1. SECTION 3.2 MEASURES OF SPREAD Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

2. Section 3.2 - Objectives 1. Compute the range of a data set 2. Compute the variance of a population and a sample 3. Compute the standard deviation of a population and a sample 4. Approximate the standard deviation using grouped data 5. Use the Empirical Rule to summarize data that are unimodal and approximately symmetric 6. Use Chebyshev’s Inequality to describe a data set 7. Compute the coefficient of variation Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

3. Compute the range of a data set Objective 1 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4. The Range The range of a data set is a measure of spread. That is, it measure how spread out the data are. The range of a data set is the difference between the largest and the smallest value. Range = Largest Value – Smallest Value Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

5. Example 3.10 The following table presents the average monthly temperature, in degrees Fahrenheit, for the cities of San Francisco and St. Louis. Compute the range for each city. JanFebMarAprMayJunJulAugSepOctNovDec San Francisco515455565860 6163625852 St. Louis303544576675797870594535 Source: National Weather Service Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6. Solution The largest value for San Francisco is 63 and the smallest is 51. The range for San Francisco is 63 – 51 = 12. The largest value for St. Louis is 79 and the smallest is 30. The range for St. Louis is 79 – 30 = 49. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

7. The Range is Not Used in Practice Although the range is easy to compute, it is not often used in practice. The reason is that the range involves only two values from the data set; the largest and smallest. The measures of spread that are most often used are the variance and the standard deviation, which use every value in the data set. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

8. Compute the variance of a population and a sample Objective 2 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

9. Variance When a data set has a small amount of spread, like the San Francisco temperatures, most of the values will be close to the mean. When a data set has a larger amount of spread, more of the data values will be far from the mean. The variance is a measure of how far the values in a data set are from the mean, on the average. The variance is computed slightly differently for populations and samples. The population variance is presented first. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

10. Definition: Population Variance Let x 1, x 2, x 3, … x N denote the values in a population of size N. Let μ denote the population mean. The population variance, denoted by σ 2, is Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

11. Procedure for Computing the Population Variance Following is the procedure for computing the population variance of a data set: Step 1:Compute the population mean μ. Step 2:For each population value x i compute x i – μ. This is called the deviation for the value x i. Step 3:Square the deviations to obtain the quantity (x i – μ ) 2. Step 4:Sum the squared deviations to obtain the quantity Σ (x i – μ ) 2. Step 5:Divide the sum obtained in Step 4 by the population size N to obtain the population variance σ 2. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

12. Example 3.11 Compute the population variance for the San Francisco temperatures. JanFebMarAprMayJunJulAugSepOctNovDec San Francisco515455565860 6163625852 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

13. Step 2:For each population value x i compute x i – μ. These values are shown in the second row below. XiXi 515455565860 6163625852 x i – μ –6.5–3.5–2.5–1.50.52.5 3.55.54.50.5-5.5 Solution Step 1: Compute the population mean μ. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

14. Solution Step 3:Square the deviations to obtain the quantity (x i – μ ) 2. These values are shown in the third row. xixi 515455565860 6163625852 x i – μ –6.5–3.5–2.5–1.50.52.5 3.55.54.50.5-5.5 (x i – μ ) 2 42.2512.256.252.250.256.25 12.2530.2520.250.2530.25 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

15. Solution Step 4: Sum the squared deviations to obtain the quantity Σ (x i – μ ) 2. Step 5:Divide the sum obtained in Step 4 by the population size N to obtain the population variance σ 2. (x i – μ ) 2 42.2512.256.252.250.256.25 12.2530.2520.250.2530.25 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

16. Sample Variance When the data values come from a sample rather than a population, the variance is called the sample variance. The procedure for computing the sample variance is a bit different from the one used to compute a population variance. In the formula, the mean μ is replaced by the sample mean and the denominator is n – 1 instead of N. The sample variance is denoted by s 2. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

17. Why Divide by n –1? When computing the sample variance, we use the sample mean to compute the deviations. For the population variance we use the population mean for the deviations. It turns out that the deviations using the sample mean tend to be a bit smaller than the deviations using the population mean. If we were to divide by n when computing a sample variance, the value would tend to be a bit smaller than the population variance. It can be shown mathematically that the appropriate correction is to divide the sum of the squared deviations by n –1 rather than n. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

18. Example 3.12 A company that manufactures batteries is testing a new type of battery designed for laptop computers. They measure the lifetimes, in hours, of six batteries, and the results are presented in the following table. Find the sample variance of the lifetimes. Battery Lifetime346542 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

19. Solution Step 1: Compute the sample mean. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

20. Solution Solution (continued): Step 2:For each population value x i compute x i –. These values are shown in the second row below. Step 3:Square the deviations to obtain the quantity (x i – ) 2. These values are shown in the third row. xixi 346542 x i ––10210–2 xixi 346542 x i ––10210–2 (x i – ) 2 104104 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

21. Solution Solution (continued): Step 4: Sum the squared deviations to obtain the quantity Σ (x i – ) 2. Step 5:Divide the sum obtained in Step 4 by the n –1 to obtain the sample variance s 2. (x i – ) 2 104104 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

