Chapter 3
Student Lecture Notes
3-1
Basic Business Statistics
(9th Edition)
Chapter 3
Numerical Descriptive Measures
© 2004 Prentice-Hall, Inc.
Chap 3-1
Chapter Topics
„
Measures of Central Tendency
„
Quartiles
„
Measures of Variation
„
„
„
Mean, Median, Mode, Geometric Mean
Range, Interquartile Range, Variance and Standard
Deviation, Coefficient of Variation
The Empirical Rule and the BienaymeChebyshev Rule
Chap 3-2
© 2004 Prentice-Hall, Inc.
Chapter Topics
„
„
„
(continued)
Shape
„
Symmetric, Skewed
„
Using Box-and-Whisker Plots
Coefficient of Correlation
Pitfalls in Numerical Descriptive Measures and
Ethical Issues
© 2004 Prentice-Hall, Inc.
Chap 3-3
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 3
Student Lecture Notes
3-2
Summary Measures
Summary Measures
Central Tendency
Mean
Variation
Quartiles
Mode
Median
Coefficient of
Variation
Range
Variance
Standard Deviation
Geometric Mean
Chap 3-4
© 2004 Prentice-Hall, Inc.
Measures of Central Tendency
Central Tendency
Mean
Median
Mode
n
∑
X =
i =1
Xi
Geometric Mean
n
X G = ( X 1 × X 2 ×L × X n )
1/ n
N
µ=
∑X
i =1
i
N
Chap 3-5
© 2004 Prentice-Hall, Inc.
Mean (Arithmetic Mean)
„
Mean (Arithmetic Mean) of Data Values
„
Sample mean
Sample Size
n
X=
„
∑ Xi
i =1
n
=
Population mean
Population Size
N
µ=
© 2004 Prentice-Hall, Inc.
∑ Xi
i =1
N
X1 + X 2 + L + X n
n
=
X1 + X 2 + L + X N
N
Chap 3-6
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 3
Student Lecture Notes
3-3
Mean (Arithmetic Mean)
(continued)
„
„
The Most Common Measure of Central
Tendency
Affected by Extreme Values (Outliers)
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
Chap 3-7
© 2004 Prentice-Hall, Inc.
Mean (Arithmetic Mean)
From a Frequency Distribution
(continued)
„
Approximating the Arithmetic Mean
„
Used when raw data are not available
c
„
X=
∑m
j =1
j
fj
n
n = sample size
c = number of classes in the frequency distribution
m j = midpoint of the jth class
f j = frequencies of the jth class
Chap 3-8
© 2004 Prentice-Hall, Inc.
Median
„
„
Robust Measure of Central Tendency
Not Affected by Extreme Values
0 1 2 3 4 5 6 7 8 9 10
Median = 5
„
0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5
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
© 2004 Prentice-Hall, Inc.
Chap 3-9
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 3
Student Lecture Notes
3-4
Mode
„
„
„
„
„
„
A Measure of Central Tendency
Value that Occurs Most Often
Not Affected by Extreme Values
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
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
No Mode
Mode = 9
Chap 3-10
© 2004 Prentice-Hall, Inc.
Geometric Mean
„
Useful in the Measure of Rate of Change of a
Variable Over Time
X G = ( X 1 × X 2 ×L × X n )
1/ n
„
Geometric Mean Rate of Return
„
Measures the status of an investment over time
RG = ⎡⎣(1 + R1 ) × (1 + R2 ) × L × (1 + Rn ) ⎤⎦
1/ n
−1
Chap 3-11
© 2004 Prentice-Hall, Inc.
Example
An investment of $100,000 declined to $50,000 at the
end of year one and rebounded back to $100,000 at end
of year two:
R1 = −0.5 (or − 50%)
R2 = 1 (or 100% )
Average rate of return:
( −0.5) + (1)
= 0.25 (or 25%)
2
Geometric rate of return:
R=
RG = ⎣⎡(1 − 0.5 ) × (1 + 1) ⎦⎤
1/ 2
= ⎡⎣( 0.5 ) × ( 2 ) ⎤⎦
1/ 2
© 2004 Prentice-Hall, Inc.
