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QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-1
Chapter 3
Describing Data Using
Numerical Measures
Business Statistics
Department of Quantitative Methods & Information Systems
Dr. Mohammad Zainal QMIS 120
After completing this chapter, you should be able to:
Compute and interpret the mean, median, and mode for a
set of data
Compute the range, variance, and standard deviation and
know what these values mean
Construct and interpret a box and whisker graph
Compute and explain the coefficient of variation and z scores
Use numerical measures along with graphs, charts, and
tables to describe data
Chapter Goals
QMIS 120, by Dr. M. Zainal Chap 3-2
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-2
Chapter Topics
Measures of Center and Location
Mean, median, mode
Other measures of Location
Weighted mean, percentiles, quartiles
Measures of Variation
Range, interquartile range, variance and standard
deviation, coefficient of variation
Using the mean and standard deviation together
Coefficient of variation, z-scores
QMIS 120, by Dr. M. Zainal Chap 3-3
Summary Measures
Center and Location
Mean
Median
Mode
Other Measures
of Location
Weighted Mean
Describing Data Numerically
Variation
Variance
Standard Deviation
Coefficient of
Variation
Range
Percentiles
Interquartile Range
Quartiles
QMIS 120, by Dr. M. Zainal Chap 3-4
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-3
Measures of Center and Location
Center and Location
Mean Median Mode Weighted Mean
N
x
n
x
x
N
i
i
n
i
i
1
1
i
ii
W
i
iiW
w
xw
w
xwX
Overview
QMIS 120, by Dr. M. Zainal Chap 3-5
Mean (Arithmetic Average)
The Mean is the arithmetic average of data
values
Population mean
Sample mean n = Sample Size
N = Population Size
n
xxx
n
x
x n
n
i
i
211
N
xxx
N
xN
N
i
i
211
QMIS 120, by Dr. M. Zainal Chap 3-6
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-4
Mean (Arithmetic Average)
The most common measure of central tendency
Mean = sum of values divided by the number of values
Affected by extreme values (outliers)
(continued)
0 1 2 3 4 5 6 7 8 9 10
Mean = 3
0 1 2 3 4 5 6 7 8 9 10
Mean = 4
35
15
5
54321
4
5
20
5
104321
QMIS 120, by Dr. M. Zainal Chap 3-7
Median
In an ordered array, the median is the “middle”
number, i.e., the number that splits the
distribution in half
The median is not affected by extreme values
0 1 2 3 4 5 6 7 8 9 10
Median = 3
0 1 2 3 4 5 6 7 8 9 10
Median = 3
QMIS 120, by Dr. M. Zainal Chap 3-8
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-5
Median
To find the median, sort the n data values
from low to high (sorted data is called a
data array)
Find the value in the i = (1/2)n position
The ith position is called the Median Index
Point
If i is not an integer, round up to next highest
integer
(continued)
Chap 3-9 QMIS 120, by Dr. M. Zainal
Median Example
Note that n = 13
Find the i = (1/2)n position:
i = (1/2)(13) = 6.5
Since 6.5 is not an integer, round up to 7
The median is the value in the 7th position:
Md = 12
(continued)
Data array:
4, 4, 5, 5, 9, 11, 12, 14, 16, 19, 22, 23, 24
Chap 3-10 QMIS 120, by Dr. M. Zainal
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-6
Shape of a Distribution
Describes how data is distributed
Symmetric: identical on both sides of its central point
Skewed: the tail on one side is longer than the tail
on the other side
QMIS 120, by Dr. M. Zainal Chap 3-11
Shape of a Distribution
Mean = Median
Mean < Median Median < Mean
Right-Skewed Left-Skewed Symmetric
(Longer tail extends to left) (Longer tail extends to right)
QMIS 120, by Dr. M. Zainal Chap 3-12
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-7
Mode
A measure of location
The value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may be no mode
There may be several modes
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 5
0 1 2 3 4 5 6
No Mode
QMIS 120, by Dr. M. Zainal Chap 3-13
Weighted Mean
Used when values are grouped by frequency or
relative importance
Days to
Complete Frequency
5 4
6 12
7 8
8 2
Example: Sample of
26 Repair Projects
QMIS 120, by Dr. M. Zainal Chap 3-14
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-8
Five houses on a hill by the beach
Review Example
$2,000 K
$500 K
$300 K
$100 K
$100 K
House Prices:
$2,000,000
500,000
300,000
100,000
100,000
Chap 3-15 QMIS 120, by Dr. M. Zainal
Summary Statistics
Chap 3-16 QMIS 120, by Dr. M. Zainal
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-9
Mean is generally used, unless
extreme values (outliers) exist
Then Median is often used, since
the median is not sensitive to
extreme values.
