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Statistics For Managers 5th Edition

Statistics For Managers 5th Edition. Chapter 3 Numerical Descriptive Measures. Chapter Topics. Measures of central tendency Mean, median, mode, weighted mean, geometric mean, quartiles Measure of variation

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Statistics For Managers 5th Edition

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  1. Statistics For Managers5th Edition Chapter 3 Numerical Descriptive Measures

  2. Chapter Topics • Measures of central tendency • Mean, median, mode, weighted mean, geometric mean, quartiles • Measure of variation • Range, interquartile range, average deviation, variance and standard deviation, coefficient of variation, standard units, Sharpe ratio, Sortino ratio • Shape

  3. Summary Measures Summary Measures Variation Central Tendency Quartile Mean Mode Coefficient of Variation Median Range Variance Standard Deviation Geometric Mean

  4. Measures of Central Tendency Central Tendency Average Median Mode Geometric Mean

  5. Mean (Arithmetic Mean) • Mean (arithmetic mean) of data values • Sample mean • Population mean Sample Size Population Size

  6. Mean (Arithmetic Mean) (continued) • The most common measure of central tendency • Most commonly used average 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 5 Mean = 6

  7. (F) (M) M • F Class Frequency Mid-Point 10 but under 20 3 15 45 150 20 but under 30 6 25 175 30 but under 40 5 35 180 40 but under 50 4 45 110 50 but under 60 2 55 20 660 S (M • F) 660 X 33 = = = n 20 Mean of Grouped Data

  8. Advantages and Disadvantages of the Arithmetic Mean • Familiar and Easy to Understand • Easy to Calculate • Always Exists • Is Unique • Lends Itself to Further Calculation • Affected by Extreme Values

  9. Median • Robust measure of central tendency • 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 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 Median = 5

  10. (F) Class Frequency 10 but under 20 20 but under 30 Median Class 30 but under 40 40 but under 50 50 but under 60 Step 1: Locate Median Term MT = = = 10 n 20 Step 2: Assign a Value to the Median Term 2 2 (MT - SFP) (10 - 9) •(i) = 30+ MD = L+ •10 = 32 FMD 5 Median of Group Data 3 6 5 4 2 20

  11. Advantages and Disadvantages of the Median • Easy to Understand • Easy to Calculate • Always Exists • Is Unique • Not Affected by Extreme Values • Only Indicates Middle Value

  12. 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 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 No Mode Mode = 9

  13. Weighted Mean • Used when observations differ in relative importance • Xw = X1W1 + X2W2 + …….. + XnWn W1 + W2 + …….. + Wn

  14. Weighted Mean If you bought 100 shares of a stock at $20 per share, 400 shares at $30 per share, and 500 shares at $40 per share, what would your average cost per share be? Xw = 100(20) + 400(30)+500(40) =$34 100 + 400 +500

  15. Geometric Mean • Useful in the measure of rate of change of a variable over time • Geometric mean rate of return • Measures the status of an investment over time

  16. 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:

  17. Quartiles • Split Ordered Data into 4 Quarters • Position of i-th Quartile • and Are Measures of Noncentral Location • = Median, A Measure of Central Tendency 25% 25% 25% 25% Data in Ordered Array: 11 12 13 16 16 17 18 21 22

  18. Measures of Variation Variation Variance Standard Deviation Coefficient of Variation Range Population Variance Population Standard Deviation Sample Variance Sample Standard Deviation Interquartile Range

  19. Range • Measure of variation • Difference between the largest and the smallest observations: • 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

  20. Interquartile Range • Measure of variation • Also known as midspread • Spread in the middle 50% • Difference between the first and third quartiles • Not affected by extreme values Data in Ordered Array: 11 12 13 16 16 17 17 18 21

  21. X (X-X) X-X 1 - 4 4 25 = = 5 X = 3 - 2 2 5 6 + 1 1 S 12 X-X 9 + 4 4 AD = = = 2.4 n 5 6 + 1 1 25 0 12 S X n Average Deviation

  22. Standard Deviation • Most important measure of variation • Shows variation about the mean • Has the same units as the original data • Sample standard deviation: • Population standard deviation:

  23. Calculation Example:Sample Standard Deviation Sample Data (Xi) : 10 12 14 15 17 18 18 24 n = 8 Mean = X = 16 A measure of the “average” scatter around the mean

  24. 2 2•F F M Classes (M - X) (M - X) (M - X) 972 10 but under 20 3 15 324 -18 384 20 but under 30 6 25 64 -8 20 30 but under 40 5 35 4 +2 576 40 but under 50 4 45 144 +12 968 50 but under 60 2 55 484 +22 2920 20 2 [ ] • F ( - ) å X 2920 M = = s = 12.39 - 19 n 1 Standard Deviation of Grouped Data

  25. Comparing Standard Deviations Data A Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21 Data B Mean = 15.5 s = .9258 11 12 13 14 15 16 17 18 19 20 21 Data C Mean = 15.5 s = 4.57 11 12 13 14 15 16 17 18 19 20 21

  26. Variance • Important measure of variation • Shows variation about the mean • Sample variance: • Population variance:

  27. Coefficient of VariationUsed to Compare Relative Variation in Two or More Data Sets • 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

  28. Comparing Coefficient of Variation • Stock A: • Average price last year = $50 • Standard deviation = $5 • Stock B: • Average price last year = $100 • Standard deviation = $5 • Coefficient of variation: • Stock A: • Stock B:

  29. 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?

  30. XB - mB 4,000,000 – 2,500,000 = = = +3 ZB sB 500,000 XP - mP 6,000,000 – 4,800,000 = = = +2 ZP sP 600,000 Standard UnitsUsed to Compare Relative Positions of Individual Observations in Two or More Data Sets Sales person Bill mB= $2,500,000 sB= $500,000 XB= $4,000,000 Sales person Paula mP=$4,800,000 sP= $600,000 XP= $6,000,000

  31. 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.

  32. 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

  33. Shape of a Distribution • Describes how data is distributed • Measures of shape • Symmetric or skewed Right-Skewed Left-Skewed Symmetric Mean < Median < Mode Mean = Median =Mode Mode <Median < Mean

  34. 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

  35. 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

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