Basic Business Statistics 9 th Edition Chapter 3

Chapter Topics n Measures of Central Tendency n Mean, Median, Mode, Geometric Mean n Quartiles n Measures of Variation n n Range, Interquartile Range, Variance and Standard Deviation, Coefficient of Variation The Empirical Rule and the Bienayme. Chebyshev Rule © 2004 Prentice-Hall, Inc. 2

Chapter Topics n n n (continued) Shape n Symmetric, Skewed n Using Box-and-Whisker Plots Coefficient of Correlation Pitfalls in Numerical Descriptive Measures and Ethical Issues © 2004 Prentice-Hall, Inc. 3

Summary Measures Central Tendency Mean Median Mode Quartiles Range Variance Geometric Mean © 2004

Measures of Central Tendency Mean Median Mode Geometric Mean © 2004 Prentice-Hall, Inc. 5

Mean (Arithmetic Mean) n Mean (Arithmetic Mean) of Data Values n Sample mean n

Mean (Arithmetic Mean) (continued) n n 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 © 2004 Prentice-Hall, Inc. 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 6 7

Mean (Arithmetic Mean) From a Frequency Distribution (continued) n Approximating the Arithmetic Mean n

Median n n Robust Measure of Central Tendency Not Affected by Extreme Values 0 1 2 3 4 5 6 7 8 9 10 Median = 5 n 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 n n 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. 9

Mode n n n 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 Mode = 9 © 2004 Prentice-Hall, Inc. 0 1 2 3 4 5 6 No Mode 10

Geometric Mean n n Useful in the Measure of Rate of Change of a

Example An investment of $100, 000 declined to $50, 000 at the end of

Quartiles n Split Ordered Data into 4 Quarters 25% n 25% 25% Position of i-th Quartile Data in Ordered Array: 11 12 13 16 16 17 18 21 22 n n and are Measures of Noncentral Location = Median, a Measure of Central Tendency © 2004 Prentice-Hall, Inc. 13

Measures of Variation Range Interquartile Range Variance Population Variance Sample Variance © 2004 Prentice-Hall, Inc. Standard Deviation Coefficient of Variation Population Standard Deviation Sample Standard Deviation 14

Range n Measure of Variation Difference between the Largest and the Smallest Observations: n Ignores How Data are Distributed n Range = 12 - 7 = 5 7 8 © 2004 Prentice-Hall, Inc. 9 10 11 12 7 8 9 10 11 12 15

Interquartile Range n n Measure of Variation Also Known as Midspread n n Spread in the middle 50% Difference between the First and Third Quartiles Data in Ordered Array: 11 12 13 16 16 17 n 17 18 21 Not Affected by Extreme Values © 2004 Prentice-Hall, Inc. 16

Variance n n Important Measure of Variation Shows Variation about the Mean n Sample

Standard Deviation n Most Important Measure of Variation Shows Variation about the Mean Has the Same Units as the Original Data n Sample Standard Deviation: n Population Standard Deviation: © 2004 Prentice-Hall, Inc. 18

Standard Deviation From a Frequency Distribution (continued) n Approximating the Standard Deviation n Used when the raw data are not available and the only source of data is a frequency distribution n © 2004 Prentice-Hall, Inc. 19

Comparing Standard Deviations 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 © 2004 Prentice-Hall, Inc. 13 14 15 16 17 18 19 Mean = 15. 5 s = 3. 338 20

Coefficient of Variation n Measure of Relative Variation n Always in Percentage (%) n Shows Variation Relative to the Mean n Used to Compare Two or More Sets of Data Measured in Different Units n n Sensitive to Outliers © 2004 Prentice-Hall, Inc. 21

Comparing Coefficient of Variation n Stock A: n n n Stock B: n n n Average price last year = $50 Standard deviation = $2 Average price last year = $100 Standard deviation = $5 Coefficient of Variation: n Stock A: n Stock B: © 2004 Prentice-Hall, Inc. 22

Shape of a Distribution n Describe How Data are Distributed n Measures of Shape n Symmetric or skewed Left-Skewed Symmetric Mean < Median < Mode Mean = Median =Mode © 2004 Prentice-Hall, Inc. Right-Skewed Mode < Median < Mean 23

Exploratory Data Analysis n Box-and-Whisker Plot n Graphical display of data using 5 -number

Distribution Shape & Box-and-Whisker Plot Left-Skewed © 2004 Prentice-Hall, Inc. Symmetric Right-Skewed 25

The Empirical Rule n For Data Sets That Are Approximately Bellshaped: n n n 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. 26

The Bienayme-Chebyshev Rule n The Percentage of Observations Contained Within Distances of k Standard Deviations Around the Mean Must Be at Least n n 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. 27

Coefficient of Correlation n Measures the Strength of the Linear Relationship between 2 Quantitative

Features of Correlation Coefficient n Unit Free n Ranges between – 1 and 1 n n n 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 © 2004 Prentice-Hall, Inc. 29

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

Pitfalls in Numerical Descriptive Measures and Ethical Issues n Data Analysis is Objective n n Data Interpretation is Subjective n n Should report the summary measures that best meet the assumptions about the data set Should be done in a fair, neutral and clear manner Ethical Issues n Should document both good and bad results n Presentation should be fair, objective and neutral n Should not use inappropriate summary measures to distort the facts © 2004 Prentice-Hall, Inc. 31

Chapter Summary n Described Measures of Central Tendency n Mean, Median, Mode, Geometric Mean n Discussed Quartiles n Described Measures of Variation n n Range, Interquartile Range, Variance and Standard Deviation, Coefficient of Variation Described the Empirical Rule and the Bienayme-Chebyshev Rule © 2004 Prentice-Hall, Inc. 32

Chapter Summary n n n (continued) Illustrated Shape of Distribution n Symmetric, Skewed n Using Box-and-Whisker Plots Discussed Correlation Coefficient Addressed Pitfalls in Numerical Descriptive Measures and Ethical Issues © 2004 Prentice-Hall, Inc. 33