−1
− 1 = 11/ 2 − 1 = 0 (or 0%)
Chap 3-12
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 3
Student Lecture Notes
3-5
Quartiles
Split Ordered Data into 4 Quarters
„
25%
25%
25%
25%
( Q1 ) ( Q2 ) ( Q3 )
Position of i-th Quartile ( Q ) = i ( n + 1)
i
4
„
Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Position of Q1 =
1( 9 + 1)
= 2.5
4
Q1 =
(12 + 13) = 12.5
2
Q1 and Q3 are Measures of Noncentral Location
„ Q = Median, a Measure of Central Tendency
2
„
Chap 3-13
© 2004 Prentice-Hall, Inc.
Measures of Variation
Variation
Range
Variance
Interquartile
Range
Standard
Deviation
Coefficient
of Variation
Population
Standard
Deviation
Sample
Standard
Deviation
Population
Variance
Sample
Variance
© 2004 Prentice-Hall, Inc.
Chap 3-14
Range
„
„
Measure of Variation
Difference between the Largest and the
Smallest Observations:
Range = X Largest − X Smallest
„
Ignores How Data are Distributed
Range = 12 - 7 = 5
Range = 12 - 7 = 5
7
8
© 2004 Prentice-Hall, Inc.
9
10
11
12
7
8
9
10
11
12
Chap 3-15
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 3
Student Lecture Notes
3-6
Interquartile Range
„
„
Measure of Variation
Also Known as Midspread
„
„
Spread in the middle 50%
Difference between the First and Third
Quartiles
Data in Ordered Array: 11 12 13 16 16 17
17 18 21
Interquartile Range = Q3 − Q1 = 17.5 − 12.5 = 5
„
Not Affected by Extreme Values
Chap 3-16
© 2004 Prentice-Hall, Inc.
Variance
„
„
Important Measure of Variation
Shows Variation about the Mean
„
Sample Variance:
n
S2 =
„
∑( X
i =1
−X)
i
2
n −1
Population Variance:
N
σ2 =
∑( X
i =1
i
−µ)
2
N
© 2004 Prentice-Hall, Inc.
Chap 3-17
Standard Deviation
„
„
„
Most Important Measure of Variation
Shows Variation about the Mean
Has the Same Units as the Original Data
„
Sample Standard Deviation:
S=
„
Population Standard Deviation:
σ=
© 2004 Prentice-Hall, Inc.
n
∑( X
i =1
−X)
i
2
n −1
N
∑( X
i =1
i
−µ)
2
N
Chap 3-18
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 3
Student Lecture Notes
3-7
Standard Deviation
From a Frequency Distribution
(continued)
„
Approximating the Standard Deviation
„
Used when the raw data are not available and the
only source of data is a frequency distribution
∑ (m
c
„
S=
j =1
− X ) fj
2
j
n −1
n = sample size
c = number of classes in the frequency distribution
m j = midpoint of the jth class
© 2004 Prentice-Hall, Inc.
f j = frequencies of the jth class
Chap 3-19
Comparing Standard Deviations
Data A
11 12
Mean = 15.5
s = 3.338
13 14
15 16
17 18 19 20 21
13 14
15 16
17 18 19 20 21
s = .9258
15 16
17 18 19 20 21
Mean = 15.5
s = 4.57
Data B
Mean = 15.5
11 12
Data C
11 12
13 14
Chap 3-20
© 2004 Prentice-Hall, Inc.
Coefficient of Variation
„
Measure of Relative Variation
„
Always in Percentage (%)
„
Shows Variation Relative to the Mean
„
„
„
Used to Compare Two or More Sets of Data
Measured in Different Units
⎛S
CV = ⎜
⎝X
⎞
⎟100%
⎠
Sensitive to Outliers
© 2004 Prentice-Hall, Inc.
Chap 3-21
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 3
Student Lecture Notes
3-8
Comparing Coefficient
of Variation
„
Stock A:
„
„
„
Stock B:
„
„
„
Average price last year = $50
Standard deviation = $2
Average price last year = $100
Standard deviation = $5
Coefficient of Variation:
„
Stock A:
„
Stock B:
© 2004 Prentice-Hall, Inc.
⎛S⎞
⎛ $2 ⎞
CV = ⎜ ⎟ 100% = ⎜
⎟ 100% = 4%
⎝X⎠
⎝ $50 ⎠
⎛S
CV = ⎜
⎝X
⎞
⎛ $5 ⎞
⎟100% = ⎜
⎟100% = 5%
⎠
⎝ $100 ⎠
Chap 3-22
Shape of a Distribution
„
Describe How Data are Distributed
„
Measures of Shape
„
Symmetric or skewed
Left-Skewed
Symmetric
Mean < Median < Mode Mean = Median =Mode
Right-Skewed
Mode < Median < Mean
Chap 3-23
© 2004 Prentice-Hall, Inc.
Exploratory Data Analysis
„
Box-and-Whisker Plot
„
Graphical display of data using 5-number summary
X smallest Q
1
4
© 2004 Prentice-Hall, Inc.
6
Median( Q2)
8
Q3
10
Xlargest
12
Chap 3-24
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 3
Student Lecture Notes
3-9
Distribution Shape &
Box-and-Whisker Plot
Left-Skewed
Q2 Q3
Q1
Symmetric
Q1 Q2Q3
Right-Skewed
Q1 Q2 Q3
Chap 3-25
© 2004 Prentice-Hall, Inc.
The Empirical Rule
„
For Data Sets That Are Approximately Bellshaped:
„
„
„
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
© 2004 Prentice-Hall, Inc.
Chap 3-26
The Bienayme-Chebyshev Rule
„
The Percentage of Observations Contained
Within Distances of k Standard Deviations
Around the Mean Must Be at Least (1 − 1/ k 2 )100%
„
„
„
„
Applies regardless of the shape of the data set
At least 75% of the observations must be contained
within distances of 2 standard deviations around the
mean
At least 88.89% of the observations must be
contained within distances of 3 standard deviations
around the mean
At least 93.75% of the observations must be
contained within distances of 4 standard deviations
around the mean
© 2004 Prentice-Hall, Inc.
Chap 3-27
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 3
Student Lecture Notes
3-10
Coefficient of Correlation
„
Measures the Strength of the Linear
Relationship between 2 Quantitative Variables
n
„
r=
∑( X − X )(Y − Y )
i
i =1
n
i
∑( Xi − X )
2
i =1
n
∑(Y − Y )
2
i
i =1
Chap 3-28
© 2004 Prentice-Hall, Inc.
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
Chap 3-29
© 2004 Prentice-Hall, Inc.
Scatter Plots of Data with
Various Correlation Coefficients
Y
Y
Y
X
r = -1
X
r = -.6
Y
© 2004 Prentice-Hall, Inc.
X
r=0
Y
r = .6
X
r=1
X
Chap 3-30
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 3
Student Lecture Notes
3-11
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
„
Ethical Issues
„
Should be done in a fair, neutral and clear manner
„
Should document both good and bad results
„
Presentation should be fair, objective and neutral
„
Should not use inappropriate summary measures
to distort the facts
Chap 3-31
© 2004 Prentice-Hall, Inc.
Chapter Summary
„
Described Measures of Central Tendency
„
Discussed Quartiles
„
Described Measures of Variation
„
„
„
Mean, Median, Mode, Geometric Mean
Range, Interquartile Range, Variance and Standard
Deviation, Coefficient of Variation
Described the Empirical Rule and the
Bienayme-Chebyshev Rule
Chap 3-32
© 2004 Prentice-Hall, Inc.
Chapter Summary
„
„
„
(continued)
Illustrated Shape of Distribution
„
Symmetric, Skewed
„
Using Box-and-Whisker Plots
Discussed Correlation Coefficient
Addressed Pitfalls in Numerical Descriptive
Measures and Ethical Issues
© 2004 Prentice-Hall, Inc.
Chap 3-33
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.