When you have completed this chapter, you will be able to: Compute and interpret the range, the mean deviation, the variance, the standard deviation, and the coefficient of variation of ungrouped data;
4 1 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 4 2 When you have completed this chapter, you will be able to: Compute and interpret the range, the mean deviation, the variance, the standard deviation, and the coefficient of variation of ungrouped data Compute and interpret the range, the variance, and the standard deviation from grouped data Explain the characteristics, uses, advantages, and disadvantages of each measure Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 4 3 Understand Chebyshev’s theorem and the normal or empirical rule, as it relates to a set of observations Compute and interpret percentiles, quartiles and the interquartile range Construct and interpret box plots Compute and describe the coefficient of skewness and kurtosis of a data distribution Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Terminology Range …is the difference between the …is the difference between the largest and the and the smallest smallest value. value. largest Only two values are used in its calculation Only two values are used in its calculation It is influenced by an extreme value It is influenced by an extreme value It is easy to compute and understand It is easy to compute and understand Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 4 4 Terminology 4 5 Mean Deviation …is the arithmetic mean of the absolute values of …is the arithmetic mean of the absolute values of the deviations from the arithmetic mean. the deviations from the arithmetic mean. MD = Σ x − µ N All values are used in the calculation All values are used in the calculation It is not unduly influenced by large or small values It is not unduly influenced by large or small values The absolute values are difficult to manipulate The absolute values are difficult to manipulate Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 4 6 The weights of a sample of crates containing books for the bookstore (in kg) are: 103 97 101 106 103 Findtherangeandthemeandeviation Copyrightâ2004byTheMcGrawưHillCompanies,Inc.Allrightsreserved. 4ư7 10397101106103 10397101106103 x à= N Findthemeanweight Findthemeandeviation 103 102 103 102 Find the range Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. MD = 510 Σ x − µ 5 106 – 97 = 9 102 N = 2.4 Terminology 4 8 Variance …is the arithmetic mean of the …is the arithmetic mean of the squared deviations squared deviations from the arithmetic mean. from the arithmetic mean. All values are used in the calculation All values are used in the calculation It is not influenced by extreme values It is not influenced by extreme values The units are awkward…the square The units are awkward…the square units. units Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. of the original of the original Computation Computing the V Variance ariance Computing the Formula … for a Population Formula σ Σ( x − µ ) = N Formula … for a Sample Formula s Σ( x − x ) = n −1 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 4 9 4 10 The ages of the Dunn family are: 2, 18, 34, 42 What is the population mean and variance? x µ = N σ Σ( x − µ ) = N Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 96 24 24 944 236 42 24 Example 4 34 For a set of observations the third quartile is 24 and the first quartile is 10. What is the interquartile range? The interquartile range is 24 10 = 14. Fifty percent of the observations will occur between 10 and 24 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 4 35 Box Plots …is a graphical display, based on quartiles, that helps to picture a set of data Five pieces of data are needed to construct a box plot: … the Minimum Value, … the First Quartile, … the Median, … the Third Quartile, and … the Maximum Value Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Example Example 4 36 Based on a sample of 20 deliveries, Buddy’s Pizza determined the following information The…minimum delivery time was 13minutes …the maximum 30 minutes The…first quartile was 15 minutes …the median 18 minutes, and … the third quartile 22 minutes Develop a box plot for the delivery times Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Solution Solution Min. Q1 Median Q3 Max 12 14 16 18 20 22 24 26 28 30 32 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 