Solution: Finding Standard Deviation from a Frequency Distribution. In Exercises 3740

Comparing Variation Which do you think has less variation: the IQ scores of students in your statistics class or the IQ scores of a simple random sample taken from the general population? Why?

Correct Statements? Which of the following statements are true? a. If each of 25 sample values is equal to 20 min, the standard deviation of the sample is 0 min. b. For any set of sample values, the standard deviation can never be a negative value. c. If the standard deviation of a sample is 3 kg, then the variance is 9 kg. d. If the variance of a sample is 16 sec 2 , then the standard deviation is 4 sec. e. If the standard deviation of a sample is 25 cm, then the variance is 5 cm 2 .

Variation and Variance In statistics, how do the terms variation and variance differ?

Symbols Identify the symbols used for each of the following: (a) sample standard deviation (b) population standard deviation (c) sample variance (d) population variance.

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Top 10 Celebrity Incomes Listed below are the earnings (in millions of dollars) of the celebrities with the 10 highest incomes in a recent year. The celebrities in order are Steven Spielberg, Howard Stern, George Lucas, Oprah Winfrey, Jerry Seinfeld, Tiger Woods, Dan Brown, Jerry Bruckheimer, J. K. Rowling, and Tom Cruise. Can this Top 10 list be used to learn anything about the standard deviation of the annual earnings of all celebrities? 332 302 235 225 100 90 88 84 75 67

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Top 10 Most Expensive Colleges Listed below are the annual tuition amounts of the 10 most expensive colleges in the United States for a recent year. The colleges listed in order are Sarah Lawrence, NYU, George Washington, Bates, Skidmore, Johns Hopkins, Georgetown, Connecticut College, Harvey Mudd, and Vassar. Can this Top 10 list tell us anything about the standard deviation of the population of all U.S. college tuitions? $54,410 $51,991 $51,730 $51,300 $51,196 $51,190 $51,122 $51,115 $51,037 $50,875

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Car Crash Test Measurements Listed below are head injury measurements from small cars that were tested in crashes. The measurements are in hic, which is a measurement of a standard head injury criterion. The data are from Data Set 13 in Appendix B, which is based on data from the National Highway Traffic Safety Administration. The listed values correspond to these cars: Chevrolet Aveo, Honda Civic, Volvo S40, VW Jetta, Hyundai Elantra, Kia Rio, and Kia Spectra. Given that all of the cars are small, does it appear that the head injury measurements are about the same for all small cars? 371 356 393 544 326 520 501

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Tests of Child Booster Seats The National Highway Traffic Safety Administration conducted crash tests of child booster seats for cars. Listed below are results from those tests, with the measurements given in hic, which is a measurement of a standard head injury criterion. According to the safety requirement, the hic measurement should be less than 1000 hic. Do the different child booster seats have much variation among their crash test measurements? 774 649 1210 546 431 612

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Harry Potter Listed below are the gross amounts (in millions of dollars) earned in box office receipts for the movie Harry Potter and the HalfBlood Prince. The movie opened on a Wednesday, and the amounts are listed in order for the first 14 days of the movie's release. If you invested in this movie, what characteristic of the data set would you care about most, and is it a measure of center or variation? 58 22 27 29 21 10 10 8 7 9 11 9 4 4

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Florida Manatee Deaths Listed below are the numbers of manatee deaths caused each year by collisions with watercraft. Manatees, also called sea cows, are large mammals that live underwater, often in or near waterways. The data are listed in order for each year of the past decade. The data are from the Florida Fish and Wildlife Conservation Commission. What important feature of the data is not revealed through the different measures of variation? 78 81 95 73 69 79 92 73 90 97

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Ghost Prices Listed below are the prices listed for Norton Ghost 14.0 software from these vendors: Newegg, Dell, Buycheapsoftware.com, PC Connection, Walmart, and Overstock.com. When trying to find the best deal, how helpful are the measures of variation? $55.99 $69.99 $48.95 $48.92 $71.77 $59.68

