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Chapter 2
Methods for
Describing Sets
of Data
ALWAYS LEARNING
Slide – 1
Contents
1. Describing Qualitative Data
2. Graphical Methods for Describing
Quantitative Data
3. Numerical Measures of Central
Tendency
4. Numerical Measures of Variability
5. Using the Mean and Standard Deviation
to Describe Data
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Slide – 2
Contents (cont)
6. Numerical Measures of Relative
Standing
7. Methods for Detecting Outliers: Box
Plots and z-scores
8. Distorting the Truth with Descriptive
Techniques
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Slide – 3
Learning Objectives
1. Describe data using graphs
2. Describe data using numerical measures
3. Describe quantitative data using
numerical measures
4. Detecting descriptive methods that
distort the truth
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Slide – 4
2.1
Describing Qualitative Data
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Slide – 5
Key Terms
A class is one of the categories into which
qualitative data can be classified.
The class frequency is the number of
observations in the data set falling into a
particular class.
The class relative frequency is the class
frequency divided by the total numbers of
observations in the data set.
The class percentage is the class relative
frequency multiplied by 100.
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Slide – 6
Data Presentation
Data
Presentation
Qualitative
Data
Quantitative
Data
Dot
Plot
Summary
Table
Bar
Graph
Pie
Chart
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Pareto
Diagram
Stem-&-Leaf
Display
Frequency
Distribution
Histogram
Slide – 7
Data Presentation
Data
Presentation
Qualitative
Data
Quantitative
Data
Dot
Plot
Summary
Table
Bar
Graph
Pie
Chart
ALWAYS LEARNING
Pareto
Diagram
Stem-&-Leaf
Display
Frequency
Distribution
Histogram
Slide – 8
Summary Table
1. Lists categories & number of elements in
category
2. Obtained by tallying responses in category
3. May show frequencies (counts), % or both
Row Is
Major
Category
Accounting
Economics
Management
Total
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Count
130
20
50
200
Tally:
|||| ||||
|||| ||||
Slide – 9
Data Presentation
Data
Presentation
Qualitative
Data
Quantitative
Data
Dot
Plot
Summary
Table
Bar
Graph
Pie
Chart
ALWAYS LEARNING
Pareto
Diagram
Stem-&-Leaf
Display
Frequency
Distribution
Histogram
Slide – 10
Bar Graph
Percent
Used
Also
Frequency
150
Equal Bar
Widths
Bar Height
Shows
Frequency or
%
100
50
0
Acct.
Econ.
Major
Zero Point
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Mgmt.
Vertical Bars
for Qualitative
Variables
Slide – 11
Data Presentation
Data
Presentation
Qualitative
Data
Quantitative
Data
Dot
Plot
Summary
Table
Bar
Graph
Pie
Chart
ALWAYS LEARNING
Pareto
Diagram
Stem-&-Leaf
Display
Frequency
Distribution
Histogram
Slide – 12
Pie Chart
1. Shows breakdown of
Majors
total quantity into
categories
Econ.
2. Useful for showing
10% 36°
relative differences
Acct.
65%
3. Angle size

Mgmt.
25%
(360°)(percent)
(360°) (10%) = 36°
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Slide – 13
Data Presentation
Data
Presentation
Qualitative
Data
Quantitative
Data
Dot
Plot
Summary
Table
Bar
Graph
Pie
Chart
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Pareto
Diagram
Stem-&-Leaf
Display
Frequency
Distribution
Histogram
Slide – 14
Pareto Diagram
Like a bar graph, but with the categories arranged
by height in descending order from left to right.
Percent
Used
Also
Frequency
150
Equal Bar
Widths
Bar Height
Shows
Frequency or
%
100
50
0
Acct.
Mgmt.
Major
Zero Point
ALWAYS LEARNING
Econ.
Vertical Bars
for Qualitative
Variables
Slide – 15
Summary
Bar graph: The categories (classes) of the
qualitative variable are represented by bars, where
the height of each bar is either the class frequency,
class relative frequency, or class percentage.
Pie chart: The categories (classes) of the
qualitative variable are represented by slices of a
pie (circle). The size of each slice is proportional to
the class relative frequency.