4 37 4 38 The following are the average rates of return for Stocks A and B over a six year period, In which of the following Stocks would you prefer to invest? Why? Stock A: Stock B:7 153 -10 18 10 -5 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 4 39 Find the Mean rate of return for each of the two stocks: Stock A: Mean = 36/6 = 6 Stock B: 15 -10 18 10 -5 Mean = 36/6 = 6 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 4 40 Find the Range of Values of each stock: Stock A: 8 – 3 = 5 Stock B: 15 -10 18 10 -5 18 – ( 10) = 28 Therefore, Stock B is riskier Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Relative Dispersion 4 41 The coefficient of variation is the ratio of the standard deviation to the arithmetic mean, expressed as a percentage: s x CV (100%) A standard deviation of 10 may be perceived as large when the mean value is 100, but only moderately large when the mean value is 500! Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 4 42 Example Rates of return over the past 6 years for two mutual funds are shown below. Fund A: 8.3, 6.0, 18.9, 5.7, 23.6, 20 Fund B: 12, 4.8, 6.4, 10.2, 25.3, 1.4 Which one has a higher level of risk? Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Solution 4 43 Solution Let us use the Excel printout that is run from the “Descriptive Statistics” submenu Fund A Fund B Mean 9.85 Mean Standard Error 5.38 Standard Error Median 13.60 Median Mode #N/A Mode Standard Deviation 13.19 Standard Deviation Sample Variance 173.88 Sample Variance Kurtosis -2.21 Kurtosis Skewness -0.44 Skewness Range 29.60 Range Minimum -6 Minimum Maximum 23.6 Maximum Sum 59.1 Sum Count Count Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 8.42 4.20 8.30 #N/A 10.29 105.81 0.90 0.61 30.1 -4.8 25.3 50.5 4 44 Solution Is Fund A riskier because its standard deviation is larger? Fund A Fund B Mean 9.85 Mean Standard Error 5.38 Standard Error Median 13.60 Median Mode #N/A Mode Standard Deviation 13.19 Standard Deviation Sample Variance 173.88 Sample Variance Kurtosis -2.21 Kurtosis Skewness -0.44 Skewness Range 29.60 Range Minimum -6 Minimum Maximum 23.6 Maximum Sum 59.1 Sum Count Count Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 8.42 4.20 8.30 #N/A 10.29 105.81 0.90 0.61 30.1 -4.8 25.3 50.5 4 45 Solution But the means of the two funds are different Fund A Fund B Mean 9.85 Mean 8.42 Standard Error 5.38 Standard Error 4.20 Median 13.60 Median 8.30 Mode #N/A Mode #N/A Standard Deviation 13.19 Standard Deviation 10.29 Sample Variance 173.88 Sample Variance 105.81 Kurtosis -2.21 Kurtosis 0.90 Fund A has a higher rate of return, Fund A has a higher rate of return, Skewness but it also has a larger sd -0.44 Skewness 0.61 but it also has a larger sd Range 29.60 Range 30.1 Therefore we need to compare the Therefore we need to compare the Minimum -6 Minimum -4.8 relative variability Maximum 23.6 Maximum 25.3 relative variability Sum 59.1 Sum 50.5 using the coefficient of using the coefficient of Count Count variation variation Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 4 46 CV s x Solution (100%) Fund A: CV = 13.19 / 9.85 = 1.34 Fund A: CV = 13.19 / 9.85 = 1.34 Fund B: CV CV = 10.29 / 8.42 = 1.22 = 10.29 / 8.42 = 1.22 Fund B: So now we say that there is So now we say that there is more variability in Fund A more variability in Fund A as compared to Fund B as compared to Fund B Therefore, Fund A is riskier Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Test your learning… … Test your learning … … n o n o k ilcick CCl www.mcgrawhill.ca/college/lind Online Learning Centre for quizzes extra content data sets searchable glossary access to Statistics Canada’s EStat data …and much more! Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 4 47 4 48 This completes Chapter 4 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. ... Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. The standard The standard deviation is deviation is 8.04 = 2.8 Interpretation and Uses Interpretation and Uses of the Standard of the Standard ... Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. S Solution (continued) 4 21 Using Step 1 Step 1 Step 2 Step 2 Determine the mean and Determine the mean and standard deviation of the sample... Find the mean, variance, and Standard Deviation Find the mean, variance, and Standard Deviation x µ = N 37 2 7 Σ ( x − x ) s2 = = n −1 s = Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved.