1. Explain what a measure of center is. (p. 108)

2. Which measure may be used as the balance point of the data set? Explain how this works. (p. 110)

3. Explain what we mean when we say that the mean is sensitive to the presence of extreme values. Explain whether the median is sensitive to extreme values. (pp. 111-112)

4. What are the three measures of center that we learned about in this section? (p. 108)

5. The number of observations in your sample data set (p. 109)

6. The number of observations in your population data set (p. 109)

7. Notation denoting “sum all the data” (p. 109)

8. Notation for what we get when we add up all the data values in the population, and divide by how many observations there are in the population (p. 109)

9. Notation for what we get when we add up all the data values in the sample, and divide by how many observations there are in the sample (p. 109)

10. The middle data value when the data are put in ascending order (p. 112)

11. The data value that occurs with the greatest frequency (p. 114)

12. The sample mean (p. 109)

1. Find the population size $N$.
2. Calculate the population mean $\mu$.

13. State exports to other countries are shown in the table for the population of all New England states, for the month of June 2014, expressed in billions of dollars.

14. The number of wins for each baseball team in the population of the American League West division for 2013 is shown in the table.

15. The table provides the motor vehicle theft rate for the population of the top 10 countries in the world for motor vehicle theft, for 2012. The theft rate equals the number of motor vehicles stolen in 2012 per 100,000 residents.

16. The National Center for Education Statistics sponsors the Trends in International Mathematics and Science Study (TIMSS). The table contains the mean science scores for the eighth-grade science test for the populations of all Asian-Pacific countries that took the exam.

17. The table contains the number of petit larceny cases for the population of all police precincts in South Manhattan in 2013.

18. The table contains the number of criminal trespass cases for the population of all police precincts in South Manhattan in 2013.

1. Find the sample size $n$.
2. Calculate the sample mean $\overline{x}$.

19. A sample of the state export data from Exercise 13 is provided in the table.

20. A sample from the baseball data in Exercise 14 is shown here.

21. A sample from the motor vehicle theft data in Exercise 15 is as follows.

22. A sample from the science score data in Exercise 16 is given here.

23. The following sample is taken from the petit larceny data in Exercise 17.

24. A sample taken from the criminal trespass data in Exercise 18 is as follows.

25. Data from Exercise 19. $\text{Extreme value}=10$

26. Data from Exercise 20. $\text{Extreme value}=20$

27. Data from Exercise 21. $\text{Extreme value}=1000$

28. Data from Exercise 22. $\text{Extreme value}=0$

29. Data from Exercise 23. $\text{Extreme value}=\mathrm{20,000}$

30. Data from Exercise 24. $\text{Extreme value}=1500$

1. Calculate the median of the data without the extreme value.
2. Find the median of the data including the extreme value. Compare your answers from (a) and (b). Note that the median did not change as much as the mean did in Exercises 25–30.

31. Data from Exercise 19. $\text{Extreme value}=10$

32. Data from Exercise 20. $\text{Extreme value}=20$

33. Data from Exercise 21. $\text{Extreme value}=1000$

34. Data from Exercise 22. $\text{Extreme value}=0$

35. Data from Exercise 23. $\text{Extreme value}=\mathrm{20,000}$

36. Data from Exercise 24. $\text{Extreme value}=1500$

37. The table contains the number of dangerous weapons cases for four police precincts in Manhattan.

38. The Recording Industry Association of America (RIAA) awards multi-platinum status for any musical recording that sells more than 2 million copies. The table contains a random sample of 10 of the musical artists with the most multi-platinum singles.

39. The table contains the unemployment rates in August 2014 for 10 countries.

40. The table contains the top 10 most downloaded free apps for the IOS platform, as reported by Apple.com, along with the app type, for June 2014. Find the mode of App Type.

41. The distribution in A

42. The distribution in B

43. The distribution in C

44. The distribution in D

45. NFL Football, Southern Style. The table contains the population of all the teams in the National Football Conference South Division, along with the number of wins in the 2013 season.

46. New England Electoral Votes. The table contains the population of all the New England states, along with their electoral votes.

47. NFL Football, Southern Style. Refer to the population data in Exercise 45. Suppose we take a sample from the population, and we get the Carolina Panthers and the Atlanta Falcons.

48. New England Electoral Votes. Refer to the population data in Exercise 46. Suppose we take a sample from the population, and get Massachusetts, Rhode Island, and Vermont.

49. Find the following measures of center for total sales.

50. Calculate the following measures of center for weeks.

51. Find the following measures of center for the darts stock returns.

52. Find the following measures of center for the DJIA.

53. Find the following measures of center for the women's ages.

54. Find the following measures of center for the women's heights.

55. Find the following measures of center for calories.

56. Find the following measures of center for the grams of saturated fat.

57. Find the sample size, $n$.

58. Calculate the sample mean trade balance, $\overline{x}$.

59. Find the median.

60. Find the modes.

61. Find the following for the number of cylinders:

62. Refer to your work in Exercise 61. Which measure of center do you think is most representative of the typical number of cylinders? Explain.

63. Find the following for the engine size:

64. Find the following for the city mpg:

65. Find the mode for country of manufacture.

66. Refer to the super speedways data. Find the following:

67. Refer to the short tracks data. Find the following:

68. Refer to the totals data. Find the following:

69. Find the mean, median, and mode for the price of these five books on the best-seller list. Suppose a salesperson claimed that the price of a typical book on the best-seller list is less than $14. How would you use these statistics to respond to this claim?