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.CEO Compensation Listed below are the recent annual compensation amounts for these chief executive officers: Mulally (Ford), Jobs (Apple), Kent (CocaCola), Otellini (Intel), and McNerney (Boeing). The data are from the Associated Press. What is particularly notable about these compensation amounts? $17,688,241 $1 $19,628,585 $12,407,800 $14,765,410

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Lead in Medicine Listed below are the lead concentrations (in g / g ) measured in different Ayurveda medicines. Ayurveda is a traditional medical system commonly used in India. The lead concentrations listed here are from medicines manufactured in the United States. The data are based on the article Lead, Mercury, and Arsenic in US and Indian Manufactured Ayurvedic Medicines Sold via the Internet, by Saper et al., Journal of the American Medical Association, Vol. 300, No. 8. What would be the values of the measures of variation if the medicines contained no lead? 3.0 6.5 6.0 5.5 20.5 7.5 12.0 20.5 11.5 17.5

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Mercury in Sushi Listed below are the amounts of mercury (in parts per million, or ppm) found in tuna sushi sampled at different stores in New York City. The study was sponsored by the New York Times, and the stores (in order) are D'Agostino, Eli's Manhattan, Fairway, Food Emporium, Gourmet Garage, Grace's Marketplace, and Whole Foods. What would be the values of the measures of variation if the tuna sushi contained no mercury? 0.56 0.75 0.10 0.95 1.25 0.54 0.88

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Years to Earn Bachelor's Degree Listed below are the lengths of time (in years) it took for a random sample of college students to earn bachelor's degrees (based on data from the National Center for Education Statistics). Based on these results, is it unusual for someone to earn a bachelor's degree in 12 years? 4 4 4 4 4 4 4.5 4.5 4.5 4.5 4.5 4.5 6 6 8 9 9 13 13 15

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Cell Phone Radiation Listed below are the measured radiation emissions (in W/kg) corresponding to these cell phones: Samsung SGHtss9, Blackberry Storm, Blackberry Curve, Motorola Moto, TMobile Sidekick, Sanyo Katana Eclipse, Palm Pre, Sony Ericsson, Nokia 6085, Apple iPhone 3G S, and Kyocero Neo E1100. The data are from the Environmental Working Group. If one of each model of cell phone is measured for radiation and the results are used to find the standard deviation, is that standard deviation equal to the standard deviation of the population of all cell phones that are in use? Why or why not? 0.38 0.55 1.54 1.55 0.50 0.60 0.92 0.96 1.00 0.86 1.46

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.JFK to LAX Flight Delays Listed below are the arrival delay times (in minutes) of randomly selected American Airline flights from New York's JFK airport to Los Angeles (LAX). Negative values correspond to flights that arrived early before the scheduled arrival time, and positive values represent lengths of delays. (The data are from the Bureau of Transportation, and more data are listed in Data Set 15 in Appendix B.) Some of the sample values are negative, but can the standard deviation ever be negative? 15 18 32 21 9 32 11 2

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Freshman 15 According to the freshman 15 legend, college freshmen gain 15 pounds (or 6.8 kilograms) during their freshman year. Listed below are the amounts of weight change (in kilograms) for a simple random sample of freshmen included in a study (Changes in Body Weight and Fat Mass of Men and Women in the First Year of College: A Study of the 'Freshman 15' by Hoffman, Policastro, Quick, and Lee, Journal of American College Health, Vol. 55, No. 1). Positive values correspond to students who gained weight and negative values correspond to students who lost weight. Is a weight gain of 15 pounds (or 6.8 kg) unusual? Why or why not? If 15 pounds (or 6.8 kg) is not unusual, does that support the legend of the freshman 15? 11 3 0 2 3 2 2 5 2 7 2 4 1 8 1 0 5 2

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Saints in Super Bowl Listed below are the numbers on the jerseys of the starting lineup for the New Orleans Saints when they recently won their first Super Bowl football game. What do the measures of variation tell us about the team? Does it make sense to compute the measures of variation for these data? 9 23 25 88 12 19 74 77 76 73 78

In Exercises 520, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as minutes) in your results. (The same data were used in Section 32 where we found measures of center. Here we find measures of variation.) Then answer the given questions.Phenotypes of Peas Biologists conducted experiments to determine whether a deficiency of carbon dioxide in the soil affects the phenotypes of peas. Listed below are the phenotype codes, where 1 = smoothyellow , 2 = smoothgreen m 3 = wrinkledyellow, and 4 = wrinkledgreen . Can the measures of center be obtained for these values? Do the results make sense? 2 1 1 1 1 1 1 4 1 2 2 1 2 3 3 2 3 1 3 1 3 1 3 2 2

In Exercises 2124, find the coefficient of variation for each of the two samples then compare the variation. (The same data were used in Section 32.). Speeding and Race Listed below are speeds (in mi/h) of cars on the New Jersey Turnpike by race of the driver. All cars are going in the same direction, and all of the cars are from New Jersey. The data are from Statlib and the authors are Joseph Kadane and John Lamberth. White drivers: 74 77 69 71 77 69 72 75 74 72 African American drivers: 79 70 71 76 76 74 71 75 74 74

In Exercises 2124, find the coefficient of variation for each of the two samples then compare the variation. (The same data were used in Section 32.)Parking Meter Theft Listed below are amounts (in millions of dollars) collected from parking meters by Brinks and others in New York City during similar time periods. A larger data set was used to convict five Brinks employees of grand larceny. The data were provided by the attorney for New York City, and they are listed on the DASL web site. Do the limited data listed here show evidence of stealing by Brinks employees? Collection contractor was Brinks: 1.3 1.5 1.3 1.5 1.4 1.7 1.8 1.7 1.7 1.6 Collection contractor was not Brinks: 2.2 1.9 1.5 1.6 1.5 1.7 1.9 1.6 1.6 1.8

In Exercises 2124, find the coefficient of variation for each of the two samples then compare the variation. (The same data were used in Section 32.)Political Contributions Listed below are contributions (in dollars) made to the two presidential candidates in the most recent election. All contributions are from the same Zip code as the author, and the data are from the Huffington Post. Obama: $275 $452 $300 $1000 $1000 $500 $100 $1061 $1200 $235 $875 $2000 $350 $210 $250 McCain: $50 $75 $240 $302 $250 $700 $350 $500 $1250 $1500 $500 $500 $40 $221 $400

In Exercises 2124, find the coefficient of variation for each of the two samples then compare the variation. (The same data were used in Section 32.)Customer Waiting Times Waiting times (in minutes) of customers at the Jefferson Valley Bank (where all customers enter a single waiting line) and the Bank of Providence (where customers wait in individual lines at three different teller windows) are listed below. Jefferson Valley (single line): 6.5 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7 7.7 Providence (individual lines): 4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0

Large Data Sets from Appendix B. In Exercises 2528, refer to the indicated data set in Appendix B. Use computer software or a calculator to find the range, variance, and standard deviation. Express answers using appropriate units, such as minutes.Earthquakes Use the magnitudes (Richter scale) of the earthquakes listed in Data Set 16 in Appendix B.

Large Data Sets from Appendix B. In Exercises 2528, refer to the indicated data set in Appendix B. Use computer software or a calculator to find the range, variance, and standard deviation. Express answers using appropriate units, such as minutes.Flight Data Refer to Data Set 15 in Appendix B and use the times required to taxi out for takeoff.

Large Data Sets from Appendix B. In Exercises 2528, refer to the indicated data set in Appendix B. Use computer software or a calculator to find the range, variance, and standard deviation. Express answers using appropriate units, such as minutes.Presidential Longevity Refer to Data Set 12 in Appendix B and use the numbers of years that U.S. presidents have lived after their first inauguration.

