Business Statistics Chapter 3

© 2003 Prentice-Hall, Inc.
Basic Business Statistics
Chapter 3
Numerical Descriptive Measures
Chapters Objectives:
Learn about Measures of Center.
How to calculate mean, median and midrange
Learn about Measures of Spread
Learn how to calculate Standard Deviation, IQR and
Range
Learn about 5 number summaries
Coefficient of Correlation

Chapter Topics
 Measures of Central Tendency
 Mean, Median, Midrange,
 Quartile
 Measure of Variation
 Range, Interquartile Range, Variance and
Standard Deviation, Shape
 Symmetric, Skewed, Using Box-and-Whisker Plots

Chapter Topics
Coefficient of Correlation
 Pitfalls in Numerical Descriptive Measures
and Ethical Issues
(continued)

Summary Measures
Central Tendency
Mean
Median
Mode
Quartile
Mid Range
Summary Measures
Variation
Variance
Standard Deviation
Range
Not very good
measure

The Second Step of Data
Analysis
 We looked at displaying data using graphs.
 But numeric summaries help us to compare variables and to
talk about relationships among variables in precise (exact)
ways.
 After drawing the graph, it is usual to calculate summary
values.
 This Lesson considers a different numerical summaries.

Summaries of Center and Spread
Can you think of a way of estimating the
 Center of this histogram or
 How wide it is (Spread or Variance)
Measure
of Center
Measure
of Spread
Mean Standard
Deviation
Media IQR
Midrange Range0
5
10
15
20
25
30
30-40 40-50 50-60 60-70 70-80 80-90 90-100
•The shape that appears over the histogram is called the normal distribution shape
•This shape is used to estimate the shape of the histogram

Finding the Central
Value
 We summarize the distribution with mean
median and midrange
 The mean median and midrange have
different idea of what is the center, and
behaves differently
 The Mode is also used however this is not
a good measure of the central value

Measures of Central Tendency
Central Tendency
Mean Median Midrange
1
1
n
i
i
N
i
i
X
X
n
X
N
µ
=
=
=
=
∑
∑

Mean (Arithmetic Mean)
 Mean (Arithmetic Mean) of Data Values
 Sample mean
 Population mean
1 1 2
n
i
i n
X
X X X
X
n n
= + + +
= =
∑ L
1 1 2
N
i
i N
X
X X X
N N
µ = + + +
= =
∑ L
Sample Size
Population Size

 The Most Common Measure of Central
Tendency
 Affected by Extreme Values (Outliers)
(continued)
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 5 Mean = 6

 Approximating the Arithmetic Mean
 Used when raw data are not available

(continued)
1
sample size
number of classes in the frequency distribution
midpoint of the th class
frequencies of the th class
c
j j
j
j
j
m f
X
n
n
c
m j
f j
=
=
=
=
=
=
∑

How to compute the Mean?
Step 1: Add all the data points. This is called the SUM or the Total
Step 2: Count the number of Data points. This is sometimes called
the count
Step 3: Divide The Total by the Number of Data points
Calculate the mean for
the following data
points. 5,20,15,3,7
Revision: 5.1 Know how to compute the mean of a collection of data
values.

Median
 Robust Measure of Central Tendency
 Not Affected by Extreme Values
 In an Ordered Array, the Median is the
‘Middle’ Number
 If n or N is odd, the median is the middle number
 If n or N is even, the median is the average of the 2
middle numbers
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5 Median = 5

Example: Find median of the following numbers
14.1 3.2 25.3 2.8 -17.5 13.9 45.8
Step 1 Order the data
1 2 3 4 5 6 7
-17.5 2.8 3.2 13.9 14.1 25.3 45.8
Step 2 There are 7 values
n is odd
Step 3 a Median is the middle number
Median is 13.9
If we added 35.7 to the above numbers (data set) . What would the median now be?
14.1 3.2 25.3 2.8 -17.5 13.9 45.8 35.7
Step 3 b Median is average of the two middle numbers
5 position 14.1
4 position 13.9
Median = '(13.9+14.1)/2= 14
1 2 3 4 5 6 7 8
-17.5 2.8 3.2 13.9 14.1 25.3 35.7 45.8
Step 1 Order the data
n is even
Step 2 There are 8 values
Calculate the median
for the following data
points. 5,20,15,3,7
Calculate the median
for the following data
points. 5,20,15,3,7,10
MEDIAN
MEDIAN between 13.9 and 14.1

