8.1 Means, Medians, and Modes 8.1 OBJECTIVES 1. Calculate the mean 2. Interpret the mean 3. Find the median 4. Interpret the median 5. Find a mode A very useful concept is the average of a group of numbers. An average is a number that is typical of a larger group of numbers. In mathematics we have several different kinds of averages that we can use to represent a larger group of numbers. The first of these is the mean. Step by Step: Finding the Mean To find the mean for a group of numbers, follow these two steps: Step 1 Step 2 Add all of the numbers in the group. Divide that sum by the number of items in the group. Example 1 Finding the Mean Find the mean of the group of numbers 12, 19, 15, and 14. Step 1 Add all of the numbers. 12 19 15 14 60 Step 2 Divide that sum by the number of items. 60 4 15 There are four items in this group. The mean of this group of numbers is 15. CHECK YOURSELF 1 Find the mean of the group of numbers 17, 24, 19, and 20. Let s apply the mean to a word problem. Example 2 Finding the Mean The ticket prices (in dollars) for the nine concerts held at the Civic Arena this school year were 33, 31, 30, 59, 30, 35, 32, 36, 56 What was the mean price for these tickets? Step 1 Add all the numbers. 33 31 30 59 30 35 32 36 56 342 615
616 CHAPTER 8 DATA ANALYSIS AND STATISTICS NOTE Divide by 9 because there are 9 ticket prices. Step 2 Divide by 9. 342 9 38 The mean ticket price was $38. CHECK YOURSELF 2 The costs (in dollars) of the six textbooks that Aaron needs for the fall quarter are 75, 69, 57, 87, 76, 80 Find the mean cost of these books. Although the mean is probably the most common way to find an average for a group of numbers, it is not always the most representative. Another kind of average is called the median. Step by Step: Finding the Median The median is the number for which there are as many instances that are above that number as there are instances below it. To find the median, follow these steps: Step 1 Step 2 Step 3 Rewrite the numbers in order from smallest to largest. Count from both ends to find the number in the middle. If there are two numbers in the middle, add them together and find their mean. Example 3 Finding the Median Find the median for the following groups of numbers. (a) 35, 18, 27, 38, 19, 63, 22 Step 1 Rewrite the numbers in order from smallest to largest. 18, 19, 22, 27, 35, 38, 63 Step 2 Count from both ends to find the number in the middle. Counting from both ends, we find that 27 is the median. There are three numbers above 27 and three numbers below it. (b) 29, 88, 74, 81, 62, 37 Step 1 Rewrite the numbers in order from smallest to largest. 29, 37, 62, 74, 81, 88 Step 2 Count from both ends to find the number in the middle. Counting from both ends, we find that there are two numbers in the middle, 62 and 74. We go on to step 3.
MEANS, MEDIANS, AND MODES SECTION 8.1 617 Step 3 If there are two numbers in the middle, find their mean. (62 74) 2 136 2 68 CHECK YOURSELF 3 Find the median for each group of numbers: (a) 8, 6, 19, 4, 21, 5, 27 (b) 43, 29, 13, 37, 29, 53 There are times in which the median is a better representative of a group of numbers than the mean is. Example 4 illustrates such a case. Example 4 Comparing the Mean and the Median The following numbers represent the hourly wage of seven employees of a local chip manufacturing plant. 12, 11, 14, 16, 32, 13, 14 (a) Find the mean hourly wage. Step 1 Add all of the numbers in the group. 12 11 14 16 32 13 14 112 Step 2 Divide that sum by the number of items in the group. 112 7 16 The mean wage is $16 an hour. (b) Find the median wage for the seven workers. Step 1 Rewrite the numbers in order from smallest to largest. 11, 12, 13, 14, 14, 16, 32 Step 2 Count from both ends to find the number in the middle. The middle number is 14. There are three numbers above it and three numbers below it. The median salary is $14 per hour. Which salary do you think is more typical of the workers? Why? CHECK YOURSELF 4 The following are Jessica s phone bills for each month of 2000. 26, 67, 31, 24, 15, 17, 41, 27, 17, 22, 26, 47 (a) Find the mean amount of her phone bills. (b) Find the median amount of her phone bills.