Pareto diagram: A bar graph with the categories
(classes) of the qualitative variable (i.e., the bars)
arranged by height in descending order from left to
right.
ALWAYS LEARNING
Slide – 16
Thinking Challenge
You’re an analyst for IRI. You want to show the
market shares held by Web browsers in 2016.
Construct a bar graph, pie chart, & Pareto
diagram to describe the data.
Browser
Firefox
Internet Explorer
Safari
Others
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Mkt. Share (%)
14
81
4
1
Slide – 17
Market Share (%)
Bar Graph Solution*
100%
80%
60%
40%
20%
0%
Firefox
Internet
Explorer
Safari
Others
Browser
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Slide – 18
Pie Chart Solution*
Market Share
Firefox,
14%
Safari, 4%
Others,
1%
Internet
Explorer,
81%
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Slide – 19
Market Share (%)
Pareto Diagram Solution*
100%
80%
60%
40%
20%
0%
Internet
Explorer
Firefox
Safari
Others
Browser
ALWAYS LEARNING
Slide – 20
2.2
Graphical Methods for
Describing Quantitative Data
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Slide – 21
Data Presentation
Data
Presentation
Qualitative
Data
Quantitative
Data
Dot
Plot
Summary
Table
Bar
Graph
Pie
Chart
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Stem-&-Leaf
Display
Histogram
Pareto
Diagram
Slide – 22
Dot Plot
1. Horizontal axis is a scale for the quantitative
variable, e.g., percent.
2. The numerical value of each measurement is
located on the horizontal scale by a dot.
ALWAYS LEARNING
Slide – 23
Data Presentation
Data
Presentation
Qualitative
Data
Quantitative
Data
Dot
Plot
Summary
Table
Bar
Graph
Pie
Chart
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Stem-&-Leaf
Display
Histogram
Pareto
Diagram
Slide – 24
Stem-and-Leaf Display
1. Divide each
observation into stem
2 144677
value and leaf value
• Stems are listed in
3 028
order in a column
• Leaf value is
4 1
placed in
corresponding stem
row to right of bar
2. Data: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41
ALWAYS LEARNING
26
Slide – 25
Data Presentation
Data
Presentation
Qualitative
Data
Quantitative
Data
Dot
Plot
Summary
Table
Bar
Graph
Pie
Chart
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Stem-&-Leaf
Display
Histogram
Pareto
Diagram
Slide – 26
We will let Minitab determine the number of classes for us.
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Slide – 27
Histogram
Class
15.5 – 25.5
25.5 – 35.5
35.5 – 45.5
Count
5
Frequency
Relative
Frequency
Percent
4
Freq.
3
5
2
3
Bars
Touch
2
1
0
0
15.5
25.5
35.5
45.5
55.5
Lower Boundary
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Slide – 28
Summary
Dot plot: The numerical value of each quantitative
measurement in the data set is represented by a dot
on a horizontal scale. When data values repeat, the
dots are placed above one another vertically.
Stem-and-leaf display: The numerical value of the
quantitative variable is partitioned into a “stem” and a
“leaf.” The possible stems are listed in order in a
column. The leaf for each quantitative measurement in
the data set is placed in the corresponding stem row.
Leaves for observations with the same stem value are
listed in increasing order horizontally.
ALWAYS LEARNING
Slide – 29
Summary
Histogram: The possible numerical values of the
quantitative variable are partitioned into class intervals,
where each interval has the same width. These
intervals form the scale of the horizontal axis. The
frequency or relative frequency of observations in each
class interval is determined. A horizontal bar is placed
over each class interval, with height equal to either the
class frequency or class relative frequency.
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Slide – 30
2.3
Numerical Measures
of Central Tendency
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Slide – 31
Two Characteristics
The central tendency of the set of
measurements–that is, the tendency of the
data to cluster, or center, about certain
numerical values.
Central Tendency
(Location)
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Slide – 32
Two Characteristics
The variability of the set of measurements–
that is, the spread of the data.