Example: Median home prices may be
reported for a region – less sensitive to
outliers
Which measure of location is the “best”?
Chap 3-17 QMIS 120, by Dr. M. Zainal
Which measure of location is the “best”?
Example: Find the mean and median for the
following two sets of measurements
2, 9, 11, 5, 6 2, 9, 110, 5, 6
Chap 3-18 QMIS 120, by Dr. M. Zainal
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-10
Measures of Variation
Variation
Variance Standard Deviation Coefficient of
Variation
Population
Variance
Sample
Variance
Population
Standard
Deviation
Sample
Standard
Deviation
Range
Interquartile
Range
QMIS 120, by Dr. M. Zainal Chap 3-19
Measures of variation give information on
the spread or variability of the data
values.
Variation
Same center,
different variation
Chap 3-20 QMIS 120, by Dr. M. Zainal
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-11
Range
Simplest measure of variation
Difference between the largest and the smallest
observations:
Range = xmaximum – xminimum
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Range = 14 - 1 = 13
Example:
QMIS 120, by Dr. M. Zainal Chap 3-21
Ignores the way in which data are distributed
Sensitive to outliers
7 8 9 10 11 12
Range = 12 - 7 = 5
7 8 9 10 11 12
Range = 12 - 7 = 5
Disadvantages of the Range
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Range = 5 - 1 = 4
Range = 120 - 1 = 119
QMIS 120, by Dr. M. Zainal Chap 3-22
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-12
Average of squared deviations of values from
the mean
Population variance:
Sample variance:
Variance
N
μ)(x
σ
N
1i
2
i2
1- n
)x(x
s
n
1i
2
i2
QMIS 120, by Dr. M. Zainal Chap 3-23
Standard Deviation
Most commonly used measure of variation
Shows variation about the mean
Has the same units as the original data
Population standard deviation:
Sample standard deviation:
N
μ)(x
σ
N
1i
2
i
1-n
)x(x
s
n
1i
2
i
QMIS 120, by Dr. M. Zainal Chap 3-24
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-13
Calculation Example: Sample Standard Deviation
Sample
Data (Xi) : 10 12 14 15 17 18 18 24
QMIS 120, by Dr. M. Zainal Chap 3-25
Comparing Standard Deviations
Mean = 15.5
s = 3.338 11 12 13 14 15 16 17 18 19 20 21
11 12 13 14 15 16 17 18 19 20 21
Data B
Data A
Mean = 15.5
s = .9258
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 4.57
Data C
Same mean, but different
standard deviations:
QMIS 120, by Dr. M. Zainal Chap 3-26
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-14
Using Microsoft Excel
Descriptive Statistics are easy to obtain
from Microsoft Excel
Use menu choice:
Data / data analysis / descriptive statistics
Enter details in dialog box
QMIS 120, by Dr. M. Zainal Chap 3-27
Using Excel
Select:
Data / data analysis / descriptive statistics
QMIS 120, by Dr. M. Zainal Chap 3-28
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-15
Enter dialog box details
Check box for summary statistics
Click OK
Using Excel (continued)
QMIS 120, by Dr. M. Zainal Chap 3-29
Excel output
Microsoft Excel
descriptive statistics output,
using the house price data:
House Prices:
$2,000,000
500,000
300,000
100,000
100,000
QMIS 120, by Dr. M. Zainal Chap 3-30
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-16
Coefficient of Variation
Measures relative variation
Always in percentage (%)
Shows variation relative to mean
Is used to compare two or more sets of data
measured in different units
100%x
sCV
100%
μ
σCV
Population Sample
QMIS 120, by Dr. M. Zainal Chap 3-31
Comparing Coefficients of Variation
Stock A:
Average price last year = $50
Standard deviation = $5
Stock B:
Average price last year = $100
Standard deviation = $5
Both stocks
have the same
standard
deviation, but
stock B is less
variable relative
to its price
10%100%$50
$5100%
x
sCVA
5%100%$100
$5100%
x
sCVB
QMIS 120, by Dr. M. Zainal Chap 3-32
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-17
Mean for grouped data
Once we group the data, we no longer know the values
of individual observations.
Thus, we find an approximation for the sum of these
values.
QMIS 120, by Dr. M. Zainal Chap 3-33
class. a offrequency theis andmidpoint theis Where
: samplefor Mean
:populationfor Mean
fm
n
mfx
N
mf
Variance and Standard
Deviation for grouped data
Also, we find an approximation for the variance and
standard deviation of grouped data.
QMIS 120, by Dr. M. Zainal Chap 3-34
class. a of frequency the is andmidpoint the is Where
1 : Sample
:Population
2
2
2
2
2
2
fm
n
n
mffm
s
N
N
mffm
22 and ss
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-18
Example: Grouped Data
The table below gives the frequency distribution of
the daily commuting times (in minutes) from home to
CBA for all 25 students in QMIS 120. Calculate the
mean and the standard deviation of the daily
commuting times.
QMIS 120, by Dr. M. Zainal Chap 3-35
f Daily commuting time (min)
4 0 to less than 10
9 10 to less than 20
6 20 to less than 30
4 30 to less than 40
2 40 to less than 50
25 Total
Example: Grouped Data
QMIS 120, by Dr. M. Zainal Chap 3-36
(continued)
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-19
Other Location Measures
Other Measures
of Location
Percentiles Quartiles
1st quartile = 25th percentile
2nd quartile = 50th percentile
= median
3rd quartile = 75th percentile
The pth percentile in a data array:
p% are less than or equal to this
value
(100 – p)% are greater than or
equal to this value
(where 0 ≤ p ≤ 100)
QMIS 120, by Dr. M. Zainal Chap 3-37
Percentiles
The pth percentile in an ordered array of n values is the
value in ith position, where
(n)100
pi
If i is not an integer,
round up to the next
higher integer value
QMIS 120, by Dr. M. Zainal Chap 3-38
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-20
Percentiles
Example: Find the 60th percentile in an ordered array of
the following 19 values.
6 7 2 3 5 1 8 6 7 3 9 4 2 1 5 4 3 9 6
Solution:
QMIS 120, by Dr. M. Zainal Chap 3-39
Quartiles
Quartiles split the ranked data into 4 equal
groups:
Note that the second quartile (the 50th percentile)
is the median
25% 25% 25% 25%
Q1 Q2 Q3
QMIS 120, by Dr. M. Zainal Chap 3-40
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-21
Quartiles
Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Example: Find the first quartile and third quartile
(n = 9)
Q1 = 25th percentile, so find i : i = (9) = 2.25
so round up and use the value in the 3rd position: Q1 = 13
Q3 = 75th percentile, so find i : i = (9) = 6.75
so round up and use the value in the 7th position: Q3 = 18
25 100
75 100
QMIS 120, by Dr. M. Zainal Chap 3-41
Box and Whisker Plot
A graphical display of data using a central “box” and extended “whiskers”:
Example:
QMIS 120, by Dr. M. Zainal Chap 3-42
25% 25% 25% 25%
* *
largest values* Smallest values*
Q1 Q2 Q3 Lower fence Upper fence
Outliers
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-22
Box and Whisker Plot
Box and whisker plot gives a graphic presentation of
data using five measures:
Q1, Q2, Q3, smallest, and largest values*.
Can help to visualize the center, the spread, and the
skewness of a data set.
Very good tool of comparing more than a distribution.
* *
QMIS 120, by Dr. M. Zainal Chap 3-43
Box and Whisker Plot
Can help in detecting outliers.
Detecting an outlier: Lower fence: Q1 – 1.5(Q3 - Q1)
Upper fence: Q3 + 1.5(Q3 - Q1)
If a data point is larger than the upper fence or smaller than the lower fence, it is considered to be an outlier.
