4 - 1 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

4 - 2 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 1. Compute and interpret the range, the mean deviation, the variance, the standard deviation, and the coefficient of variation of ungrouped data 2. Compute and interpret the range, the variance, and the standard deviation from grouped data When you have completed this chapter, you will be able to: 3. Explain the characteristics, uses, advantages, and disadvantages of each measure

4 - 3 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved 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

4 - 4 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. T erminology Range …is the difference between the largest and the smallest value. Only two values are used in its calculation. It is influenced by an extreme value. It is easy to compute and understand. Only two values are used in its calculation. It is influenced by an extreme value. It is easy to compute and understand.

4 - 5 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. T erminology Mean Deviation …is the arithmetic mean of the absolute values of the deviations from the arithmetic mean. All values are used in the calculation. It is not unduly influenced by large or small values. The absolute values are difficult to manipulate. All values are used in the calculation. It is not unduly influenced by large or small values. The absolute values are difficult to manipulate.

4 - 6 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The weights of a sample of crates containing books for the bookstore (in kg) are: Find the range and the mean deviation.

4 - 7 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Find the mean weight Find the mean deviation Find the range – 97 = 9 =   

4 - 8 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. T erminology V ariance …is the arithmetic mean of the squared deviations from the arithmetic mean. All values are used in the calculation. It is not influenced by extreme values. The units are awkward…the square of the original units. All values are used in the calculation. It is not influenced by extreme values. The units are awkward…the square of the original units. C omputation

4 - 9 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Formula Computing the V ariance … for a Population Formula … for a Sample

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The ages of the Dunn family are: 2, 18, 34, 42 What is the population mean and variance ?

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Population Standard Deviation … is the square root of the population variance From previous example…    2  236 = E xample

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. E XAMPLE The hourly wages earned by a sample of five students are: $7, $5, $11, $8, $6. Find the mean, variance, and Standard Deviation. =  =          =  s2s2 s =

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. A sample of ten movie theatres in a metropolitan area tallied the total number of movies showing last week. Compute the mean number of movies showing per theatre. The Mean of Grouped Data From chapter 3….

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Continued… 6610Total to under to under to under to under to under 3 (f)(x) Class Midpoint Frequency f Movies Showing The Mean of Grouped Data The Mean of Grouped Data N fx  

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. =  Continued… (f)(x) Class Midpoint Frequency f Movies Showing 6610Total Now: Compute the variance and standard deviation. The Mean of Grouped Data The Mean of Grouped Data N fx   Formula N fx  

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Sample Variance for Grouped Data The formula for the sample variance for grouped data is: f is class frequency and X is class midpoint where 1 )(     n n fxfx fxfx s

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 6610Total to under to under to under to under to under 3 (f)(x) Class Midpoint Frequency f Movies Showing (x2)f(x2)f Sample Variance for Grouped Data

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. (f)(x) Class Midpoint Frequency f Movies Showing (x2)f(x2)f 6610Total508 1 )(     n n fxfx fxfx s = = 8.04 Sample Variance for Grouped Data The variance is The standard deviation is 8.04 = 2.8

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Interpretation and Uses of the Standard Deviation Chebyshev’s Theorem: For any set of observations, the minimum proportion of the values that lie within k s tandard d eviations of the mean is at least: where k 2 is any constant greater than k  Formula

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Suppose that a wholesale plumbing supply company has a group of 50 sales vouchers from a particular day. The amount of these vouchers are: How well does this data set fit Chebychev’s Theorem? S olution

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using S olution (continued) Determine the mean and standard deviation of the sample Step 1 Mean = $319 SD = $ Input k =2 into Chebyshev’s theorem Step = 1 – ¼ = 3/4 i.e. At least.75 of the observations will fall within 2SD of the mean. Step 3

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Using the mean and SD, find the range of data values within 2 SD of the mean Step 3 Mean = $319 SD = $ = (2)101.78, (101.78) = (115.44, ) Now, go back to the sample data, and see what proportion of the values fall between and S olution (continued) Proportion ( - 2 S, + 2 S ) xx