618 CHAPTER 8 DATA ANALYSIS AND STATISTICS Another measure used as an average is called the mode. Definitions: Mode NOTE A set with two different modes is called bimodal. The mode of a set of data is the item or number that appears most frequently. Example 5 Finding a Mode Find the mode for the set of numbers given. 22, 24, 24, 24, 24, 27, 28, 32, 32 The mode, 24, is the number that appears most frequently. CHECK YOURSELF 5 Find the mode for the set of numbers given. 7, 7, 7, 9, 11, 13, 13, 15, 15, 15, 15, 21 One advantage of the mode is that it can be used with data that are not a set of numbers. Example 6 Finding a Mode NOTE If each eye color appeared three times, there would be no mode! Not every data set has a mode. Following are the eye colors from a class of 12 students. Which color is the mode? blue, brown, hazel, blue, brown, brown, brown, brown, blue, brown, hazel, green Because brown occurs most frequently, it is the mode. CHECK YOURSELF 6 The following types of computers were available in the lab. Which type was the mode? Apple, IBM, Compaq, Dell, Apple, IBM, Apple, Compaq, Dell, Apple, IBM, Apple, Dell, Apple, Compaq CHECK YOURSELF ANSWERS 1. 20 2. $74 3. (a) 8; (b) 33 4. (a) $30; (b) $26 5. 15 6. Apple
Name 8.1 Exercises Section Date Find the mean for each set of numbers. 1. 6, 9, 10, 8, 12 2. 13, 15, 17, 17, 18 ANSWERS 1. 3. 13, 15, 17, 19, 24, 26 4. 41, 43, 56, 67, 69, 72 2. 3. 5. 12, 14, 15, 16, 16, 16, 17, 22, 25, 27 6. 21, 25, 27, 32, 36, 37, 43, 44, 44, 51 4. 5. 7. 5, 8, 9, 11, 12 8. 7, 18, 11, 7, 12 6. 7. 9. 9, 8, 11, 14, 8 10. 21, 23, 25, 27, 22, 20 Find the median for each set of numbers. 11. 2, 3, 5, 6, 10 12. 12, 13, 15, 17, 18 8. 9. 10. 11. 12. 13. 23, 24, 27, 31, 36, 38, 41 14. 1, 4, 9, 16, 25, 36, 49 13. 14. 15. 46, 13, 47, 25, 68, 51, 71 16. 26, 71, 33, 69, 71, 25, 75 Find the mode for each set of numbers. 17. 17, 13, 16, 18, 17 18. 41, 43, 56, 67, 69, 72 15. 16. 17. 18. 19. 19. 21, 44, 25, 27, 32, 36, 37, 44 20. 9, 8, 10, 9, 9, 10, 8 20. 21. 12, 13, 7, 14, 4, 11, 9 22. 8, 2, 3, 3, 4, 9, 9, 3 Solve the following applications. 23. Temperature. High temperatures of 86, 91, 92, 103, and 98 were recorded for the first 5 days of July. What was the mean high temperature? 21. 22. 23. 619
ANSWERS 24. 25. 26. 27. 28. 29. 24. Travel. A salesperson drove 238, 159, 87, 163, and 198 miles (mi) on a 5-day trip. What was the mean number of miles driven per day? 25. Mileage rating. Highway mileage ratings for seven new diesel cars were 43, 29, 51, 36, 33, 42, and 32 miles per gallon (mi/gal). What was the mean rating? 26. Enrollments. The enrollments in the four elementary schools of a district are 278, 153, 215, and 198 students. What is the mean enrollment? 30. 27. Test scores. To get an A in history, you must have a mean of 90 on five tests. Your scores thus far are 83, 93, 88, and 91. How many points must you have on the final test to receive an A? (Hint: First find the total number of points you need to get an A.) 28. Test scores. To pass biology, you must have a mean of 70 on six quizzes. So far your scores have been 65, 78, 72, 66, and 71. How many points must you have on the final quiz to pass biology? 29. Test scores. Louis had scores of 87, 82, 93, 89, and 84 on five tests. Tamika had scores of 92, 83, 89, 94, and 87 on the same five tests. Who had the higher mean score? By how much? 30. Heating bills. The Wong family had heating bills of $105, $110, $90, and $67 in the first 4 months of 1999. The bills for the same months of 2000 were $110, $95, $75, and $76. In which year was the mean monthly bill higher? By how much? 620
