Transcript Basic Business Statistics, 8th Edition

Basic Business Statistics
(8th Edition)
Chapter 3
Numerical Descriptive Measures
© 2002 Prentice-Hall, Inc.
Chap 3-1
Chapter Topics

Measures of central tendency

Mean, median, mode, geometric mean, midrange

Quartile

Measure of variation


Range, Interquartile range, variance and standard
deviation, coefficient of variation
Shape

Symmetric, skewed, using box-and-whisker plots
© 2002 Prentice-Hall, Inc.
Chap 3-2
Chapter Topics


(continued)
Coefficient of correlation
Pitfalls in numerical descriptive measures and
ethical considerations
© 2002 Prentice-Hall, Inc.
Chap 3-3
Summary Measures
Summary Measures
Central Tendency
Mean
Quartile
Mode
Median
Range
Variation
Coefficient of
Variation
Variance
Geometric Mean
© 2002 Prentice-Hall, Inc.
Standard Deviation
Chap 3-4
Measures of Central Tendency
Central Tendency
Average
Median
Mode
n
X 
X
i 1
N

i 1
Geometric Mean
X G   X1  X 2 
n
X
i
 Xn 
1/ n
i
N
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Chap 3-5
Mean (Arithmetic Mean)

Mean (arithmetic mean) of data values

Sample mean
Sample Size
n
X

X
i 1
i
n
 Xn
Population mean
Population Size
N

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X1  X 2 

n
X
i 1
N
i
X1  X 2 

N
 XN
Chap 3-6
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
Mean = 5
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0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 6
Chap 3-7
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
two middle numbers
© 2002 Prentice-Hall, Inc.
Chap 3-8
Mode






A measure of central tendency
Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may 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 = 9
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0 1 2 3 4 5 6
No Mode
Chap 3-9
Geometric Mean

Useful in the measure of rate of change of a
variable over time
X G   X1  X 2 

 Xn 
1/ n
Geometric mean rate of return

Measures the status of an investment over time
RG  1  R1   1  R2  
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 1  Rn 
1/ n
1
Chap 3-10
Example
An investment of $100,000 declined to $50,000 at the
end of year one and rebounded to $100,000 at end of
year two:
X1  $100,000
X 2  $50,000
X 3  $100,000
Average rate of return:
(50%)  (100%)
X
 25%
2
Geometric rate of return:
RG  1   50%    1  100%   
1/ 2
  0.50    2  
1/ 2
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1
 1  1  1  0%
1/ 2
Chap 3-11
Quartiles

Split Ordered Data into 4 Quarters
25%
25%
 Q1 

25%
 Q2 
Position of i-th Quartile
25%
Q3 
i  n  1
 Qi  
4
Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Position of Q1 
1 9  1
4
 2.5
Q1
12  13


 12.5
2
Q1 and Q3 Are Measures of Noncentral Location
 Q = Median, A Measure of Central Tendency
2

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Chap 3-12
Measures of Variation
Variation
Variance
Range
Population
Variance
Sample
Variance
Interquartile Range
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Standard Deviation
Coefficient
of Variation
Population
Standard
Deviation
Sample
Standard
Deviation
Chap 3-13
Range


Measure of variation
Difference between the largest and the
smallest observations:
Range  X Largest  X Smallest

Ignores the way in which data are distributed
Range = 12 - 7 = 5
Range = 12 - 7 = 5
7
8
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9
10
11
12
7
8
9
10
11
12
Chap 3-14
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
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Chap 3-15
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
 
2
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 X
i 1
i

N
2
Chap 3-16
Standard Deviation



Most important measure of variation
Shows variation about the mean
Has the same units as the original data

Sample standard deviation:
n
S

Population standard deviation:

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 X
i 1
X
i
2
n 1
N
 X
i 1
i

2
N
Chap 3-17
Comparing Standard Deviations
Data A
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 3.338
Data B
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = .9258
Data C
11 12 13 14 15 16 17 18 19 20 21
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Mean = 15.5
s = 4.57
Chap 3-18
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
S
CV  
X
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
100%

Chap 3-19
Comparing Coefficient
of Variation

Stock A:



Stock B:



Average price last year = $50
Standard deviation = $5
Average price last year = $100
Standard deviation = $5
Coefficient of variation:

Stock A:

Stock B:
© 2002 Prentice-Hall, Inc.
S
CV  
X

 $5 
100%  
100%  10%

 $50 
S
CV  
X

 $5 
100%  
100%  5%

 $100 
Chap 3-20
Shape of a Distribution

Describes how data is distributed

Measures of shape

Symmetric or skewed
Left-Skewed
Mean < Median < Mode
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Symmetric
Mean = Median =Mode
Right-Skewed
Mode < Median < Mean
Chap 3-21
Exploratory Data Analysis

Box-and-whisker plot

Graphical display of data using 5-number summary
X smallest Q
1
4
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6
Median( Q2)
8
Q3
10
Xlargest
12
Chap 3-22
Distribution Shape and
Box-and-Whisker Plot
Left-Skewed
Q1
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Q2 Q3
Symmetric
Q1Q2Q3
Right-Skewed
Q1 Q2 Q3
Chap 3-23
Coefficient of Correlation

Measures the strength of the linear
relationship between two quantitative
variables
n
r
 X
i 1
n
 X
i 1
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i
i
 X Yi  Y 
X
2
n
 Y  Y 
i 1
2
i
Chap 3-24
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 positive linear
relationship
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Chap 3-25
Scatter Plots of Data with
Various Correlation Coefficients
Y
Y
Y
X
r = -1
X
r = -.6
Y
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X
r=0
Y
r = .6
X
r=1
X
Chap 3-26
Pitfalls in
Numerical Descriptive Measures

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 fair, neutral and clear manner
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Chap 3-27
Ethical Considerations
Numerical descriptive measures:



Should document both good and bad results
Should be presented in a fair, objective and
neutral manner
Should not use inappropriate summary
measures to distort facts
© 2002 Prentice-Hall, Inc.
Chap 3-28
Chapter Summary

Described measures of central tendency

Mean, median, mode, geometric mean, midrange

Discussed quartile

Described measure of variation


Range, interquartile range, variance and standard
deviation, coefficient of variation
Illustrated shape of distribution

Symmetric, skewed, box-and-whisker plots
© 2002 Prentice-Hall, Inc.
Chap 3-29
Chapter Summary


(continued)
Discussed correlation coefficient
Addressed pitfalls in numerical descriptive
measures and ethical considerations
© 2002 Prentice-Hall, Inc.
Chap 3-30