25% 25% 25% 25%
* *
QMIS 120, by Dr. M. Zainal Chap 3-44
Q1 Q2 Q3 Lower fence Upper fence
Outliers
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-23
Constructing the Box and Whisker Plot
Lower 1st Median 3rd Upper
Limit Quartile Quartile Limit
* *
The lower limit is
Q1 – 1.5 (Q3 – Q1)
The upper limit is
Q3 + 1.5 (Q3 – Q1)
The center box extends from Q1 to Q3
The line within the box is the median
The whiskers extend to the smallest and largest values within
the calculated limits
Outliers are plotted outside the calculated limits
QMIS 120, by Dr. M. Zainal Chap 3-45
Shape of Box and Whisker Plots
The Box and central line are centered between the endpoints if data is symmetric around the median
(A Box and Whisker plot can be shown in either vertical or horizontal format)
QMIS 120, by Dr. M. Zainal Chap 3-46
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-24
Distribution Shape and Box and Whisker Plot
Right-Skewed Left-Skewed Symmetric
Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3
QMIS 120, by Dr. M. Zainal Chap 3-47
Box-and-Whisker Plot Example
Below is a Box-and-Whisker plot for the following data:
0 2 2 2 3 3 4 5 6 11 27
This data is right skewed, as the plot depicts
0 2 3 6 11 27
Min Q1 Q2 Q3 Max Outlier
*
Upper limit = Q3 + 1.5 (Q3 – Q1)
= 6 + 1.5 (6 – 2) = 12
27 is above the
upper limit so is
shown as an outlier
QMIS 120, by Dr. M. Zainal Chap 3-48
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-25
Interquartile Range
Can eliminate some outlier problems by using
the interquartile range
Eliminate some high-and low-valued
observations and calculate the range from the
remaining values.
Interquartile range = 3rd quartile – 1st quartile
QMIS 120, by Dr. M. Zainal Chap 3-49
Interquartile Range Example
Median
(Q2) X
maximum X minimum Q1 Q3
Example:
25% 25% 25% 25%
12 30 45 57 70
Interquartile range
= 57 – 30 = 27
QMIS 120, by Dr. M. Zainal Chap 3-50
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-26
A standardized data value refers to
the number of standard deviations a
value is from the mean
Standardized data values are
sometimes referred to as z-scores
Standardized Data Values
QMIS 120, by Dr. M. Zainal Chap 3-51
where:
x = original data value
μ = population mean
σ = population standard deviation
z = standard score
(number of standard deviations x is from μ)
Standardized Population Values
σ
μx z
QMIS 120, by Dr. M. Zainal Chap 3-52
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-27
where:
x = original data value
x = sample mean
s = sample standard deviation
z = standard score
(number of standard deviations x is from μ)
Standardized Sample Values
s
xx z
QMIS 120, by Dr. M. Zainal Chap 3-53
IQ scores in a large population have a bell-shaped distribution with mean μ = 100 and standard deviation σ = 15
Find the standardized score (z-score) for a person with an IQ of 121.
Standardized Value Example
Answer:
QMIS 120, by Dr. M. Zainal Chap 3-54
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-28
If the data distribution is bell-shaped, then
the interval:
contains about 68% of the values in
the population or the sample
The Empirical Rule
1σμ
μ
68%
1σμ
QMIS 120, by Dr. M. Zainal Chap 3-55
contains about 95% of the values in
the population or the sample
contains about 99.7% of the values
in the population or the sample
The Empirical Rule
2σμ
3σμ
3σμ
99.7% 95%
2σμ
QMIS 120, by Dr. M. Zainal Chap 3-56
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-29
Example: The age
distribution of a sample
of 5000 persons is bell-
shaped with a mean of
40 years and a standard
deviation of 12 years.
Determine the
approximate percentage
of people who are 16 to
64 years old.
The Empirical Rule
QMIS 120, by Dr. M. Zainal Chap 3-57
Sometimes a data set may have one or more
observation that is unusually small or large
value.
An experienced statistician may face the
following situations and need to take an action
Detecting Outliers
QMIS 120, by Dr. M. Zainal Chap 3-58
Outlier Action
A data value that was incorrectly recorded Correct it before any further
analysis
A data value that was incorrectly included Remove it before any further
analysis
A data value that belongs to the data set and
correctly recorded
Keep it !
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-30
This extreme value is called an outlier and can
be detected using the z-score and the empirical
rule for data with bell-shape distribution.