ANSWERS Monthly energy use, in kilowatt-hours (kwh), by appliance type for four typical U.S. families is shown below. Wong McCarthy Abramowitz Gregg Family Family Family Family Electric range 97 115 80 96 Electric heat 1200 1086 1103 975 Water heater 407 386 368 423 Refrigerator 127 154 98 121 Lights 75 99 108 94 Air conditioner 123 117 96 120 Color TV 39 45 21 47 31. Heating. What is the mean number of kilowatt-hours used each month by the four families for heating their homes? 31. 32. 33. 34. 35. 36. 37. 38. 39. 32. Heating. What is the mean number of kilowatt-hours used each month by the four families for hot water? 33. Heating. What is the mean number of kilowatt-hours used per appliance by the McCarthy family? 34. Heating. What is the mean number of kilowatt-hours used per appliance by the Gregg family? Use your calculator for exercises 35 and 36. 35. Utility bills. Fred kept the following records of his utility bills for 12 months: $53, $51, $43, $37, $32, $29, $34, $41, $58, $55, $49, and $58. What was the mean monthly bill? 36. Test scores. The following scores were recorded on a 200-point final examination: 193, 185, 163, 186, 192, 135, 158, 174, 188, 172, 168, 183, 195, 165, 183. What was the mean of the scores? 37. The following are eye colors from a class of eight students. Which color is the mode? Hazel, green, brown, brown, blue, green, hazel, green 38. The weather in Philadelphia over the last seven days was as follows: Rain, sunny, cloudy, rain, sunny, rain, rain. What type of weather was the mode? 39. List the advantages and disadvantages of the mean, median, and mode. 621
ANSWERS 40. 41. 42. 43. 40. In a certain math class, you take four tests and the final, which counts as two tests. Your grade is the average of the six tests. At the end of the course, you compute both the mean and the median. (a) You want to convince the teacher to use the mean to compute your average. Write a note to your teacher explaining why this is a better choice. Choose numbers that make a convincing argument. (b) You want to convince the teacher to use the median to compute your average. Write a note to your teacher explaining why this is a better choice. Choose numbers that make a convincing argument. 41. Create a set of five numbers such that the mean is equal to the median. 42. Create a set of five numbers such that the mean is greater than the median. 43. Create a set of five numbers such that the mean is less than the median. Answers 1. 9 3. 19 5. 18 7. 9 9. 10 11. 5 13. 31 15. 47 17. 17 19. 44 21. No mode 23. 94 25. 38 mi/gal 27. 95 points 29. Louis s mean score was 87, Tamika s was 89. Tamika s average score was 2 points higher than Louis 31. 1091 kwh 33. 286 kwh 35. $45 37. Green 39. 41. Answer will vary 43. Answer will vary 622
Using Your Calculator to Find an Average Most electronic calculators have statistical functions that allow you to calculate means, medians, and other statistical values. Because these features vary so much from one calculator to another, you will need to consult your owner s manual to learn how to access these features. In this section, we will focus on using a calculator to compute the mean. Example 1 Calculating a Mean Find the mean for the set of numbers below. 