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Section 3.2 Measures of Dispersion HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Compute the range, variance, and standard deviation. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Range Range Range = Maximum Data Value − Minimum Data Value HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.10: Calculating the Range The following data were collected from samples of call lengths (in minutes) observed for two different mobile phone users. Calculate the range of each data set. a. 2, 25, 31, 44, 29, 14, 22, 11, 40 b. 2, 2, 44, 2, 2, 2, 2, 2 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.10: Calculating the Range (cont.) Solution a. The maximum value is 44 minutes and the minimum value is 2 minutes, so the range is as follows. Range Maximum Data Value Minimum Data Value 44 2 42 minutes HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.10: Calculating the Range (cont.) b. The maximum value for the second data set is also 44 minutes and the minimum value is also 2 minutes, so the range is calculated in the same way. Range Maximum Data Value Minimum Data Value 44 2 42 minutes HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Standard Deviation Standard Deviation The standard deviation is a measure of how much we might expect a typical member of the data set to differ from the mean. The population standard deviation is given by s HAWKES LEARNING SYSTEMS Students Matter. Success Counts. x m 2 i N Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Standard Deviation Standard Deviation (cont.) where xi is the ith value in the population, μ is the population mean, and N is the number of values in the population. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Standard Deviation Standard Deviation (cont.) The sample standard deviation is given by s x x 2 i n 1 where xi is the ith data value, x ̄ is the sample mean, and n is the number of data values in the sample. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.11: Calculating Standard Deviation Calculate the sample standard deviation of the following data collected regarding the numbers of hours students studied for a physics exam. 5, 8, 7, 6, 9 Solution Let’s calculate the sample standard deviation by hand using the following formula. s HAWKES LEARNING SYSTEMS Students Matter. Success Counts. x x 2 i n 1 Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.11: Calculating Standard Deviation (cont.) To start, we need the mean and sample size of the data set. We calculate that x ̄ = 7 and n = 5. Next, we need to subtract the mean from each number in the sample, and then square each of these differences. Let’s use a chart to keep everything organized. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.11: Calculating Standard Deviation (cont.) Deviations and Squared Deviations of the Data xi (xi – x̄) (xi – x̄)2 5 8 7 6 9 5 7 = 2 87=1 77=0 6 7 = 1 97=2 4 1 0 1 4 Next, find the sum of the squared deviations by adding up the values in the last column. x x i HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 2 4 1 0 1 4 10 Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.11: Calculating Standard Deviation (cont.) Finally, substitute the appropriate values into the sample standard deviation formula as follows. s x x 2 i n 1 10 51 2.5 1.581139 1.6 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.11: Calculating Standard Deviation (cont.) Alternate Calculator Method To find the sample standard deviation on a TI-83/84 Plus calculator, follow the steps below. • Press . • Choose option 1:Edit and press . • Enter the data in L1. • Press again. • Choose CALC. • Choose option 1:1-Var Stats. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.11: Calculating Standard Deviation (cont.) • Press twice. (Note: If your data are not in L1, before pressing the second time, enter the list where your data are located, such as L3 or L5.) The fourth value in the output, seen in the screenshot below, gives the sample standard deviation, which is Sx = 1.58113883 ≈ 1.6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Standard Deviation Where Does the Formula Come From? You can think of the standard deviation as a sort-of average distance that values in a set lie from the mean. While not the actual average, the standard deviation is usually very close to the average and, conceptually, that is a good way to think about standard deviation as we discuss how the formula is derived. To derive the formula for standard deviation, we will use a method similar to finding average distance, or deviation, from the mean. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Standard Deviation Where Does the Formula Come From? (cont.) First, we must know the actual deviation from the mean for every data value in the set. The deviation is simply the difference between each value and the mean. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Standard Deviation Where Does the Formula Come From? (cont.) Next, we need to find the sum of all the deviations. Here we run into a problem because the sum is zero. In fact, the sum of the deviations from the mean for any data set is always equal to zero because the positive deviations cancel out the negative deviations. To help eliminate the problem with the negative values canceling the positive values, we can make all the deviations positive by squaring each one. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Standard Deviation Where Does the Formula Come From? (cont.) Now, find the average by summing up all the squared deviations and dividing by the population size just as you would find a traditional mean. To find the standard deviation, we must take one additional step and “undo” the previous squaring by taking the square root. The completed population standard deviation formula is as follows. s HAWKES LEARNING SYSTEMS Students Matter. Success Counts. x m 2 i N Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.12: Using a TI-83/84 Plus Calculator to Find Standard Deviation Use a TI-83/84 Plus calculator to find the standard deviation of the data shown below, given the following conditions. 11, 18, 25, 51, 44, 29, 30, 17, 29, 47, 52, 60 a. Assume that the values represent the ages (in years) of patients randomly sampled from an urgent care clinic. b. Assume that the values represent the ages (in years) of all patients seen by Dr. Dabbs one afternoon. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.12: Using a TI-83/84 Plus Calculator to Find Standard Deviation (cont.) Solution • Press . • Choose 1:Edit. • Enter the data in L1. • Press again. • Choose CALC. • Choose option 1:1-Var Stats. • Press twice. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.12: Using a TI-83/84 Plus Calculator to Find Standard Deviation (cont.) A list of numerical summaries will be generated for the data. The beginning of the list is shown below. Use these values to find the answers for this example. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.12: Using a TI-83/84 Plus Calculator to Find Standard Deviation (cont.) a. We are told that the values in this case represent a random sample of patients. We will then need to use the sample standard deviation, denoted on the calculator by Sx. From the list, we see that s ≈ 15.9 years. b. In this scenario, the values represent all patients seen in one afternoon. The population standard deviation is most appropriate here, and it is denoted on the calculator by x. From the list, we see that ≈ 15.2 years. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.13: Interpreting Standard Deviations Mark is looking into investing a portion of his recent bonus into the stock market. While researching different companies, he discovers the following standard deviations of one year of daily stock closing prices. Profacto Corporation: Standard deviation of stock prices = $1.02 Yardsmoth Company: Standard deviation of stock prices = $9.67 What do these two standard deviations tell you about the stock prices of these companies? HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.13: Interpreting Standard Deviations (cont.) Solution A smaller standard deviation indicates that the data values are closer together, while a larger standard deviation indicates that the data values are more spread out. In this example, the standard deviation of stock prices for the Profacto Corporation is considerably smaller than that of the Yardsmoth Company. Hence, there is less variability in the daily closing prices of the Profacto stock than in the Yardsmoth stock prices. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.13: Interpreting Standard Deviations (cont.) If Mark wants a stable long-term investment, then Profacto appears to be the better choice. If, however, Mark is looking to make a quick profit and is willing to take the risk, then the Yardsmoth stock would seem to better suit his purposes. Note that looking at the standard deviations is just one component of evaluating market prices. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Standard Deviation Coefficient of Variation The coefficient of variation for a set of data is the ratio of the standard deviation to the mean as a percentage. For a population, it is given by s CV 100% m Where is the population standard deviation and m is the population mean. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Standard Deviation Coefficient of Variation (cont.) For a sample, it is given by s CV 100% x where s is the sample standard deviation and x ̄ is the sample mean. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.14: Calculating and Interpreting Coefficient of Variation Suppose that Graph A from Figure 3.1 represents average amounts of annual rainfall for a sample of farms in the United States and Graph B represents prices per 20 acres of farmland for the same farms. The mean and standard deviation of Data Set A are 26.08 inches and 7.55 inches, respectively, whereas the mean and standard deviation of Data Set B are $117,000 and $42,931, respectively. Which of the two graphs has the larger standard deviation relative to its mean? HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.14: Calculating and Interpreting Coefficient of Variation (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.14: Calculating and Interpreting Coefficient of Variation (cont.) Solution s Data Set A: CV 100% x 7.55 100% 28.9% 26.08 s Data Set B: CV 100% x 42,931 100% 36.7% 117,000 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.14: Calculating and Interpreting Coefficient of Variation (cont.) Now, it is clear to see that Data Set B has the larger standard deviation relative to its own mean. Thus, our comparison of CV shows that there is more variability in Data Set B than in Data Set A. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Variance Variance The variance is the square of the standard deviation. The population variance is given by s 2 x m 2 i N where xi is the ith value in the population, μ is the population mean, and N is the number of values in the population. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Variance Variance (cont.) The sample variance is given by s 2 x x 2 i n 1 where xi is the ith data value, x ̄ is the sample mean, and n is the number of data values in the sample. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.15: Calculating Variance Calculate the variance of the data shown below, given the following conditions. 3, 2, 5, 6, 4 a. Assume that the data represent the actual weight changes (in pounds) for a sample of fitness club members during the month of April. b. Assume that the data represent the actual weight changes (in pounds) of every member of a book club during the month of April. Use a TI-83/84 Plus calculator to perform the calculation. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.15: Calculating Variance (cont.) Solution a. As these data represent a sample, we need to calculate the sample variance. The formula that we need is given below. 2 xi x 2 s n 1 To start, we need the mean and sample size of the data set. We calculate that x ̄ = 4 and n = 5. Next, we need to subtract the mean from each number in the sample, and then square each of these differences. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.15: Calculating Variance (cont.) Let’s use a chart to keep everything organized. Deviations and Squared Deviations of the Data xi (xi – x̄) (xi – x̄)2 3 2 5 6 4 3 4 = 1 2 4 = 2 54=1 64=2 44=0 1 4 1 4 0 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.15: Calculating Variance (cont.) Next, find the sum of the squared deviations by adding up the values in the last column. x x i 2 1 4 1 4 0 10 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.15: Calculating Variance (cont.) Finally, substitute the appropriate values into the formula for sample variance as follows. s HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 2 x x 2 i n 1 10 51 2.5 Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.15: Calculating Variance (cont.) b. To calculate a variance on a TI-83/84 Plus calculator, you must actually calculate the standard deviation and then square that value to get the variance. The steps for calculating the standard deviation of a data set were presented in Example 3.12, and are repeated below. • Press . • Choose 1:Edit. • Enter the data in L1. • Press again. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.15: Calculating Variance (cont.) • Choose CALC. • Choose option 1:1-Var Stats. • Press twice. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.15: Calculating Variance (cont.) The calculator actually presents the values for both the population and sample standard deviations. Since the data set in this scenario represents a population (all members of the book club), we are looking for the population variance. Thus, we need to square the population standard deviation, given by = 1.414213562. 1.414214 2 2 2.0 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Empirical Rule Empirical Rule for Bell-Shaped Distributions • Approximately 68% of the data values lie within one standard deviation of the mean. • Approximately 95% of the data values lie within two standard deviations of the mean. • Approximately 99.7% of the data values lie within three standard deviations of the mean. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.16: Applying the Empirical Rule for Bell-Shaped Distributions The distribution of weights of newborn babies is bellshaped with a mean of 3000 grams and standard deviation of 500 grams. a. What percentage of newborn babies weigh between 2000 and 4000 grams? b. What percentage of newborn babies weigh less than 3500 grams? c. Calculate the range of birth weights that would contain the middle 68% of newborn babies’ weights. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.16: Applying the Empirical Rule for Bell-Shaped Distributions (cont.) Solution a. Since we know that the distribution of the data is bell-shaped, we can apply the Empirical Rule. We need to know how many standard deviations 2000 grams and 4000 grams are from the mean. By subtracting, we can find how far each of these figures is from the mean. Then, dividing by the standard deviation, we can convert these differences into numbers of standard deviations. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.16: Applying the Empirical Rule for Bell-Shaped Distributions (cont.) Here are the calculations. 2000 3000 1000 and 4000 3000 1000 1000 1000 2 2 500 500 Thus, these weights lie two standard deviations above and below the mean. According to the Empirical Rule, approximately 95% of values lie within two standard deviations of the mean. Therefore, we can say that approximately 95% of newborn babies weigh between 2000 and 4000 grams. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.16: Applying the Empirical Rule for Bell-Shaped Distributions (cont.) b. To begin, let’s find out how many standard deviations a weight of 3500 grams is away from the mean by performing the same calculation as before. 3500 3000 500 500 1 500 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.16: Applying the Empirical Rule for Bell-Shaped Distributions (cont.) Thus it is one standard deviation above the mean. The Empirical Rule says that 68% of data values lie within one standard deviation of the mean. Because of the symmetry of the distribution, half of this 68% is above the mean and half is below. Putting the upper 34% together with the 50% of data that is below the mean, we have that approximately 50% 34% 84% of newborn babies weigh less than 3500 grams. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.16: Applying the Empirical Rule for Bell-Shaped Distributions (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.16: Applying the Empirical Rule for Bell-Shaped Distributions (cont.) c. From the Empirical Rule, we know that 68% of the data values lie within one standard deviation of the mean for bell-shaped distributions. The standard deviation of this distribution is 500; thus, by adding 500 to and subtracting 500 from the mean of the distribution, we will get the range of birth weights that contain the middle 68% of newborn babies’ weights. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.16: Applying the Empirical Rule for Bell-Shaped Distributions (cont.) Upper end: 3000 + 500 = 3500 Lower end: 3000 500 = 2500 Thus, 68% of newborn babies weigh between 2500 and 3500 grams. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Chebyshev’s Theorem Chebyshev’s Theorem The proportion of data that lie within K standard 1 deviations of the mean is at least 1 2 , for K > 1. K When K = 2 and K = 3, Chebyshev’s Theorem says the following. 1 3 • K = 2: At least 1 2 75% of the data lie 2 4 within two standard deviations of the mean. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Chebyshev’s Theorem Chebyshev’s Theorem (cont.) 1 8 • K = 3: At least 1 2 88.9% of the data 3 9 lie within three standard deviations of the mean. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.17: Applying Chebyshev’s Theorem Suppose that in one small town, the average household income is $34,200, with a standard deviation of $2200. What percentage of households earn between $27,600 and $40,800? Solution Since we are not told in the problem whether the distribution of the data is bell-shaped, we cannot apply the Empirical Rule here. However, we can apply Chebyshev’s Theorem to find a minimum estimate. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.17: Applying Chebyshev’s Theorem (cont.) In order to do so, we need to know how many standard deviations $27,600 and $40,800 are from the mean. By subtracting, we can find how far each of these figures is from the mean. Then, dividing by the standard deviation, we can convert these differences into numbers of standard deviations. Here are the calculations. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.17: Applying Chebyshev’s Theorem (cont.) $27,600 $34,200 $6600 $6600 3 $2200 and $40,800 $34,200 $6600 $6600 3 $2200 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3.17: Applying Chebyshev’s Theorem (cont.) Thus these incomes lie three standard deviations above and below the mean. Chebyshev’s Theorem can then be applied for K = 3. Using the calculation previously shown in the box with the theorem, we can say that at least 88.9% of the household incomes lie within this range. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.