22. Compute the standard deviation of a population and a sample Objective 3 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

23. Standard Deviation Because the variance is computed using squared deviations, the units of the variance are the squared units of the data. For example, in Battery Lifetime example, the units of the data are hours, and the units of variance are squared hours. In most situations, it is better to use a measure of spread that has the same units as the data. We do this simply by taking the square root of the variance. This quantity is called the standard deviation. The standard deviation of a sample is denoted s, and the standard deviation of a population is denoted by σ. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

24. Example 3.12 Recall that in the Battery Lifetime example, the sample variance was computed as s 2 = 2. Find the sample standard deviation. Battery Lifetime346542 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

25. Solution The sample standard deviation, s, is the square root of the sample variance. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

26. Standard Deviation on the TI-84 PLUS The following steps will compute the standard deviation for both sample data and population data on the TI-84 PLUS Calculator: Enter the data into L1 in the data editor. Run the 1-Var Stats command (the same command used for means and medians), selecting L1 as the location of the data. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

27. Example – 3.14 (TI-84 PLUS Calculator) Using the TI-84 PLUS Calculator, find the sample variance. A company that manufactures batteries is testing a new type of battery designed for laptop computers. They measure the lifetimes, in hours, of six batteries, and the results are presented in the following table. Solution: Running the 1-Var Stats command, we find the sample standard deviation to be s = 1.414213562. We square this quantity to find the sample variance: s 2 = (1.414213562) 2 = 2. Battery Lifetime346542 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

28. Standard Deviation & Resistance Recall that a statistic is resistant if its value is not affected much by extreme data values. The standard deviation is not resistant. That is, the standard deviation is affected by extreme data values. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

29. Approximate the standard deviation using grouped data Objective 4 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

30. Approximating the Standard Deviation Sometimes we don’t have access to the raw data in a data set, but we are given a frequency distribution. In these cases we can approximate the standard deviation. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

31. Approximating the Standard Deviation Following is the procedure for approximating the standard deviation: Step 1:Compute the midpoint of each class and approximate the mean of the frequency distribution. Step 2:For each class, subtract mean from the class midpoint to obtain (Midpoint – Mean). Step 3:For each class, square the differences obtained in Step 2 to obtain (Midpoint – Mean) 2, and multiply by the frequency to obtain (Midpoint – Mean) 2 x (Frequency). Step 4:Add the products (Midpoint – Mean) 2 x (Frequency) over all classes. Step 5:To compute the population variance, divide the sum obtained in Step 4 by n. To compute the sample variance, divide the sum obtained in Step 4 by n –1. Step 6:Take the square root of the variance obtained in Step 5. The result is the standard deviation. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

32. Example 3.16 The following table presents the number of text messages sent via cell phone by a sample of 50 high school students. Approximate the sample standard deviation number of messages sent. Number of Text Messages SentFrequency 0 – 4910 50 – 995 100 – 14913 150 – 19911 200 – 2497 250 – 2994 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

33. Solution Step 1:Compute the midpoint of each class. Recall from Section 3.1 that the sample mean was computed as 137. Number of Text Messages SentClass Midpoint 0 – 4925 50 – 9975 100 – 149125 150 – 199175 200 – 249225 250 – 299275 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

34. Solution Step 2:For each class, subtract mean from the class midpoint to obtain (Midpoint – Mean). Number of Text Messages SentClass Midpoint (Midpoint – Mean) 0 – 4925–112 50 – 9975–62 100 – 149125–12 150 – 19917538 200 – 24922588 250 – 299275138 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

35. Solution Step 3: For each class, square the differences obtained in Step 2 to obtain (Midpoint – Mean) 2, and multiply by the frequency to obtain (Midpoint – Mean) 2 x (Frequency). Number of Text Messages SentFrequency(Midpoint – Mean) (Midpoint – Mean) 2 x (Frequency) 0 – 4910–112125,440 50 – 995–6219,220 100 – 14913–121,872 150 – 199113815,884 200 – 24978854,208 250 – 299413876,176 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

36. Solution Step 4: Add the products (Midpoint – Mean) 2 x (Frequency) over all classes. (Midpoint – Mean) 2 x (Frequency) 125,440 19,220 1,872 15,884 54,208 76,176 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

37. Solution Step 5:Since we are computing the sample variance, we divide the sum obtained in Step 4 by n –1. Step 6:Take the square root of the variance to obtain the standard deviation. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

38. Grouped Data on the TI-84 PLUS The same procedure used to compute the mean for grouped data in a frequency distribution may be used to compute the standard deviation. Enter the midpoint for each class into L1 and the corresponding frequencies in L2. Next, select the 1-Var stats command and enter L1 in the List field and L2 in the FreqList field, if using Stats Wizards. If you are not using Stats Wizards, you may rund1-Var Stats command followed by L1, comma, L2. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

