Measures of Central Tendency and Range

8-2 Measures of Central Tendency and Range MAIN IDEA Describe a set of data using mean, median, mode, and range. The number of pennies in each cup represents Jack s scores for five math quizzes. New Vocabulary measures of central tendency mean median mode Math Online glencoe.com Extra Examples Personal Tutor Self-Check Quiz 9 7 8 6 10 Move the pennies among the cups so that each cup has the same number of pennies. 1. What was Jack s average score for the five quizzes? 8 points 2. If Jack scores 14 points on the next quiz, how many pennies would be in each cup? 9 A number used to describe the center of a set of data is a measure of central tendency. The most common of these measures is the mean. Mean Words Key Concept The mean of a set of data is the sum of the data divided by the number of items in the data set. The mean is also referred to as average. Examples data set: 1 cm, 1 cm, 5 cm, 2 cm, 2 cm, 4 cm, 2 cm, 5 cm mean: 1+1+5+2+2+4+2+5 or 2.75 cm 8 Find the Mean 1 QUIZ SCORES The table shows the quiz scores for 16 students. Find the mean. 47 + 40 +... + mean = 16 714 = _ or.625 16 sum of data number of data items Quiz Scores 47 40 43 45 49 41 49 43 41 49 50 41 a. MONEY Adam earned $14, $10, $12, $15, and $13 by doing chores around the house. What is the mean amount Adam earned doing these chores? $12.80 402 Chapter 8 Statistics: Analyzing Data

Two other common measures of central tendency are median and mode. Key Concept Vocabulary Link Everyday Use the middle paved or planted section of a highway, as in median strip. Math Use the middle number of the ordered data. Words Example Mode Words Example In a data set that has been ordered from least to greatest, the median is the middle number if there is an odd number of data items. If there is an even number of data items, the median is the mean of the two numbers closest to the middle. data set: 7 yd, 11 yd, 15 yd, 17 yd, 20 yd, 20 yd median: _ 15 + 17 or 16 yd 2 The median divides the data in half. The mode of a set of data is the number that occurs most often. If there are two or more numbers that occur most often, all of them are modes. data set: 50 mi, 45 mi, 45 mi, 52 mi, 49 mi, 56 mi, 56 mi modes: 45 mi and 56 mi Find the Mean,, and Mode 2 MOVIE RENTALS The number of DVDs rented during one week at Star Struck Movie Rental is shown in the table. What are the mean, median, and mode of the data? Star Struck Movie Rental Daily DVD Rentals S M T W TH F S 55 34 35 34 57 78 106 55 + 34 + 35 + 34 + 57 + 78 + 106 mean: = _ 399 or 57 7 7 median: 34, 34, 35, 55, 57, 78, 106 First, write the data in order. median mode: 34 It is the only value that occurs more than once. The mean is 57 DVDs, the median is 55 DVDs, and the mode is 34 DVDs. b. BICYCLES The sizes of the bicycles owned by the students in Ms. Garcia s class are listed in the table. What are the mean, median, and mode of the data? 24.25 in., 24 in., 24 in. Students Bicycle Sizes (in.) 20 24 20 26 24 24 24 26 24 29 26 24 c. FOOTBALL The points scored in each game by Darby Middle School s football team for 9 games are 21, 35, 14, 17, 28, 14, 7, 21, and 14. Find the mean, median, and mode. 19, 17, 14 Lesson 8-2 Measures of Central Tendency and Range 403

3 The maximum length in feet of several whales is listed below. Comparing Measures Another way to solve Example 3 is to find the measures before 98 is added to the data set. Then find the measures after 98 is added to the data set. Then compare. 46, 53, 33, 53, 79 If the maximum length of the Blue Whale, 98 feet, is added to this list, which of the following statements would be true? A The mode would decrease. C The mean would increase. B The median would decrease. D The mean would decrease. Read the Item You are asked to identify which statement would be true if the data value 98 was added to the data set. Solve the Item Use number sense to eliminate possibilities. The mode, 53, will remain unchanged since the new data value occurs only once. So, eliminate answer choice A. Since the new data value is greater than each value in the data set, the median will not decrease. So, eliminate answer choice B. The remaining two answer choices refer to the mean. Since 98 is greater than each value in the data set, the mean will increase, not decrease. So, the answer is C. d. If the maximum length of the Orca Whale, 30 feet, is added to the list in Example 3, which of the following statements would be true? F The mode would decrease. H The mean would increase. J G The median would increase. J The mean would decrease. In addition to the mean, median, and mode, you can also use the range to describe a set of data. Below are some guidelines for using these measures. Mean,, Mode, and Range Concept Summary Measure Mean Mode Range Most Useful When... data set has no outliers data set has outliers there are no big gaps in the middle of the data data set has many identical numbers describing the spread of the data 404 Chapter 8 Statistics: Analyzing Data

e. The mean is $21.50, the median is $18.50, the mode is $40, and the range is $28. The mean or median could be used to represent the data. 4 PLANTS The line plot shows the height of desert cacti. Would the mean, median, mode, or range best represent the heights? 1 + 2 + 2 +... + 30 mean: 14 7th term + 8th term median: 2 mode: 5 range: 30-1 or 29 Choose Mean,, Mode, or Range When there is an odd number of data, the median is the middle number of the ordered set. When there is an even number of data, the median is the mean of the two middle numbers. 2 8 10 12 14 or 8.8 = _ 5 + 5 or 5 2 The mean of 8.8 misrepresents the score. The median or mode represents the height of the cacti well. e. GAMES The table shows the cost of various board games. Would the mean, median, mode, or range best represent the costs? Explain. 0 4 6 16 18 20 22 24 26 28 30 Board Game Costs ($) 12 15 40 22 14 40 15 17 20 18 40 19 16 21 19 16 indicates multi-step problem Examples 1, 2 (pp. 402 403) 1. 52.3; 57; 59 2. 9.5; 9.5; none 3. 26.5, 25.5, 23 and 25 Example 3 (p. 404) Find the mean, median, and mode for each set of data. Round to the nearest tenth if necessary. 1. Miles traveled on the weekend: 29, 14, 80, 59, 78, 30, 59, 69, 55, 50 2. Team Number of Wins Eagles 10 Hawks 8 Zipps 9 Falcons 11 4. MULTIPLE CHOICE During the week, the daily low temperatures were 52 F, 45 F, 51 F, 45 F, and 48 F. If Saturday s low temperature of 51 F is added, which statement about the data set would be true? C A The mean would decrease. C The mode would increase. B The median would decrease. D The mode would decrease. 3. 22 23 24 25 26 27 28 29 30 31 32 Example 4 (p. 405) 5. SHOES The line plot shows the price of athletic shoes. Which measure best describes the data: mean, median, mode, or range? Explain. See margin. 45 50 55 60 65 70 75 80 85 90 95 100 Lesson 8-2 Measures of Central Tendency and Range 405

HOMEWORK HELP For Exercises 6 11 30 32 12 13 HELP See Examples 1, 2 3 4 Exercise Levels A: 6 13 B: 14 24 C: 25 29 11. 46, 45, Find the mean, median, and mode for each set of data. Round to the nearest tenth if necessary. 7. 87; 90; 80 and 93 8. 92.7; 91.5; 90 6. Number of dogs groomed each week: 65, 56, 57, 75, 76, 66, 64 65.6; 65; none 7. Daily number of boats in a harbor: 93, 84, 80, 91, 94, 90, 78, 93, 80 8. Scores earned on a math test: 95, 90, 92, 94, 91, 90, 98, 88, 89, 100 9. Prices of books: $10, $18, $11, $6, $6, $5, $10, $11, $46, $7, $6, $8 $12; $9; $6 10. Cost Number of Coats $75 8 $80 3 $85 6 $79.40; $80; $75 12. MUSIC The line plot shows the number of weeks that songs have been on the Top 20 Country Songs list. Would the mean, median, mode, or range best represent the data? Explain. See margin. 11. Country Songs Number of Weeks in Top 20 36 38 40 42 46 48 50 52 54 56 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 13. SPACE Twenty-seven countries have sent people into space. The table shows the number of individuals from each country. Which measure best describes the data: mean, median, mode, or range? Explain. See margin. Real-World Link The International Space Station measures 356 feet by 290 feet, and contains almost an acre of solar panels. Source: The World Almanac 17 19. See margin. People in Space 267 1 9 8 1 1 1 1 1 97 1 1 1 3 1 1 2 1 11 2 1 1 5 1 1 1 1 Source: The World Almanac Find the mean, median, and mode for each set of data. Round to the nearest tenth if necessary. 14. 6.4, 6.5, 7.2 15. $3.50, $3.50, $3.50 14. Weight in ounces of various insects: 6.1, 5.2, 7.2, 7.2, 3.6, 9.0, 6.5, 7.4, 5.4 15. Prices of magazines: $3.50, $3.75, $3.50, $4.00, $3.00, $3.50, $3.25 16. Daily low temperatures: -2 F, -8 F, -2 F, 0 F, -1 F, 1 F, -2 F, -1 F -1.9 F, -1.5 F, -2 F REASONING Determine whether each statement is always, sometimes, or never true about the data set {8, 12, 15, 23}. Explain your reasoning. 17. If a value greater than 23 is added, the mean will increase. 18. If a value less than or equal to 8 is added, the mean will decrease. 19. If a value between 8 and 23 is added, the mean will remain unchanged. 406 Chapter 8 Statistics: Analyzing Data