How to compute the Midrange
 The range of the data is defined as the difference
between the Maximum and Minimum (Range =
Max – Min)
 What would the Range be in the previous
example?
 What would the midrange be? (max + min)/2
 DISADVANTAGE: - one extreme value can make
the midrange very large, so it does not really
represent the data.
 If you deleted 45.8 and added 1000 to the previous
example what would the midrange be?
63.3
14.15
491.25
Calculate the midrange and range for the following data points.
5,20,15,3,7

Summary Measures of center
 Measures of center are the most commonly
used (and misused) summaries of data.
 If you understand how they differ and what
they really say you can avoid
misunderstanding data or being fooled by
inappropriate summaries.

Mode
 A Measure of Central Tendency
 Value that Occurs Most Often
 There May Not Be a Mode
 There May Be Several Modes
 Used for Either Numerical or Categorical Data
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode

Why the mode is not really a measure of
center
 Often listed among measures of center.
 Is it really a central value?
1. The mode of a categorical variable is the category with the highest
frequency.

It is not a center in because the categories of a categorical
variable have no specific order

we can choose to place the modal category in a bar chart on the
right, on the left, or in the middle..
2. When a continuous distribution is unimodal and symmetric, the
single, central mode is often near to other measures of center.
3. Modes of data measured on quantitative variables are rarely useful
as measures of center.

It is a coincidence when two quantitative measurements agree
exactly,

counting such occurrences tells us little about the variable. As a
consequence, statistics packages rarely compute or report
modes.

Quartiles
 Split Ordered Data into 4 Quarters
 Position of i-th Quartile
 Q1 and Q3 are Measures of Noncentral Location
 Q2= Median, a Measure of Central Tendency
 The calculation will be covered in the IQR slides.
25% 25% 25% 25%
( )1Q ( )2Q ( )3Q
( )
( )1
4
i
i n
Q
+
=
Do not use formula in the Business Statistics BookDo not use formula in the Business Statistics Book

Know the differing properties of the
mean, median, and midrange
 The most common summary value describing a
distribution of data values is some measure of a typical
or central value.
 This can be the mean, median or midrange
You have calculate the 3 measures of center (mean, median and
midrange) for the following data points. 5,20,15,3,7.
1. Change 15 to 19 and recalculate the mean, median and midrange.
Which measures of center have changed?
2. Change 20 to 200 and recalculate the mean, median and midrange.
Which measures of center have changed?
The median is not effected by outliers like the midrange and Mean
The mean is effected by very small changes to data

Why Measures Spread?
 Spread measures how often something varies.
 Imagine you go to a restaurant and received such very good
meal, you return for the same meal the next day.
 However, this time the food tastes very bad. You never go back.
The standard of quality has varied.
 The restaurant management needs to be able to control the
quality of food.
 A key aspect of management is to reduce variability, and
provide a consistent quality of service or product
 The first step to control variability is to measure it.
 Then you can work out what causes the variability
 Then work out how to reduce variability.

© 2003 Prentice-Hall, Inc. Days of Week
The following shows the number of Cakes made by a Bakery in Phnom Penh.
It has two bakeries producing identical cakes
Day 1 Day 2 Day 3 Day 4 Day 5
Center
(Mean)
Spread
(Std Dev)
Bakery A 550 750 800 750 500 670 135
Bakery B 625 670 650 675 630 650 23
Total 1175 1420 1450 1425 1130
The bakery has to produce at least 1200 cakes to meet daily customer demand
The bakery has to produce can not produce more than 1400 cakes because
customers will not buy them and they will not be eaten
500
550
600
650
700
750
800
Day 1 Day 2 Day 3 Day 4 Day 5
NoCakesbaked
Bakery A
Bakery B
Mean A
Mean B
Spread A is 135.
Variance is very
high
Spread B is 23
The two bakeries have been baking too many or too few cakes each
day. Why?