Variation
(Dispersion)
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Slide – 33
Mean
The mean of a set of quantitative data is the
sum of the measurements divided by the
number of measurements contained in the
data set.
n
xi

x  i 1
n
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Slide – 34
Summation Notation
To learn how to work with the Summation
To view a pre-recorded lecture based on this
ALWAYS LEARNING
Slide – 35
Example
Calculate the mean of the following six sample
measurements:
10.3, 4.9, 8.9, 11.7 , 6.3 , 7.7
n
x
x
i 1
n
i
10.3  4.9  8.9  11.7  6.3  7.7

6
49.8

 8.3
6
ALWAYS LEARNING
Slide – 36
Symbols for the Sample and
Population Mean
In this text, we adopt a general policy of
using Greek letters to represent population
numerical descriptive measures and Roman
letters to represent corresponding
descriptive measures for the sample. The
symbols for the mean are
Sample mean  x
Population mean  
ALWAYS LEARNING
Slide – 37
Median
1. Measure of central tendency
2. Middle value in ordered sequence
If n is odd, middle value of sequence
If n is even, average of 2 middle values
3. Position of median in sequence
n 1
Positioning Point 
2
4. Not affected by extreme values
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Slide – 38
Median Example Odd-Sized Sample
Raw Data: 24.1 22.6 21.5 23.7 22.6
Ordered: 21.5 22.6 22.6 23.7 24.1
Position:
1
2
3
4
5
Median = 22.6
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Slide – 39
Median Example Even-Sized Sample
Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7
Ordered:
4.9 6.3 7.7 8.9 10.3 11.7
Position:
1
2
3
4
5
6
7.7  8.9
Median 
 8.3
2
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Slide – 40
Skewed
A data set is said to be skewed if one tail of the
distribution has more extreme observations than
the other tail.
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Slide – 41
Shape
1. Describes how data are distributed
2. Measures of Shape
Left-Skewed
Mean Median
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Symmetric
Mean = Median
Right-Skewed
Median Mean
Slide – 42
Mode
1. Measure of central tendency
2. Value that occurs most often
3. Not affected by extreme values
4. May be no mode or several modes
5. May be used for quantitative or
qualitative data
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Slide – 43
Mode Example
No Mode
Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7
One Mode
Raw Data: 6.3 4.9 8.9 6.3 4.9 4.9
More Than 1 Mode
Raw Data: 21 28
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28
41
43
43
Slide – 44
Thinking Challenge
You’re a financial analyst
for Prudential-Bache
Securities. You have
collected the following
closing stock prices of
new stock issues: 17, 16,
21, 18, 13, 16, 12, 11.
Describe the stock prices
in terms of central
tendency.
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Slide – 45
Solution
17, 16, 21, 18, 13, 16, 12, 11
17  16  21  18  13  16  12  11
x
8
 15.5
Raw Data: 17 16 21 18 13 16 12 11
Ordered:
11 12 13 16 16 17 18 21
Median = 16
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Slide – 46
Solution (cont)
Mode
Raw Data: 17 16 21 18 13 16 12 11
Mode = 16
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Slide – 47
Suggested Exercises
Work out the following exercises from the
Textbook :
2.37, 2.38, 2.41, 2.46, 2.49, 2.51, 2.55
These exercises will not be collected or graded,
but let me know as questions arise.
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Slide – 48
2.4
Numerical Measures
of Variability
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Slide – 49
Range
1. Measure of dispersion
2. Difference between largest & smallest
observations
Range = xlargest – xsmallest
3. Ignores how data are distributed
7 8 9 10
7 8 9 10
Range = 10 – 7 = 3
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Range = 10 – 7 = 3
Slide – 50
Let’s examine the two datasets below:
Dataset 1:
Values
Frequencies
Dataset 2:
-10,000
1
0 10,000
Values
99
Frequencies
Range = 10000 – (-10000) = 20000
1
-10,000
50
0 10,000
1
50
Range = 10000 – (-10000) = 20000
Which dataset is more variable?
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Slide – 51
Variance &
Standard Deviation
1. Measures of dispersion
2. Most common measures
3. Consider how data are distributed
4. Show variation about mean (x or μ)
x = 8.3
4
6
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8 10 12
Slide – 52
Sample Variance Formula
n
s2 
 x
i 1
i
 x
2
n 1
x1  x    x2  x 

2
2

  xn  x 
2
n 1
Standard Deviation is the positive
square root of Variance.