45, 48, 53, 59, 67, 76 When entering these numbers in your calculator, keep in mind the order of operations. There are two different techniques you may use. Method 1 Add the numbers 45 48 53 59 67 76 348 then divide the sum by 6 (there are six numbers) 348 6 58 Method 2 Use parentheses to find the mean. (45 48 53 59 67 76) 6 58 CHECK YOURSELF 1 Find the mean for the set of numbers below. 132, 144, 156, 158, 279, 337 Unfortunately, not all calculations result in answers that are whole numbers. Example 2 Finding a Mean Find the mean of the set of numbers below. 13, 15, 16, 19, 26, 28, 38 Entering this as a single expression, we have (13 15 16 19 26 28 38) 7 22 Your calculator probably has a display something like 22.14285714. Remember that this is just another (although closer) approximation. 623
624 CHAPTER 8 DATA ANALYSIS AND STATISTICS We could approximate the answer as 22, as we ve indicated. We could also subtract 22 from the calculator display, leaving only the decimal approximation, approximately 0.142857142 If we multiply this by the divisor, 7, we will get the remainder, which is 1. The exact answer is 22 1 7 CHECK YOURSELF 2 Find an approximate and exact average for the set of numbers below. 17, 23, 33 CHECK YOURSELF ANSWERS 1. 201 2. Approximately 24, exactly 24 1 3
Calculator Exercises In exercises 1 to 10, find the mean for each set of numbers. 1. 48, 50, 51, 52, 49, 50 2. 20, 18, 17, 24, 22, 19 3. 108, 113, 109, 113, 110, 101, 112, 114 4. 211, 213, 215, 208, 209, 220, 215, 221 5. 1560, 1540, 1570, 1555, 1565, 1545, 1557 6. 346, 351, 353, 347, 341, 382, 373, 363 7. 2357, 2361, 2372, 2371, 2357, 2375, 2364, 2371 8. 16,430, 16,211, 16,149, 16,232, 16,317, 16,113 9. 24,637, 24,251, 24,454, 24,580, 24,324, 24,478 10. 311,431, 286,356, 356,090, 292,007, 301,857, 299,005 Name Section ANSWERS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Date In exercises 11 to 16, find the approximate and exact mean for each set of numbers. 11. 18, 21, 20, 22 12. 36, 41, 43, 39, 40, 37, 39 11. 12. 13. 125, 121, 129, 126, 128, 123 14. 356, 371, 366, 373, 359, 363 15. 1898, 1913, 1875, 1937 16. 15,865, 16,270, 16,090, 15,904 17. The revenue for the leading apparel companies in the United States in 1997 is given in the following table. 13. 14. 15. 16. Company Nike $9187 Vanity Fair $5222 Liz Claiborne $2413 Reebok $3637 Fruit of the Loom $2140 Nine West $1865 Kellwood $1521 Warmaio $1437 Jones Apparel $1387 Revenue (in millions) What is the mean revenue taken in by these companies? 17. 625
ANSWERS 18. 19. 20. 18. Unemployment in the United States (in thousands) Year Employed Unemployed 1989.................... 117,342 6,528 1990.................... 118,793 7,047 1991.................... 117,718 8,628 1992.................... 118,482 9,613 1993.................... 120,259 8,940 1994.................... 123,060 7,996 1995.................... 124,900 7,404 1996.................... 126,708 7,236 1997.................... 129,558 6,739 Find the mean number of employed and unemployed per year from 1989 to 1997. The work stoppages (strikes and lockouts) in the United States from 1990 to 1997 are given in the following table. Year No. of Stoppages Work Days Idle 1990 185 5926 1991 392 4584 1992 364 3989 1993 182 3981 1994 322 5020 1995 192 5771 1996 273 4889 1997 339 4493 19. Find the mean number of work stoppages per year from 1990 to 1997. 20. Find the mean number of work days idle from 1990 to 1997. Answers 1. 50 3. 110 5. 1556 7. 2366 9. 24,454 11. 20; 20 1 13. 125; 125 1 15. 1906; 1905 3 4 17. $3,201,000,000 19. 281 3 4 626