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Proportion of the values that fall between and We find that or 96% of the data values are in this range – certainly at least 75% as the theorem suggests! S olution (continued)

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Interpretation and Uses of the Standard Deviation Empirical Rule : For any symmetrical, bell-shaped distribution: …About 68% of the observations will lie within 1s of the mean …About 95% of the observations will lie within 2s of the mean …Virtually all the observations will be within 3s of the mean

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.        Bell-Shaped Curve …showing the relationship between  and 

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. How well does this data set fit the Empirical Rule? S olution Suppose that a wholesale plumbing supply company has a group of 50 sales vouchers from a particular day. The amount of these vouchers are:

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. First check if the histogram has an approximate mound-shape Not bad…so we’ll proceed! We need to calculate the mean and standard deviation S olution

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Mean: $319 Standard Deviation: $ Calculate the intervals: ),(  sxsx = ( , ) )(217.22,   ) 2,2(sxsx = 319 -(2)101.78, (101.78) = 319-(3)101.78, (101.78) =  )3,3 ( sxs x ) (13.66, Interval Empirical Rule Actual # values Actual percentage , %31/50 62% , %48/50 96% 13.66, %49/50 98% Interval Empirical Rule Actual # values Actual percentage , %31/50 62% , %48/50 96% 13.66, %49/50 98% =(115.44, )

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. …The coefficient of skewness can range from up to Skewness …is the measurement of the lack of symmetry of the distribution  σ Mean  Median SK 1 = 3 …A value of 0 indicates a symmetric distribution. It is computed as follows:

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Skewness Following are the earnings per share for a sample of 15 software companies for the year The earnings per share are arranged from smallest to largest. $ Find the coefficient of skewness. Mean = 4.95 Median = 3.18 SD = 5.22 SK 1 = 3( )/5.22 = =  σ Mean  Median SK 1 = 3

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Positively Skewed Distribution Mean and Median are to the right of the Mode Skewed Right Mode< Median< Mean

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Negatively Skewed Distribution Mean and Median are to the left of the Mode Skewed left < Mode < Median Mean

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. …is the distance between the third quartile Q3 and the first quartile Q1. E xample Interquartile Range Interquartile Range This distance will include the middle 50 percent of the observations. Interquartile Range = Q3 - Q1

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. For a set of observations the third quartile is 24 and the first quartile is 10. What is the interquartile range? E xample The interquartile range is = 14. Fifty percent of the observations will occur between 10 and 24.

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 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 Box Plots …is a graphical display, based on quartiles, that helps to picture a set of data E xample

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 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. E xample S olution

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved Min. Q1 Median Q3 Max. S olution

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 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:

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Find the Mean rate of return for each of the two stocks: Stock A: Stock B: Mean =36/6 = 6 Mean =36/6 = 6

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 8 – 3 = 5 18 – ( -10) = 28 Find the Range of Values of each stock: Stock A: Stock B: Therefore, Stock B is riskier.

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Relative Dispersion The coefficient of variation is the ratio of the standard deviation to the arithmetic mean, expressed as a percentage: 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! CV s x  (100%)

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 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 S olution Which one has a higher level of risk?

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Let us use the Excel printout that is run from the “Descriptive Statistics” sub-menu S olution

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Is Fund A riskier because its standard deviation is larger? S olution

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. But the means of the two funds are different. Fund A has a higher rate of return, but it also has a larger sd. Therefore we need to compare the relative variability using the coefficient of variation. Fund A has a higher rate of return, but it also has a larger sd. Therefore we need to compare the relative variability using the coefficient of variation. S olution

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Fund A: CV = / 9.85 = 1.34 Fund B: CV = / 8.42 = 1.22 Fund A: CV = / 9.85 = 1.34 Fund B: CV = / 8.42 = 1.22 So now we say that there is more variability in Fund A as compared to Fund B So now we say that there is more variability in Fund A as compared to Fund B Therefore, Fund A is riskier. S olution CV s x  (100%)

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Test your learning … Click on… Online Learning Centre for quizzes extra content data sets searchable glossary access to Statistics Canada’s E-Stat data …and much more!

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. This completes Chapter 4