Statistics For Business 3

Basic Business Statistics C hap 3-1

CHAPTER 2 NUMERICAL DESCRIPTIVE MEASURES

© 2003 Prentice-Hall, Inc.

Chapter Topics C hap 3-2

Measures of Central Tendency 

Mean, Median, Mode, Geometric Mean

Quartile Measure of Variation 

Range, Interquartile Range, Variance and Standard Deviation, Coefficient of Variation

Shape 

Symmetric, Skewed, Using Box-and-Whisker Plots

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Chapter Topics C hap 3-3

(continued )

The Empirical Rule and the Bienayme-Chebyshev

Rule Coefficient of Correlation Pitfalls in Numerical Descriptive Measures and

Ethical Issues

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Summary Measures C hap 3-4

Summary Measures

Central Tendency Mean

Median

Mode

Quartile Range Variance

Geometric Mean

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Variation Coefficient of Variation

Standard Deviation

Measures of Central Tendency C hap 3-5

Central Tendency

Mean

Median X = ( X × X 1

2

Mode

× ... × X n )1/ n

n

X 

X i 1

i

n N



X i 1

i

N

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Geometric Mean

Mean (Arithmetic Mean) C hap 3-6

Mean (Arithmetic Mean) of Data Values 

Sample mean

Sample Size

n

X



 PopulationX mean

i 1

i

n

Population Size

N

 © 2003 Prentice-Hall, Inc.

X i 1

N

X1  X 2  L  X n  n

i

X1  X 2  L  X N  N

Mean (Arithmetic Mean) C hap 3-7

The Most Common Measure of Central

(continued ) 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

Mean (Arithmetic Mean) C hap 3-8

(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 © 2003 Prentice-Hall, Inc.

Median C hap 3-9

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

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Mode C hap 3-10

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 7 8 9 10 11 12 13 14

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Mode = 9

0 1 2 3 4 5 6

No Mode

Geometric Mean C hap 3-11

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   © 2003 Prentice-Hall, Inc.

1/ n

1

Quartiles C hap 3-12

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

1 9  1 12  13  Position Q1  Q1  Location  12.5 and ofare Measuresof2.5 Noncentral 4 2  = Median, a Measure of Central Tendency

Q1 Q2

Q3

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Measures of Variation C hap 3-13

Variation

Variance Range

Population Variance Sample Variance

Interquartile Range © 2003 Prentice-Hall, Inc.

Standard Deviation Population Standard Deviation Sample Standard Deviation

Coefficient of Variation

Range C hap 3-14

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 7 12

8

9

10

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11

Range = 12 - 7 = 5 7 12

8

9

10

11

Interquartile Range C hap 3-15

Measure of Variation Also Known as Midspread 

Spread in the middle 50%

Difference between the First and Third Quartiles

DataAffected in Ordered 11 Values 12 13 16 16 17 Not byArray: Extreme

17 18 21

Interquartile Range  Q3  Q1  17.5  12.5  5 © 2003 Prentice-Hall, Inc.

Variance C hap 3-16

Important Measure of Variation Shows Variation about the Mean 

Sample Variance:

n

S2  

 X i 1

X

i

n 1

Population Variance: N

  2

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 X i 1

i



N

2

2

Standard Deviation C hap 3-17

Most Important Measure of Variation Shows Variation about the Mean Has the Same Units as the Original Data 

Sample Standard Deviation: n

S 

 X i 1

Population Standard Deviation:

X

i

n 1

N

 © 2003 Prentice-Hall, Inc.

 X i 1

2

i



N

2

Standard Deviation C hap 3-18

Approximating the Standard Deviation 

(continued )

Used when the raw data are not available and the only source of data is a frequency distribution c



S

 m 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 f  frequencies of the jth class

j Inc. © 2003 Prentice-Hall,

Comparing Standard Deviations C hap 3-19

Data A 11

12

13

14

15

16

17

18

19

20 21

Data B 11

12

13

14

15

16

17

18

19

20 21

Mean = 15.5 s = .9258

20 21

Mean = 15.5 s = 4.57

Data C 11

12

13

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14

15

16

17

18

19

Mean = 15.5 s = 3.338

Coefficient of Variation C hap 3-20

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   100%  Sensitive X Outliers  to  © 2003 Prentice-Hall, Inc.

Comparing Coefficient of Variation C hap 3-21

Stock A:  Average price last year = $50  Standard deviation = $2 Stock B:  Average price last year = $100  Standard deviation = $5 Coefficient of Variation:  Stock A: 

Stock B:

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 S  $2  CV   100%   100%  4%  X  $50 

 S  $5  CV   100%   100%  5%  X  $100 

Shape of a Distribution C hap 3-22

Describe How Data are Distributed Measures of Shape 

Symmetric or skewed

Left-Skewed

Symmetric

Mean < Median < Mode Mean = Median =Mode

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Right-Skewed Mode < Median < Mean

Exploratory Data Analysis C hap 3-23

Box-and-Whisker  Graphical display of data using 5-number summary

X smallest Q 1

4 © 2003 Prentice-Hall, Inc.

6

Median( Q2)

8

Q3

10

Xlargest

12

Distribution Shape & Box-and-Whisker C hap 3-24

Left-Skewed

Q1

Q2 Q3

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Symmetric

Q1Q2Q3

Right-Skewed

Q1 Q2 Q3

Exploratory Data Analysis ◆

Stem-and-leaf display: An exploratory data analysis technique that simultaneously rank orders quantitative data and provides insight about the shape of the distribution.

© 2003 Prentice-Hall, Inc.

Chap 3-25

Stem-and-leaf display NUMBER OF QUESTIONS ANSWERED CORRECTLY ON AN APTITUDE TEST 112 72 69 97 107 73 92 76 86 73 126 128 118 127 124 82 104 132 134 83 92 108 96 100 92 115 76 91 102 81 95 141 81 80 106 84 119 113 98 75 68 98 115 106 95 100 85 94 106 119

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Chap 3-26

Stem-and-leaf display Number of questions Stem-and-Leaf Plot Frequency 2.00 6.00 8.00 11.00 9.00 7.00 4.00 2.00 1.00 Stem width: Each leaf:

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Stem & Leaf 6 . 89 7 . 233566 8 . 01123456 9 . 12224556788 10 . 002466678 11 . 2355899 12 . 4678 13 . 24 14 . 1 10.00 1 case(s) Chap 3-27

The Empirical Rule C hap 3-28

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

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The Bienayme-Chebyshev Rule C hap 3-29

The Percentage of Observations Contained Within

Distances of k Standard Deviations Around the Mean 2 Must Be at Least 1  1/ k   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

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Coefficient of Correlation C hap 3-30

Measures the Strength of the Linear Relationship

between 2 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

i

2

Features of Correlation Coefficient C hap 3-31

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 © 2003 Prentice-Hall, Inc.

Scatter Plots of Data with Various Correlation Coefficients Y

Y

r = -1

X

Y

Y

r = -.6

X

r=0

Y

r = .6 © 2003 Prentice-Hall, Inc.

C hap 3-32

X

r=1

X

X

Pitfalls in Numerical Descriptive Measures and Ethical Issues C hap 3-33

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

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