Large Data Sets from Appendix B. In Exercises 2528, refer to the indicated data set in Appendix B. Use computer software or a calculator to find the range, variance, and standard deviation. Express answers using appropriate units, such as minutes. IQ Scores Refer to Data Set 6 in Appendix B and use the listed IQ scores.

Estimating Standard Deviation with the Range Rule of Thumb. In Exercises 2932, refer to the data in the indicated exercise. After finding the range of the data, use the range rule of thumb to estimate the value of the standard deviation. Compare the result to the standard deviation computed with all of the data.Exercise 25

Estimating Standard Deviation with the Range Rule of Thumb. In Exercises 2932, refer to the data in the indicated exercise. After finding the range of the data, use the range rule of thumb to estimate the value of the standard deviation. Compare the result to the standard deviation computed with all of the data.Exercise 26

Estimating Standard Deviation with the Range Rule of Thumb. In Exercises 2932, refer to the data in the indicated exercise. After finding the range of the data, use the range rule of thumb to estimate the value of the standard deviation. Compare the result to the standard deviation computed with all of the data.Exercise 27

Estimating Standard Deviation with the Range Rule of Thumb. In Exercises 2932, refer to the data in the indicated exercise. After finding the range of the data, use the range rule of thumb to estimate the value of the standard deviation. Compare the result to the standard deviation computed with all of the data.Exercise 28

Identifying Unusual Values with the Range Rule of Thumb. In Exercises 3336, use the range rule of thumb to determine whether a value is unusual. Pulse Rates of Females Based on Data Set 1 in Appendix B, females have pulse rates with a mean of 77.5 beats per minute and a standard deviation of 11.6 beats per minute. Is it unusual for a female to have a pulse rate of 99 beats per minute? (All of these pulse rates are measured at rest.) Explain.

Identifying Unusual Values with the Range Rule of Thumb. In Exercises 3336, use the range rule of thumb to determine whether a value is unusual.Pulse Rates of Males Based on Data Set 1 in Appendix B, males have pulse rates with a mean of 67.3 beats per minute and a standard deviation of 10.3 beats per minute. Is it unusual for a male to have a pulse rate of 45 beats per minute? (All of these pulse rates are measured at rest.) Explain.

Identifying Unusual Values with the Range Rule of Thumb. In Exercises 3336, use the range rule of thumb to determine whether a value is unusual.Volumes of Coke Based on Data Set 19 in Appendix B, cans of regular Coke have volumes with a mean of 12.19 oz and a standard deviation of 0.11 oz. Is it unusual for a can to contain 11.9 oz of Coke? Explain.

Identifying Unusual Values with the Range Rule of Thumb. In Exercises 3336, use the range rule of thumb to determine whether a value is unusual.Weights of Pepsi Based on Data Set 19 in Appendix B, cans of regular Pepsi have weights with a mean of 0.82410 lb and a standard deviation of 0.00570 lb. Is it unusual for a can to contain 0.8133 lb of Pepsi? Explain.

Finding Standard Deviation from a Frequency Distribution. In Exercises 3740, find the standard deviation of sample data summarized in a frequency distribution table by using the formula below, where x represents the class midpoint, f represents the class frequency, and n represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 34 with the original list of data values: (Exercise 37) 11.1 years (Exercise 38) 9.0 years (Exercise 39) 13.4 (Exercise 40) 9.7 years.Age of Best Actress When Oscar Was Won Frequency 2029 27 3039 34 4049 13 5059 2 6069 4 7079 1 8089 1

Finding Standard Deviation from a Frequency Distribution. In Exercises 3740, find the standard deviation of sample data summarized in a frequency distribution table by using the formula below, where x represents the class midpoint, f represents the class frequency, and n represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 34 with the original list of data values: (Exercise 37) 11.1 years (Exercise 38) 9.0 years (Exercise 39) 13.4 (Exercise 40) 9.7 years.Age of Best Actor When Oscar Was Won Frequency 2029 1 3039 26 4049 35 5059 13 6069 6 7079 1

