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it has 6 questions total in this assignment , and the course is introduction to statistic, we use the tool is minitab 19。You will need Microsoft WORD to work on the Assignments. If you do not have Microsoft WORD installed on your
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Chapter 2
Methods for
Describing Sets
of Data
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 4
2.1
Describing Qualitative Data
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Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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.
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 6
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Count
130
20
50
200
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Mgmt.
Vertical Bars
for Qualitative
Variables
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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°
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 13
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Mkt. Share (%)
14
81
4
1
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 17
Market Share (%)
Bar Graph Solution*
100%
80%
60%
40%
20%
0%
Firefox
Internet
Explorer
Safari
Others
Browser
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 18
Pie Chart Solution*
Market Share
Firefox,
14%
Safari, 4%
Others,
1%
Internet
Explorer,
81%
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 19
Market Share (%)
Pareto Diagram Solution*
100%
80%
60%
40%
20%
0%
Internet
Explorer
Firefox
Safari
Others
Browser
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 20
2.2
Graphical Methods for
Describing Quantitative Data
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 21
Data Presentation
Data
Presentation
Qualitative
Data
Quantitative
Data
Dot
Plot
Summary
Table
Bar
Graph
Pie
Chart
ALWAYS LEARNING
Stem-&-Leaf
Display
Histogram
Pareto
Diagram
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 23
Data Presentation
Data
Presentation
Qualitative
Data
Quantitative
Data
Dot
Plot
Summary
Table
Bar
Graph
Pie
Chart
ALWAYS LEARNING
Stem-&-Leaf
Display
Histogram
Pareto
Diagram
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
26
Slide – 25
Data Presentation
Data
Presentation
Qualitative
Data
Quantitative
Data
Dot
Plot
Summary
Table
Bar
Graph
Pie
Chart
ALWAYS LEARNING
Stem-&-Leaf
Display
Histogram
Pareto
Diagram
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 26
We will let Minitab determine the number of classes for us.
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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.
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 30
2.3
Numerical Measures
of Central Tendency
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Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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)
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 32
Two Characteristics
The variability of the set of measurements–
that is, the spread of the data.
Variation
(Dispersion)
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Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 34
Summation Notation
To learn how to work with the Summation
Notation, read this note.
To view a pre-recorded lecture based on this
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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.
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
28
41
43
43
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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.
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 46
Solution (cont)
Mode
Raw Data: 17 16 21 18 13 16 12 11
Mode = 16
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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.
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 48
2.4
Numerical Measures
of Variability
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Range = 10 – 7 = 3
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 50
Range can be misleading
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?
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
8 10 12
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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.
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 53
Sample Variance Formula
From page 8 of the note on Summation Notation:
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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)
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 55
Symbols for Variance and Standard
Deviation
s2 = Sample variance
s = Sample standard deviation
 2 = Population variance
 = Population standard deviation
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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.
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 62
2.5
Using the Mean and Standard
Deviation to Describe Data
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 63
Using the Mean and Standard Deviation
to Describe Data: Chebyshev’s Rule
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Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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.
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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)
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
Slide – 67
Another Chebyshev’s Theorem Example
The example, Lumber Company, illustrates
a more sophisticated use of Chebyshev’s
Theorem.
ALWAYS LEARNING
Copyright © 2018, 2014, and 2011 Pearson Education, Inc.
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% …
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