Different Summaries of Spread
 Summaries of Spread include
 Range
 Interquartile Range (IQR) and
 Standard Deviation
 These summaries highlight different aspects of the
distribution and have different values
 But they all summarize how the values are spread
out.
 We have already seen that Range = Max - Min

Measures of Variation
Variation
Variance Standard Deviation Coefficient
of Variation
Population
Variance
Sample
Variance
Population
Standard
Deviation
Sample
Standard
Deviation
Range
Interquartile Range

Range
 Difference between the Largest and the
Smallest Observations:
 Ignores How Data are Distributed
Largest SmallestRange X X= −
7 8 9 10 11
12
Range = 12 - 7 = 5
7 8 9 10 11
12
Range = 12 - 7 = 5

 Also Known as Midspread
 Spread in the middle 50%
 Difference between the First and Third
Quartiles
3 1Interquartile Range 17.5 12.5 5Q Q= − = − =
Interquartile Range
Data in Ordered Array: 11 12 13 16 16 17 17 18 21
1317 4

5 Steps to calculate the
Interquartile Range (IQR)
 Step 1 Put the data in Order
 Step 2 Divide the dataset in two. Dataset 1 and Dataset 2.
 NOTE: If n is odd the median should be included in both data sets
 Step 3 Calculate Q1 (25th
Percentile)
 If n of Dataset 1 is odd, the Q1 is the middle number of Dataset A
 If n of Dataset 1 is even, the median is the average of the 2 middle
numbers
 Step 4 Calculate Q3 (75th percentile)
 As per step 3
 Step 5 Find The Interquartile Range (IQR)
 IQR= Q3-Q1

5 Steps to Calculate Q1, Q3 and IQR when n is odd
In order to calculate IQR we need to to first of all calculate Q1 and
Q3.
We are going to calculate the Q1, Q3 and IQR for the following
dataset: 7,8,25,11,13,5,6
We require to know: Median (Q2 or 50th
percentile), Q1 (25th
Percentile), Q3 (75th
percentile)
Step 1 Order Data
1
2 3 4 5 6 7
5 6 7 8 11 13 25

Step 2 Divide the dataset in two
Divide the above data set into Dataset 1 and Dataset 2.
NOTE: If n is odd the median should be included in both data sets
Dataset 1 is used to calculate Q1 and Dataset 2 is used to calculate Q3
Dataset 1 For Q1 Data Set 2 For Q3
1 2 3 4 1 2 3 4
5 6 7 8 8 11 13 25
Step 3 Calculate Q1 (25th
Percentile)
N is even so the Q1 is the average of the 2 middle numbers
which is position 2 and 3
Q1 = (6+7)/2 =6.5
(This is the same method used to calculate the median for a
even number of data points)
5 Steps Calculate Q1, Q3 and IQR when n is odd (Cont)

Step 4 Calculate Q3 (75th percentile)
N is even for dataset 2, the median is the average of the 2 middle
numbers which is position 2 and 3
Thus, Q3 = (11+13)/2 =12
Step 5 Find The Interquartile Range (IQR)
IQR= Q3-Q1 = 12-6.5 =5.5
5 Steps to Calculate Q1, Q3 and IQR when n is odd(Cont)
1 2 3 4
8 11 13 25
Calculate the Q1, Q3 and IQR for the following data points.
5,20,15,3,7,16,4

5 Steps to Calculate Q1, Q3 and IQR when n is EVEN
When n is even we we use almost the same process to calculate Q1
Q3 and IQR. In order to calculate IQR we need to to first of all
calculate Q1 and Q3.
We are going to calculate the Q1, Q3 and IQR for the following
dataset: 7,8,25,11,13,5,6 and 30
We require to know: Q1 (25th Percentile), Q3 (75th
Step 1 Order Data
1
2 3 4 5 6 7 8
5 6 7 8 11 13 25 30

Step 2 Divide the dataset in two
Divide the above data set into Dataset 1 and Dataset 2.
Dataset 1 is used to calculate Q1 and Dataset 2 is used to calculate Q3
Dataset 1 For Q1 Dataset 2 For Q3
1 2 3 4 1 2 3 4
5 6 7 8 11 13 25 30
Step 3 Calculate Q1 (25th
percentile)
Q1 = (6+7)/2 =6.5
(This is the same method used to calculate the median for
a even number of data points)
5 Steps: Calculate Q1, Q3 and IQR when n is even (Cont)