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Slide – 53
Sample Variance Formula
From page 8 of the note on Summation Notation:
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Slide – 54
Sample Variance Formula
Here is the sample variance formula in English:
1. Calculate the Sum of the data values
2. Calculate the Sum of Squares of the data
values
3. Divide the Square of Sum by the number of
data values and subtract the result from the
Sum of Squares
4. Divide the result of Step 3 by (the number of
data values minus 1)
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Slide – 55
Symbols for Variance and Standard
Deviation
s2 = Sample variance
s = Sample standard deviation
 2 = Population variance
 = Population standard deviation
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Slide – 56
Example
Calculate the variance and standard
deviation. 10.3, 4.9, 8.9, 11.7, 6.3, 7.7
Solution
The first step is finding the mean. Which we
calculated earlier to be 8.3.
(10.3  8.3)  (4.9  8.3)  …  (7.7  8.3)
s 
6 1
s 2  6.368
2
2
2
2
s  6.368  2.52
ALWAYS LEARNING
Slide – 57
Thinking Challenge
You’re a financial analyst
for Prudential-Bache
Securities. You have
collected the following
closing stock prices of
new stock issues: 17, 16,
21, 18, 13, 16, 12, 11.
What are the variance
and standard deviation
of the stock prices?
ALWAYS LEARNING
Slide – 58
Thinking Challenge Solution
Sample Variance
17 16 21 18 13
16
12
11
The mean = 15.5
(17  15.5)  (16  15.5)  …  (11  15.5)
s 
8 1
 11.14
2
2
2
2
s  11.14  3.337
ALWAYS LEARNING
Slide – 59
Minitab Calculations
This tutorial shows how to calculate the mean,
median, range, and standard deviation of a data
set, when the data structure is that of a “linear
array.” What that means is the following: All of your
data values are in one column of your
spreadsheet, as opposed to a “frequency
distribution.”
It also illustrates the calculation of some other
descriptive measures that we have not yet
discussed. Remember to view the tutorial again
after finishing slide #91.
ALWAYS LEARNING
Slide – 60
An example
The following example, Psychology Final,
illustrates how to calculate the mean and the
standard deviation from a frequency
distribution.
It also includes some other calculations. We
have not yet discussed the underlying
concepts. As such, at this stage you should
simply note that this is an example you
should come back to.
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Slide – 61
Suggested Exercises
Work out the following exercises from the
Textbook :
2.57, 2.61, 2.63, 2.68
These exercises will not be collected or graded,
but let me know as questions arise.
ALWAYS LEARNING
Slide – 62
2.5
Using the Mean and Standard
Deviation to Describe Data
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Slide – 63
Using the Mean and Standard Deviation
to Describe Data: Chebyshev’s Rule
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Slide – 64
Interpreting Standard Deviation:
Chebyshev’s Theorem
x  3s
x  2s
x s
x
xs
x  2s
x  3s
No useful information
At least 3/4 of the data
At least 8/9 of the data
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Slide – 65
Chebyshev’s Theorem Example
Previously we found the mean
closing stock price of new stock
issues is 15.5 and the standard
deviation is 3.34.
Use this information to form an
interval that will contain at least
75% of the closing stock prices
of new stock issues.
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Slide – 66
Chebyshev’s Theorem Example
At least 75% of the closing stock prices of new
stock issues will lie within 2 standard deviations of
the mean.
x = 15.5
s = 3.34
(x – 2s, x + 2s) = (15.5 – 2∙3.34, 15.5 + 2∙3.34)
= (8.82, 22.18)
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Slide – 67
Another Chebyshev’s Theorem Example
The example, Lumber Company, illustrates
a more sophisticated use of Chebyshev’s
Theorem.
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Slide – 68
Interpreting Standard Deviation:
Empirical Rule
Applies to data sets that are mound shaped and
symmetric (How do you know if your data set is
mound shaped and symmetric?)
Approximately 68% of the measurements lie in the
interval x  s to x  s
Approximately 95% of the measurements lie in the
interval x  2s to x  2s
Approximately 99.7% …