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Statistics for Managers 4th Edition Chapter 3 Numerical Descriptive Measures Chap 3-1 Chapter Topics Measures of central tendency Measure of variation Mean, median, mode, geometric mean, Quartile Range, interquartile range, average deviation, variance and standard deviation, coefficient of variation, standard units, Sharpe ratio, Sortino ratio Shape Chap 3-2 Chapter Topics (continued) Ethical considerations Chap 3-3 Summary Measures Summary Measures Central Tendency Mean Quartile Mode Median Range Variation Coefficient of Variation Variance Geometric Mean 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 Chap 3-5 Mean (Arithmetic Mean) Mean (arithmetic mean) of data values Sample mean Sample Size n X X i 1 i n X1 X 2 n Xn Population mean Population Size 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 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 6 Chap 3-7 Mean of Grouped Data Class 10 but under 20 20 but under 30 30 but under 40 40 but under 50 50 but under 60 (F) (M) Frequency Mid-Point M • F 45 3 15 150 6 25 175 5 35 180 4 45 110 2 55 660 20 S M • F) 660 = 33 X = = n 20 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 two middle numbers Chap 3-9 Median of Group Data Class 10 but under 20 20 but under 30 30 but under 40 40 but under 50 50 but under 60 (F) Frequency 3 6 5 4 2 Median Class 20 Step 1: Locate Median Term MT = n = 2 20 2 = 10 Step 2: Assign a Value to the Median Term (10 - 9) (MT - SFP) •(i)= 30+ MD = L+ FMD 5 •10 = 32 Chap 3-10 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 0 1 2 3 4 5 6 No Mode Chap 3-11 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 1 Rn 1/ n 1 Chap 3-12 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 1 1 1 1 0% 1/ 2 Chap 3-13 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 Chap 3-14 Measures of Variation Variation Variance Range Population Variance Sample Variance Interquartile Range Standard Deviation Coefficient of Variation Population Standard Deviation Sample Standard Deviation Chap 3-15 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 9 10 11 12 7 8 9 10 11 12 Chap 3-16 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 3-17 Average Deviation X (X-X) X-X 1 -4 4 3 -2 2 6 +1 1 9 +4 4 6 +1 1 25 0 12 X = S X = 25 = 5 n 5 S X-X AD = n 12 = = 2.4 5 Chap 3-18 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 X i 1 i N 2 Chap 3-19 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: X i 1 X i 2 n 1 N X i 1 i 2 N Chap 3-20 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 Mean = 15.5 s = 4.57 Chap 3-21 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 100% Chap 3-22 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: S CV X $5 100% 100% 10% $50 S CV X $5 100% 100% 5% $100 Chap 3-23 Using z scores to evaluate performance (Example) The industry in which sales rep Bill works has average annual sales of $2,500,000 with a standard deviation of $500,000. The industry in which sales rep Paula works has average annual sales of $4,800,000 with a standard deviation of $600,000. Last year Rep Bill’s sales were $4,000,000 and Rep Paula’s sales were $6,000,000. Which of the representatives would you hire if you had one sales position to fill? Chap 3-24 Standard Units Sales person Bill Sales person Paula B= $2,500,000 P=$4,800,000 B= $500,000 P= $600,000 XB= $4,000,000 XP= $6,000,000 ZB = ZP = XB - B 4,000,000 – 2,500,000 = = +3 B 500,000 XP - P 6,000,000 – 4,800,000 = = +2 P 600,000 Chap 3-25 SHARPE RATIO Sharpe ratio = (Prr – RFrr)/Srr Where: Prr = Annualized average return on the portfolio RFrr = Annualized average return on risk free proxy Srr = Annualized standard deviation of average returns Sharpe R = (10.5 – 2.5)/ 3.5 = 2.29 Generally, the higher the better. Chap 3-26 SORTINO RATIO Sortino Ratio = (Prr – RFrr)/Srr(downside) Where: Prr = Annualized rate of return on portfolio RFrr= Annualized risk free annualized rate of return on portfolio Srr(downside) = downside semi-standard deviation Sortino = (10.5-2.5)/ 2.5 = 3.20 Doesn’t penalize for positive upside returns which the Sharpe ratio does Chap 3-27 Shape of a Distribution Describes how data is distributed Measures of shape Symmetric or skewed Left-Skewed Mean < Median < Mode Symmetric Mean = Median =Mode Right-Skewed Mode < Median < Mean Chap 3-28 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 Chap 3-29 Chapter Summary Described measures of central tendency Mean, median, mode, geometric mean Discussed quartile Described measure of variation Range, interquartile range, average deviation, variance, and standard deviation, coefficient of variation, standard units, Sharp ratio, Sortino ratio Illustrated shape of distribution Symmetric, skewed Chap 3-30

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