Step 4 Calculate Q3 (75th percentile)
Thus, Q3 = (13+25)/2 =19
Step 5 Find The Interquartile Range (IQR)
IQR= Q3-Q1 =19-6.5=12.5
5 Steps to Calculate Q1, Q3 and IQR when n is even (Cont)
1 2 3 4
11 13 25 30
Calculate the Q1, Q3 and IQR for the following data points.
5,20,15,3,7,16,4,30

( )
2
2 1
N
i
i
X
N
µ
σ =
−
=
∑
 Important Measure of Variation
 Shows Variation about the Mean
 Sample Variance:
 Population Variance:
( )
2
2 1
1
n
i
i
X X
S
n
=
−
=
−
∑
Variance

Standard Deviation
 Most Important Measure of Variation
 Shows Variation about the Mean
 Has the Same Units as the Original Data
 Sample Standard Deviation:
 Population Standard Deviation:
( )
2
1
1
n
i
i
X X
S
n
=
−
=
−
∑
( )
2
1
N
i
i
X
N
µ
σ =
−
=
∑

How to Calculate the Standard Deviation?
Step 1 Step 2 Step 3 Step 4
Count Data (y) Mean (y-Mean)
1 3.4 33.35 -29.95 897
2 20 33.35 -13.35 178
3 63 33.35 29.65 879
4 47 33.35 13.65 186
Total 133.4 2141 Step 5
Mean = SUM/Count =133.4/4=33.35 33.35
Step 6 Total Step 5/( Count -1)=2141/(4-1) = 713.6
Step 7 Square root of 713.6= 26.7
Steps to Calc Standard
Deviation
1: Calculate the Mean
=Sum/Count
Deviation
=Sum/Count
2: Enter the mean on each
line on the table
Deviation
=Sum/Count
line on the table
3: Take the each data
point from the mean
Deviation
=Sum/Count
line on the table
point from the mean
4: Square the answer to
Step 3 (Multiply each data
point by each other)
Deviation
=Sum/Count
line on the table
point from the mean
5: Sum the all the data
points
6: Divide the total for Step
5 by (Count -1)
7: Calculate the square
root of Step 6
Deviation
=Sum/Count
line on the table
point from the mean
points
6: Divide the total for Step
5 by (Count -1)
Steps to Calculate
Standard Deviation
=Sum/Count
line on the table
point from the mean
points
Calculate the Standard
Deviation for the following
data points. 5,20,15,3,7,16,4

Histograms to display measures of spread
 When the distribution of data is unimodal and
symmetric, the middle 38% of the distribution
is about one standard deviation wide.
 This rule of thumb provides a way to display
the standard deviation, but it only applies to
unimodal, symmetric distributions.
 We can display common measures of spread
in a Histogram.

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

Shape of a Distribution
 Describe How Data are Distributed
 Measures of Shape
 Symmetric or skewed
Mean = Median =ModeMean < Median < Mode Mode < Median < Mean
Right-SkewedLeft-Skewed Symmetric

Describing Distribution:
Shape, Center and Spread
Start by displaying the distribution of your data. You should describe:
 Symmetry vs. skewness * (Look at the tails)
 Single vs. multiple modes
 Possible outliers separated from the main body of the data.
... But remember that not all distributions have simple shapes.
Symmetry Skewness
Unimodel Bimodel multimodal Uniform
Outliers

Identifying Distribution Shape
When looking at the distribution of a quantitative
variable, consider:
 Symmetry,
 How many modes it has,
 Whether it has any outliers or gaps.

Summary: Describing a Distribution
 You should always begin analyzing data by graphing the
distributions of the variables.
 We describe the shape of the distribution of a quantitative
variable by noting whether it is symmetric or skewed,
unimodal, bimodal, or multimodal.
 Following the general rule that we first describe a
pattern and then look for deviations from that
pattern, we look at distributions to see if there are straggling
outliers or gaps in the distribution shape.
 Each of these descriptions is quite general and vague, so
people may disagree on the "close calls." Nevertheless, we
usually begin describing data by summarizing the distribution
shapes.