Finding Standard Deviation from a Frequency Distribution. In Exercises 3740, find the standard deviation of sample data summarized in a frequency distribution table by using the formula below, where x represents the class midpoint, f represents the class frequency, and n represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 34 with the original list of data values: (Exercise 37) 11.1 years (Exercise 38) 9.0 years (Exercise 39) 13.4 (Exercise 40) 9.7 years.Verbal IQ Score of Subject Exposed to Lead Frequency 5059 4 6069 10 7079 25 8089 43 9099 26 100109 8 110119 3 120129 2

Finding Standard Deviation from a Frequency Distribution. In Exercises 3740, find the standard deviation of sample data summarized in a frequency distribution table by using the formula below, where x represents the class midpoint, f represents the class frequency, and n represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 34 with the original list of data values: (Exercise 37) 11.1 years (Exercise 38) 9.0 years (Exercise 39) 13.4 (Exercise 40) 9.7 years.Years President Lived After First Inauguration Frequency 04 8 59 2 1014 5 1519 7 2024 4 2529 6 3034 0 3539 1

The Empirical Rule Based on Data Set 1 in Appendix B, blood platelet counts of women have a bellshaped distribution with a mean of 280 and a standard deviation of 65. ( All units are 1000 cells / L . ) Using the empirical rule, what is the approximate percentage of women with platelet counts a. within 2 standard deviations of the mean, or between 150 and 410? b. between 215 and 345?

The Empirical Rule Based on Data Set 3 in Appendix B, body temperatures of healthy adults have a bellshaped distribution with a mean of 98.20 F and a standard deviation of 0.62 F . Using the empirical rule, what is the approximate percentage of healthy adults with body temperatures a. within 1 standard deviation of the mean, or between 97.58 F and 98.82 F ? b. between 96.34 F and 100.06 F ?

Chebyshev's Theorem Based on Data Set 1 in Appendix B, blood platelet counts of women have a bellshaped distribution with a mean of 280 and a standard deviation of 65. ( All units are 1000 cells / L . ) Using Chebyshev's theorem, what do we know about the percentage of women with platelet counts that are within 2 standard deviations of the mean? What are the minimum and maximum platelet counts that are within 2 standard deviations of the mean?

Chebyshev's Theorem Based on Data Set 3 in Appendix B, body temperatures of healthy adults have a bellshaped distribution with a mean of 98.20 F and a standard deviation of 0.62 F . Using Chebyshev's theorem, what do we know about the percentage of healthy adults with body temperatures that are within 3 standard deviations of the mean? What are the minimum and maximum body temperatures that are within 3 standard deviations of the mean?

Why Divide by n 1 ? Let a population consist of the values 2 min, 3 min, 8 min. (These are departure delay times taken from American Airlines flights from New York's JFK airport to Los Angeles. See Data Set 15 in Appendix B.) Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.) a. Find the variance 2 of the population {2 min, 3 min, 8 min}. b. After listing the nine different possible samples of two values selected with replacement, find the sample variance s 2 (which includes division by n 1 ) for each of them then find the mean of the nine sample variances s 2 .c. For each of the nine different possible samples of two values selected with replacement, find the variance by treating each sample as if it is a population (using the formula for population variance, which includes division by n), then find the mean of those nine population variances. d. Which approach results in values that are better estimates of 2 : part (b) or part (c)? Why? When computing variances of samples, should you use division by n or n 1 ? e. The preceding parts show that s 2 is an unbiased estimator of 2 . Is s an unbiased estimator of ? Explain.

Mean Absolute Deviation Use the same population of {2 min, 3 min, 8 min} from Exercise 45. Show that when samples of size 2 are randomly selected with replacement, the samples have mean absolute deviations that do not center about the value of the mean absolute deviation of the population. What does this indicate about a sample mean absolute deviation being used as an estimator of the mean absolute deviation of a population?