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IE 340/440 PROCESS IMPROVEMENT THROUGH PLANNED EXPERIMENTATION Numerical Descriptive Measures Dr. Xueping Li University of Tennessee © 2003 Prentice-Hall, Inc. Chap 3-1 Chapter Topics 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 © 2003 Prentice-Hall, Inc. Chap 3-2 Chapter Topics (continued) The Empirical Rule and the BienaymeChebyshev Rule Coefficient of Correlation Pitfalls in Numerical Descriptive Measures and Ethical Issues © 2003 Prentice-Hall, Inc. Chap 3-3 Summary Measures Summary Measures Central Tendency Mean Quartile Mode Median Range Variation Coefficient of Variation Variance Geometric Mean © 2003 Prentice-Hall, Inc. Standard Deviation Chap 3-4 Measures of Central Tendency Central Tendency Mean 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 © 2003 Prentice-Hall, Inc. Chap 3-5 Mean (Arithmetic Mean) Mean (Arithmetic Mean) of Data Values Sample mean Sample Size n X X i 1 i n Population mean N © 2003 Prentice-Hall, Inc. X i 1 N i X1 X 2 n Xn Population Size 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 © 2003 Prentice-Hall, Inc. 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 6 Chap 3-7 Mean (Arithmetic Mean) (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. Chap 3-8 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 © 2003 Prentice-Hall, Inc. Chap 3-9 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 7 8 9 10 11 12 13 14 Mode = 9 © 2003 Prentice-Hall, Inc. 0 1 2 3 4 5 6 No Mode Chap 3-10 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 © 2003 Prentice-Hall, Inc. 1 Rn 1/ n 1 Chap 3-11 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) R 0.25 (or 25%) 2 Geometric rate of return: RG 1 0.5 1 1 1/ 2 0.5 2 1/ 2 © 2003 Prentice-Hall, Inc. 1 1 11/ 2 1 0 (or 0%) Chap 3-12 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 1 9 1 Position of Q1 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 © 2003 Prentice-Hall, Inc. Chap 3-13 Measures of Variation Variation Variance Range Population Variance Sample Variance Interquartile Range © 2003 Prentice-Hall, Inc. Standard Deviation Coefficient of Variation Population Standard Deviation Sample Standard Deviation 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 © 2003 Prentice-Hall, Inc. 9 10 11 12 7 8 9 10 11 12 Chap 3-15 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 © 2003 Prentice-Hall, Inc. Chap 3-16 Variance Important Measure of Variation Shows Variation about the Mean Sample Variance: n S 2 X i 1 X i 2 n 1 Population Variance: N 2 © 2003 Prentice-Hall, Inc. X i 1 i N 2 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: © 2003 Prentice-Hall, Inc. n X i 1 X i 2 n 1 N X i 1 i 2 N Chap 3-18 Standard Deviation (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 © 2003 Prentice-Hall, Inc. f j frequencies of the jth class Chap 3-19 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 © 2003 Prentice-Hall, Inc. Mean = 15.5 s = 4.57 Chap 3-20 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 © 2003 Prentice-Hall, Inc. Chap 3-21 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: © 2003 Prentice-Hall, Inc. S CV X $2 100% 100% 4% $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 Mean < Median < Mode © 2003 Prentice-Hall, Inc. Symmetric Mean = Median =Mode Right-Skewed Mode < Median < Mean Chap 3-23 Exploratory Data Analysis 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 Chap 3-24 Distribution Shape & Box-and-Whisker Left-Skewed Q1 © 2003 Prentice-Hall, Inc. Q2 Q3 Symmetric Q1Q2Q3 Right-Skewed Q1 Q2 Q3 Chap 3-25 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 © 2003 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 © 2003 Prentice-Hall, Inc. Chap 3-27 Coefficient of Correlation Measures the Strength of the Linear Relationship between 2 Quantitative Variables n r X i 1 n X i 1 © 2003 Prentice-Hall, Inc. i i X Yi Y X 2 n Y Y i 1 2 i Chap 3-28 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 © 2003 Prentice-Hall, Inc. Chap 3-29 Scatter Plots of Data with Various Correlation Coefficients Y Y Y X r = -1 X r = -.6 Y © 2003 Prentice-Hall, Inc. X r=0 Y r = .6 X r=1 X Chap 3-30 Pitfalls in Numerical Descriptive Measures and Ethical Issues Data Analysis is Objective Data Interpretation is Subjective 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 Should document both good and bad results Presentation should be fair, objective and neutral Should not use inappropriate summary measures to distort the facts © 2003 Prentice-Hall, Inc. Chap 3-31 Chapter Summary Described Measures of Central Tendency Mean, Median, Mode, Geometric Mean Discussed Quartiles Described Measures of Variation Range, Interquartile Range, Variance and Standard Deviation, Coefficient of Variation Illustrated Shape of Distribution Symmetric, Skewed, Using Box-and-Whisker Plots © 2003 Prentice-Hall, Inc. Chap 3-32 Chapter Summary (continued) Described the Empirical Rule and the Bienayme-Chebyshev Rule Discussed Correlation Coefficient Addressed Pitfalls in Numerical Descriptive Measures and Ethical Issues © 2003 Prentice-Hall, Inc. Chap 3-33

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