Exploratory Data Analysis
 Box-and-Whisker
 NOTE: Different from Basic Business Statistics Book!
 Graphical display of data using 5-number summary
Median( )
4 6 8 10 12
Top of
Whiskers
Bottom of
Whiskers
1Q 3Q
2Q

Distribution Shape &
Box-and-Whisker
Right-SkewedLeft-Skewed Symmetric
1Q 1Q 1Q2Q 2Q 2Q3Q 3Q3Q

The Empirical Rule
 For Most Data Sets, Roughly 68% of the
Observations Fall Within 1 Standard Deviation
Around the Mean
 Roughly 95% of the Observations Fall Within
2 Standard Deviations Around the Mean
 Roughly 99.7% of the Observations Fall
Within 3 Standard Deviations Around the
Mean

Coefficient of Correlation
 Measures the Strength of the Linear
Relationship between 2 Quantitative Variables

( )( )
( ) ( )
1
2 2
1 1
n
i i
i
n n
i i
i i
X X Y Y
r
X X Y Y
=
= =
− −
=
− −
∑
∑ ∑

Features of Correlation
Coefficient
 Unit Free
 Ranges between –1 and 1
 The Closer to –1, the Stronger the Negative
Linear Relationship
 The Closer to 1, the Stronger the Positive
Linear Relationship
 The Closer to 0, the Weaker Any Linear
Relationship

Scatter Plots of Data with
Various Correlation Coefficients
Y
X
Y
X
Y
X
Y
X
Y
X
r = -1 r = -.6 r = 0
r = .6 r = 1

Scatter Plots
 Best way to see relationships between two Quantitative Variables
 If there is a relationship it is called an association. Scatter plots are
the best way to look for associations
Relationship of Class Attended and Statitstics Score
20
25
30
35
40
45
50
55
60
65
70
4 6 8 10 12 14
Number Classes Attended
Score

Scatter Plots
 Only Quantitative Variables can be used
 The following is the relationship between the Mango Taste
Score and the Number of Weeks it grown for
Weeks Grown on Tree
Time taken to grow the best Mango
0
20
40
60
80
100
6 8 10 12 14 16 18
TasteScore

Direction: Positive
0
10
20
30
40
50
60
70
80
90
100
0 1 2 3 4 5 6
Scatter Plots : DIRECTION and FORM
Negtive Direction
0
10
20
30
40
50
60
70
80
90
0 1 2 3 4 5 6
•Direction: +Ve
•Form: Linear (Straight Line)
•Direction: +Ve
•Form: Linear (Straight Line)

Form describes the shape
 Describe the Form if there is one. EG Straight, or
arched or curved
A Curve
0
2
4
6
8
10
12
14
16
18
20
0 1 2 3 4 5 6
An Arch
0
20
40
60
80
100
6 8 10 12 14 16 18
TasteScore

Scatter: The following Scatter Plots have no scatter
0
50
100
150
200
250
0 5 10 15 20
0
5
10
15
20
25
30
0 2 4 6 8 10 12
•The third thing to look for is scatter.
•These graphs have no Scatter because the dots follow each other in a line.

Scatter: The following Scatter Plots have scatter
•These graphs have Scatter because the dots are like a cloud.
•Always look for something you do not expect to find
•One Graph has an outlier which are very important unexpected points
that may occur
0
2
4
6
8
10
12
0 10 20 30 40 50
0
10
20
30
40
50
60
70
80
90
0 10 20 30 40 50 60

How to Measure Scatter: Correlation Coefficient, r
I X Y
1 2 5 -4 -10 40
2 4 10 -2 -5 10
3 6 15 0 0 0
4 8 20 2 5 10
5 10 25 4 10 40
Total 30 75 100
Mean 6 15
Standard Deviation 3.162278 7.905694
Sx = 3.162278 Sy = 7.905694
1
905694.7*162278.3*)15(
100
)1(
)(*)(
=
−
=
−
−−
=
∑
SxSyn
yyxx
CorrCoef
xx −
xx − yy −
)(*)( yyxx −−

