished
through manipulation of the scale of the graph.
Example 1 :
Union Membership The following relative frequency histogram
represents the proportion of employed people aged 25 to 64 years old
who were members of a union.
Los Angeles Mission College
Prepared by DW
(a) Describe how this graph is misleading. What might a reader conclude
from the graph?
The vertical axis starts a 0.08 instead of 0. Readers may think the
proportion of those employed aged 45 to 54 years who are union
members is much higher than for those aged 35 to 44 years.
(b) Redraw the histogram with a starting point of zero on the vertical
axis so that it is not misleading
0.18
Union Membership
Proportion Employed
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
Los Angeles Mission College
25
35
45
Age
55
65
Prepared by DW
Example 2 :
Inauguration Cost The following is a USA Today-type graph. Explain
how it is misleading.
The lengths of the bars are not proportional. For example, the bar
representing the cost of Clinton’s inauguration should be slightly more
than 9 times the one for Carter’s cost and twice as long as the bar
representing Reagan’s cost.
Los Angeles Mission College
Prepared by DW