70. Linear transformations. Add $10 to the price of each book.

71. Linear transformations. Multiply the price of each book by 5.

72. Find the mode for the following variables:

73. Explain whether it makes sense to find the mean or median of the variable author.

74. Find the mode for violation type. Does this mean that most violations are of this type?

75. Calculate the mode for borough.

76. Does the idea of the mean or median of these two variables make any sense? Explain clearly why not.

77. What are the mean, median, and mode of the model year?

78. Calculate a new statistic “age of the car in 2015” as follows: take the model year and subtract it from 2015.

79. What will be the mean, median, and mode of the car ages in 2025?

80. Five friends have just had dinner at the local pizza joint. The total bill came to $30.60. What is the mean cost of each person's meal?

81. Lindsay just bought four shirts at the boutique in the mall, costing a total of $84.28. What was the mean cost of each shirt?

82. The mean cost of a sample of five items is $20. The cost of four of the items is as follows: $25, $15, $15, $20. What is the cost of the 5th item?

83. The mean size of four downloaded music files is 3 Mb (megabytes). The size of three of the files is as follows: 5 Mb, 2 Mb, 3 Mb. What is the length of the 4th music file?

84. The median number of students in a sample of seven statistics classes is 25. The ordered values are: 20, 22, 24, __, 27, 27, 28. What is the missing value?

85. The median number of academic credits taken in a sample of six students is 15. The ordered values are: 12, 12, 14, __, 17, 17. What is the missing value?

86. Find the following sample statistics.

87. What do these statistics tell us about the skewness of the distribution?

88. Linear Transformations. If we take each cereal rating and subtract 5 from it, how would that affect the mean, median, and mode? Would it affect each of the measures equally?

89. Linear Transformations. If we cut each of the cereals' ratings in half, how would that affect the mean, median, and mode? Would it affect each of the measures equally?

90. Examine Figure 10.

91. Find the following medians:

92. Find the following modes:

93. What if the fastest pulse rate for the men was a typo and should have been an unspecified lower pulse rate? Describe how and why this change would have affected the following, if at all. Would they increase, decrease, or remain unchanged? Or is there insufficient information to tell what would happen? Explain your answers.

94. Trimmed Mean. Because the mean is sensitive to extreme values, the trimmed mean was developed as another measure of center. To find the 10% trimmed mean for a data set, omit the largest 10% of the data values and the smallest 10% of the data values, and calculate the mean of the remaining values. Because the most extreme values are omitted, the trimmed mean is less sensitive, or more robust (resistant), than the mean as a measure of center. For the data in the table, calculate the following:

The data represent the number of business establishments in a sample of states.

95. Challenge Exercise. In general, would you expect the trimmed mean to be larger, smaller, or about the same as the mean for data sets with the following shapes?

96. Midrange. Another measure of center is the midrange.

Because the midrange is based on the maximum and minimum values in the data set, it is not a robust statistic, but it is sensitive to extreme values. Calculate the midrange for the following data:

97. Harmonic Mean. The harmonic mean is a measure of center most appropriately used when dealing with rates, such as miles per hour (mph). The harmonic mean is calculated as

where $n$ is the sample size and the $x$'s represent rates, such as the speeds in mph. Emily walked five miles today, but her walking speed slowed as she walked farther. Her walking speed was 5 mph for the first mile, 4 mph for the second mile, 3 mph for the third mile, 2 mph for the fourth mile, and 1 mph for the fifth mile. Calculate her harmonic mean walking speed over the entire five miles.

98. Challenge Exercise. The (arithmetic) mean for Emily's five-mile walk in Exercise 97 is 3 mph. Explain clearly why the value you calculated for the harmonic mean in Exercise 97 makes more sense than this arithmetic mean of 3 mph. (Hint: Consider time.)

99. Geometric Mean. The geometric mean is a measure of center used to calculate growth rates. Suppose that we have $n$ positive values; then the geometric mean is the $n$th root of the product of the $n$ values. Jamal has been saving money in an account that has had 4% growth, 6% growth, and 10% growth over the last three years. Calculate the average growth rate over these three years. (Hint: Find the geometric mean of 1.04, 1.06, and 1.10 and subtract 1.)

100. Construct your own data set with $n=10$, where the mean, the median, and the mode are all the same. Yes, just make up your own list of numbers, as long as the mean, median, and mode are all the same. Draw a dotplot. Comment on the skewness of the distribution.

101. Construct your own data set with $n=10$, where the mean is greater than the median, which is greater than the mode. Draw a dotplot. Comment on the skewness of the distribution.

102. Construct your own data set with $n=10$, where the mode is greater than the median, which is greater than the mean. Draw a dotplot. Comment on the skewness of the distribution.

103. Construct your own data set with $n=3$. Let the mean and median be equal. Now, alter the three data values so that the mean of the altered data set has increased, while the median of the altered data set has decreased.

104. Insert three points on the line by clicking just below it: two near the left side and one near the middle.

105. Explain why each of the measures behaves the way it does in the previous exercise.

106. Find the mean and median weekly sales.

107. Suppose we remove the biggest seller for the week, Minecraft for PS3, from the data. Given what you have learned about the sensitivity of the mean to the presence of extreme values, which measure do you expect will change the most, the mean or the median?

108. Recalculate the mean and the median of weekly sales, this time omitting Minecraft for PS3. Was your intuition in Exercise 107 confirmed?

109. Compute the mean and median total sales for the 30 games.

110. Identify the video game with the largest total sales. Omit this video game, and recompute the mean and median total sales. Which measure of center was more sensitive to the removal of the extreme value?

111. Find the mode for each of the following variables:

112. Compute the mean, median, and mode for the variable weeks on list.

113. What if we add a certain unknown amount $x$ to each value in the variable weeks on list? Describe what will happen to the following measures of center.