Detecting Outliers
QMIS 120, by Dr. M. Zainal Chap 3-59
3σμ
99.7% Outliers Outliers
Regardless of how the data are distributed, at least (1 - 1/k2) of the values will fall within k standard deviations of the mean
Examples:
(1 - 1/12) = 0% ……..... k=1 (μ ± 1σ)
(1 - 1/22) = 75% …........ k=2 (μ ± 2σ)
(1 - 1/32) = 89% ………. k=3 (μ ± 3σ)
Tchebysheff’s Theorem
within At least
QMIS 120, by Dr. M. Zainal Chap 3-60
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-31
Tchebysheff’s Theorem
QMIS 120, by Dr. M. Zainal Chap 3-61
k = 2 k = 3
Example: The average systolic blood pressure for 4000
women who were screened for high blood pressure was
found to be 187 with standard deviation of 22. Using
Chebyshev’s theorem, find at least what percentage of
women in this group have a systolic blood pressure
between 143 and 231.
Tchebysheff’s Theorem
QMIS 120, by Dr. M. Zainal Chap 3-62
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-32
Tchebysheff’s Theorem
QMIS 120, by Dr. M. Zainal Chap 3-63
(continued)
So far we have studied numerical methods to describe data with one variable.
Often decision makers are interested in the relationship between two variables.
To do so, we will use descriptive measure that is called covariance.
Covariance assigns a numerical value to the linear relationship between two variables (see scatter diagram)
Measures of association
between two variables
QMIS 120, by Dr. M. Zainal Chap 3-64
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-33
It is given by:
A big disadvantage of the covariance is that it depends on the units of measurement for x and y.
Measures of association
between two variables
QMIS 120, by Dr. M. Zainal Chap 3-65
N
yx
n
yyxx
yixi
ii
xy
xy
: covariance Population
1S :covariance Sample
For the same data set, we will have two different covariance values depending on the units (i.e. height in meters or centimeters will make a big difference).
Pearson’s correlation coefficient is a good remedy to that problem as it can go only from -1 to 1.
Measures of association
between two variables
QMIS 120, by Dr. M. Zainal Chap 3-66
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-34
It is given by
Measures of association
between two variables
QMIS 120, by Dr. M. Zainal Chap 3-67
N
yx
n
yyxx
yixi
ii
xy
xy
: covariance Population
1S :covariance Sample
Example: A golfer is interested in investigating the relationship, if any, between driving distance and 18-hole score.
Example
QMIS 120, by Dr. M. Zainal Chap 3-68
Average Driving Distance (m)
Average 18-Hole Score
277.6 69
259.5 71
269.1 70
267.0 70
255.6 71
272.9 69
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-35
Example
QMIS 120, by Dr. M. Zainal Chap 3-69
Problems
Consider the following sample of four purchases of one stock in
the KSE. Find the average cost of the stock.
Purchase Price Quantity
1 .300 5,000
2 .325 15,000
3 .350 10,000
4 .295 20,000
QMIS 120, by Dr. M. Zainal Chap 3-70
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-36
Problems
Construct a box plot for the following data:
340 300 400 360 320 290 260 330
QMIS 120, by Dr. M. Zainal Chap 3-71
Problems
Example: You are given 8 measurements: 3, 5, 4, 6, 12, 5, 6, 7.
Find
a) The mean b) The median c) The mode
QMIS 120, by Dr. M. Zainal Chap 3-72
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-37
Problems
Find the covariance and Pearson’s correlation coefficient for
the following data
QMIS 120, by Dr. M. Zainal Chap 3-73
Week Number of commercials (x) Sales in $ (y)
1 2 50
2 5 57
3 1 41
4 3 54
5 4 54
6 1 38
7 5 63
8 3 48
9 4 59
10 2 46
Mean 3 51
St. Dev. 1.49 7.93
Measures of association between two variables
QMIS 120, by Dr. M. Zainal Chap 3-74
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-38
Chapter Summary
Described measures of center and location
Mean, median, mode, weighted mean
Discussed percentiles and quartiles
Created Box and Whisker Plots
Illustrated distribution shapes
Symmetric, skewed
QMIS 120, by Dr. M. Zainal Chap 3-75
Chapter Summary
Described measure of variation
Range, interquartile range, variance,
standard deviation, coefficient of variation
Discussed Tchebysheff’s Theorem
Calculated standardized data values
(continued)
QMIS 120, by Dr. M. Zainal Chap 3-76
QMIS 120, by Dr. M. Zainal
Chapter 3 Student Lecture Notes 3-39
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.
Chap 3-77 QMIS 120, by Dr. M. Zainal