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P91, #11

P91, #11. P. 97, #19 a,b. Q 1 = P 25 i =(25/100)(9)=2.25, round to 3 Q 1 =45. Q 3 = P 75 i =(75/100)(9)=6.75, round to 7 Q 3 =55. P. 97, #19 c. Variation in air quality in Anaheim higher, means similar. Last Time:. Sample Standard Deviation. Population Standard Deviation.

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P91, #11

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  1. P91, #11

  2. P. 97, #19 a,b Q1 = P25 i=(25/100)(9)=2.25, round to 3 Q1=45 Q3 = P75 i=(75/100)(9)=6.75, round to 7 Q3=55

  3. P. 97, #19 c Variation in air quality in Anaheim higher, means similar

  4. Last Time: Sample Standard Deviation Population Standard Deviation

  5. Using the Standard Deviation • Chebyshev’s Theorem • Empirical Rule • Z scores

  6. Chebyshev’s Theorem At least [1 – (1/z)2] of the data values must be within z standard deviations of the mean, where z is any value greater than 1. Since [1 – (½)2] = 1 – ¼ = ¾, 75% of the data values must lie within two standard deviations of the mean Since [1 – (1/3)2] = 1 – 1/9 = 8/9, 88.9% of the data values must lie within three standard deviations of the mean

  7. Application of Chebyshev From 1926 to 2005 the average annual total return on large company stocks was 12.3% The standard deviation of the annual returns was 20.2% Source: A Random Walk Down Wall Street by Burton G. Malkiel, 2007 edition

  8. Application of Chebyshev, cont. Chebyshev’s Theorem states that there is at least a 75% chance that a randomly chosen year will have a return between -28.1% and 52.7%. – 2(s) = 12.3% - 2(20.2%) = -28.1% + 2(s) = 12.3% + 2(20.2%) = 52.7% Alternatively, there is up to a 25% chance the value will fall outside that range.

  9. Application of Chebyshev, cont. From 1926 to 2005 the average annual total return on long-term government bonds was 5.8% The standard deviation of the annual returns was 9.2% What range would capture at least 75% of the values? Source: A Random Walk Down Wall Street by Burton G. Malkiel, 2007 edition

  10. Application of Chebyshev, cont. – 2(s) = 5.8% - 2(9.2%) = -12.6% + 2(s) = 5.8% + 2(9.2%) = 24.2% The corresponding range for U.S. Treasury Bills is -2.4% to 10%

  11. Source: http://disciplinedinvesting.blogspot.com/2007/02/stocks-versus-bonds.html

  12. Source: http://disciplinedinvesting.blogspot.com/2007/02/stocks-versus-bonds.html

  13. Empirical Rule The empirical rule applies when the values have a bell-shaped distribution. • Approximately 68% of the values will be within one standard deviation of the mean • Approximately 95% of the values will be within two standard deviations of the mean • Virtually all of the values will be within three standard deviations of the mean

  14. Source: http://fisher.osu.edu/~diether_1/b822/riskret_2up.pdf

  15. Z Score The distance, measured in standard deviations, between some value and the mean. Also referred to as the “standardized value”

  16. Z Score Z1980 = (32.5-12.3)/20.2 = 1 Z1990 = (-3.1-12.3)/20.2 = -0.8 Z2000 = (-9.1-12.3)/20.2 = -1.1 Z2007 = (5.5-12.3)/20.2 = -0.3

  17. Outliers Observations with extremely small or extremely large values. Values more than three standard deviations from the mean are typically considered outliers. The S&P 500 index fell by 37% in 2008. Should it be considered an outlier? Z2008 = (-37-12.3)/20.2 = -2.4

  18. Distribution Shape - Skewness A distribution is skewed when one side of a distribution has a longer tail than the other side. The distribution is symmetric when the two sides of the distribution are mirror images of each other.

  19. Distribution Shape - Skewness Mean = Median, Skewness =0

  20. Distribution Shape - Skewness Mean < Median, Skewness < 0

  21. Distribution Shape - Skewness Mean > Median, Skewness > 0

  22. Numerical Measures of Association • Covariance • Correlation Coefficient

  23. Covariance Sample covariance: Population covariance:

  24. Covariance , Sxy > 0 II I III IV Mean of X = 5.5, Mean of Y = 7.6

  25. Covariance , Sxy < 0 II I III IV Mean of X = 5.5, Mean of Y = 7.6

  26. Covariance, Sxy = 0 II I III IV Mean of X = 5.5, Mean of Y = 7.6

  27. Covariance, cont. The sign of the covariance indicates if the relationship is positive (direct) or negative (inverse). However, the size of the covariance is not a good indicator of the strength of the relationship because it is sensitive to the units of measurement used.

  28. Correlation Coefficient Pearson Product Moment Correlation Coefficient Population: Sample:

  29. Correlation Coefficient, cont. • Properties of the correlation coefficient: • Value is independent of the unit of measurement • Sign indicates whether relationship is positive or negative • Value can range from -1 to 1 • A value of -1 or 1 indicates a perfect linear relationship

  30. Correlation Coefficient, r=1

  31. Numerical Example Mean of x = 3, Mean of y = 7 Sum of squared deviations, x = 14 Sum of squared deviations, y = 42 Sum of the product of deviations = -24

  32. Numerical Example, cont.

  33. Numerical Example, cont.

  34. Numerical Example Mean of x = 3, Mean of y = 5 Sum of squared deviations, x = 14 Sum of squared deviations, y = 18 Sum of the product of deviations = 13

  35. Numerical Example, cont.

  36. Numerical Example, cont.

  37. Weighted Mean

  38. Example Fiji Stock Market, Day 1 Unweighted mean = (1+2+3)/3 = 2 Weighted mean = [(1)(50)+(2)(200)+(3)(50)]/(50+200+50)= 600/300=2

  39. Example Fiji Stock Market, Day 2 Unweighted mean = (2+1+4)/3 = 2.33 Weighted mean = [(2)(50)+(1)(200)+(4)(50)]/(50+200+50)= 500/300=1.67

  40. http://www.moneychimp.com/articles/volatility/standard_deviation.htmhttp://www.moneychimp.com/articles/volatility/standard_deviation.htm

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