Plot Chart and describe scatter plot. Calculate
Correlation Coefficient
0
10
20
30
0 5 10 15
X
Y
These data points go in straight
line with no scatter in a positive
direction (positively correlated)
1
905694.7*162278.3*)15(
100
)1(
)(*)(
=
−
=
−
−−
=
∑
SxSyn
yyxx
CorrCoef
Step 1: Draw Scatter Plot
Step 2: Write Description
Step 3: If there is a linear
(Straight line) relationship
calculate

Example 2
 For the following data, plot a scatter plot and
describe it. Calculate Correlation Coefficient
n X Y X-Mean X Y-Mean X Product
1 4 22 -2 7.6 -15.2
2 5 23 -1 8.6 -8.6
3 6 12 0 -2.4 0
4 7 13 1 -1.4 -1.4
5 8 2 2 -12.4 -24.8
Total 30 72 -50
Mean 6 14.4

Answer: Example 2
n X Y
1 4 22 -2 7.6 -15.2
2 5 23 -1 8.6 -8.6
3 6 12 0 -2.4 0
4 7 13 1 -1.4 -1.4
5 8 2 2 -12.4 -24.8
Total 30 72 -50
Mean 6 14.4
These data points go in straight
line with some scatter in a negative
direction (negatively correlated)
0
5
10
15
20
25
3 4 5 6 7 8 9
x
y
92.0
1214.9*5811.1*)15(
50
)1(
)(*)(
−=
−
−
=
−
−−
=
∑
SxSyn
yyxx
CorrCoef
xx − yy −
)(*)( yyxx −−

Example 3
1 1 5 -2 -10.2 20.4
2 2 21 -1 5.8 -5.8
3 3 25 0 9.8 0
4 4 19 1 3.8 3.8
5 5 6 2 -9.2 -18.4
Total 15 76 0
Mean 3 15.2

Answer: Example 3
n X Y
1 1 5 -2 -10.2 20.4
2 2 21 -1 5.8 -5.8
3 3 25 0 9.8 0
4 4 19 1 3.8 3.8
5 5 6 2 -9.2 -18.4
Total 15 76 0
Mean 3 15.2
0
10
20
30
0 2 4 6
X
Y
These data points are in an arched
shape with a little scatter.
0
1214.9*5811.1*)15(
0
)1(
)(*)(
=
−
=
−
−−
=
∑
SxSyn
yyxx
CorrCoef
xx − yy −
)(*)( yyxx −−

Example 4
1 1 17 -4.4 5.8 -25.52
2 3 1 -2.4 -10.2 24.48
3 6 22 0.6 10.8 6.48
4 7 3 1.6 -8.2 -13.12
5 10 13 4.6 1.8 8.28
Total 27 56 0.6
Mean 5.4 11.2

Answer: Example 4
1 1 17 -4.4 5.8 -25.52
2 3 1 -2.4 -10.2 24.48
3 6 22 0.6 10.8 6.48
4 7 3 1.6 -8.2 -13.12
5 10 13 4.6 1.8 8.28
Total 27 56 0.6
Mean 5.4 11.2
0
10
20
30
0 5 10 15
X
Y
0.005
9.011*3.5071*1)(5
6.0
1)SxSy(n
)y(y*)x(x
CorrCoef
=
−
=
−
−−
=
∑
These data points have a lot of
scatter and they are not correlated

Pitfalls in Numerical Descriptive
Measures and Ethical Issues
 Data Analysis is Objective
 Should report the summary measures that best meet the
assumptions about the data set
 Data Interpretation is Subjective
 Should be done in a fair, neutral and clear manner
 Ethical Issues
 Should document both good and bad results
 Presentation should be fair, objective and neutral
 Should not use inappropriate summary measures to distort
the facts

Chapter Summary
 Described Measures of Central Tendency
 Mean, Median, Midrange
 Discussed Quartiles
 Described Measures of Variation
 Range, Interquartile Range, Variance and
Standard Deviation
 Illustrated Shape of Distribution
 Symmetric, Skewed, Using Box-and-Whisker Plots

Chapter Summary
 Described the Empirical
 Discussed Correlation Coefficient
 Addressed Pitfalls in Numerical Descriptive
Measures and Ethical Issues
(continued)

Business Statistics Chapter 3

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Business Statistics Chapter 3