39. Example 3.16 using TI-84 PLUS The output for Example 3.16 on the TI-84 PLUS Calculator is presented below. The value of s represents the approximate sample standard deviation. In this example s = 77.30142. Therefore the approximate standard deviation is 77.30142. Class MidpointFrequency 2510 755 12513 17511 2257 2754 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

40. Use the Empirical Rule to summarize data that are unimodal and approximately symmetric Objective 5 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

41. Bell-Shaped Histogram Many histograms have a single mode near the center of the data, and are approximately symmetric. Such histograms are often referred to as bell-shaped. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

42. The Empirical Rule When a data set has a bell-shaped histogram, it is often possible to use the standard deviation to provide an approximate description of the data using a rule known as The Empirical Rule. When a population has a histogram that is approximately bell-shaped, then:  Approximately 68% of the data will be within one standard deviation of the mean. In other words, approximately 68% of the data will be in the interval µ - σ to µ + σ.  Approximately 95% of the data will be within two standard deviations of the mean. In other words, approximately 95% of the data will be in the interval µ - 2 σ to µ + 2 σ.  All, or almost all, of the data will be within three standard deviations of the mean. In other words, all, or almost all, of the data will be in the interval µ - 3 σ to µ + 3 σ. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

43. The Empirical Rule Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

44. Example 3.17 The following table presents the U.S. Census Bureau projection for the percentage of the population aged 65 and over for each state and the District of Columbia. Compute the population mean and standard deviation and use The Empirical Rule to describe the data. Alabama14.1Rhode Island14.1Nevada12.3Kentucky13.1 Arkansas14.3Tennessee13.3New Mexico14.1Maryland12.2 Connecticut14.4Vermont14.3North Dakota15.3Minnesota12.4 Florida17.8West Virginia16Oregon13Montana15 Idaho12Alaska8.1South Carolina13.6New Hampshire12.6 Iowa14.9California11.5Texas10.5New York13.6 Louisiana12.6Delaware14.1Virginia12.4Ohio13.7 Massachusetts13.7Georgia10.2Wisconsin13.5Pennsylvania15.5 Mississippi12.8Illinois12.4Arizona13.9South Dakota14.6 Nebraska13.8Kansas13.4Colorado10.7Utah9 New Jersey13.7Maine15.6D.C.11.5Washington12.2 North Carolina12.4Michigan12.8Hawaii14.3Wyoming14 Oklahoma13.8Missouri13.9Indiana12.7 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

45. Solution We first note that the histogram is approximately bell-shaped. We may use the TI-84 PLUS Calculator – or other technology – to compute the population mean and population standard deviation. Mean: µ = 13.24901961 Standard Deviation: σ = 1.682711694 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

46. Solution Next, we compute the quantities: Approximately 68% of the data values are between these. Approximately 95% of the data values are between these. Almost all of the data values are between these. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

47. Solution 8.20 9.88 11.57 14.93 16.61 18.30 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

48. Use Chebyshev’s Inequality to describe a data set Objective 6 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

49. Any Data Set When a distribution is bell-shaped, we use The Empirical Rule to approximate the proportion of data within one or two standard deviations. Another rule called Chebyshev’s Inequality holds for any data set. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

50. Chebyshev’s Inequality In any data set, the proportion of the data that is within K standard deviations of the mean is at least 1– 1/K 2. Specifically, by setting K = 2 or K = 3, we obtain the following results.  At least 3/4 (75%) of the data are within two standard deviations of the mean.  At least 8/9 (89%) of the data are within three standard deviations of the mean. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

51. Example 3.20 As part of a public health study, systolic blood pressure was measured for a large group of people. The mean was 120 and the standard deviation was 10. What information does Chebyshev’s Inequality provide about these data? Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

52. Solution We compute the following: We conclude:  At least 3/4 (75%) of the people had systolic blood pressures between 100 and 140.  At least 8/9 (89%) of the people had systolic blood pressures between 90 and 150. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

53. Compute the coefficient of variation Objective 7 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

54. Coefficient of Variation The coefficient of variation (CV for short) tells how large the standard deviation is relative to the mean. It can be used to compare the spreads of data sets whose values have different units. The coefficient of variation is found by dividing the standard deviation by the mean. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

55. Example 3.21 National Weather service records show that over a thirty-year period, the annual precipitation in Atlanta, Georgia had a mean of 49.8 inches with a standard deviation of 7.6 inches, and the annual temperature had a mean of 62.2 degrees Fahrenheit with a standard deviation of 1.3 degrees. Compute the coefficient of variation for precipitation and for temperature. Which has greater spread relative to its mean? Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

56. Solution We compute the following: The CV for precipitation is larger than the CV for temperature. Therefore precipitation has a greater spread relative to its mean. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

57. Do You Know… How to compute the range of a data set? How to compute the variance of a population and a sample and the appropriate notation? How to compute the standard deviation of a population and a sample and the appropriate notation? How to approximate the standard deviation using grouped data? How to use the Empirical Rule to summarize data? How to use Chebyshev’s Inequality to describe a data set? How to compute the coefficient of variation? Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.