21. Sample answer: A length of 960 inches is much greater than the other pieces of data. So, if this piece of data is added, the mean will increase. 22. or mode; Mean, median, or mode; Sample answer: The length of the Brachiosaurus is an outlier, so the mean misrepresents the data if this value is included. DINOSAURS For Exercises 20 22, use the lengths of the dinosaurs shown below. Tyrannosaurus Rex Length: 480 inches Corythosaurus Length: 396 inches Parasaurolophus Length: 480 inches 20. What is the mean length of the dinosaurs? 452 inches 21. One of the largest dinosaurs ever is the Brachiosaurus. Its length was 960 inches. If this data value is added to the lengths of the dinosaurs above, how will it affect the mean? Explain your reasoning. 22. Which measure best describes the data if the length of the Brachiosaurus is included: mean, median, mode, or range? If the length of the Brachiosaurus is not included? Explain any similarities or differences. 23. SPORTS The table shows the points scored Hawks Lacrosse Team by a lacrosse team so far this season. The Points Scored team will play 14 games this season. How 11 15 12 10 10 10 13 many points need to be scored during the 14 13 13 10 15 12 last game so that the average number of points scored this season is 12? Explain. 10 points; See margin for explanation. EXTRA PRACTICE See pages 687, 711. 24. FIND THE DATA Refer to the Data File on pages 16 19. Choose some data and then describe it using the mean, median, mode, and range. See students work. H.O.T. Problems 25. Sample answer: 4, 5, 2, 2, 3, 3, 1, 0, 1, 2, 4, 68, 5; the mean, 7.7, is not the best representation since it is greater than all the data items except one. 25. OPEN ENDED Give an example of a set of data in which the mean is not the 26. range; mean, median, and mode are used to describe the center of data. Range describes the spread of the data. 28. CHALLENGE Without calculating, would the mean, median, or mode be most best representation of the data set. Explain why not. 26. Which One Doesn t Belong? Identify the term that does not have the same characteristic as the other three. Explain your reasoning. mean median range mode 27. REASONING Determine whether the median is sometimes, always, or never part of the data set. Explain your reasoning. See margin. affected by eliminating 1,000 from the data shown? Which would be the least affected? Explain your reasoning. See margin. 50, 100, 75, 60, 75, 1,000, 90, 100 29. WR ITING IN MATH According to the U.S. Census Bureau, the typical number of family members per household is 2.59. State whether this measure is a mean or mode. Explain how you know. See margin. Lesson 8-2 Measures of Central Tendency and Range 407

30. The table below shows the number of soup labels collected in one week by each homeroom in grade 7. Classroom Number of Soup Labels Mr. Martin 138 Ms. Davis 125 Mr. Cardona 89 Mrs. Turner 110 Mr. Wilhelm 130 Mrs. LaBash? Which number could be added to the set of data in order for the mode and median of the set to be equal? C A 89 C 125 B 110 D 130 31. An antique dealer purchased 5 antiques for a total of $850.00. He later bought another antique for $758.00. What is the mean cost of all the antiques? H F $151.60 H $268.00 G $170.00 J $321.60 32. Gina found the mean and median of the following list of numbers. 5, 7, 7 If the number 11 was added to this list, which of the following statements would be true? A A The mean would increase. B The mean would decrease. C The median would increase. D The median would decrease. 33. TEMPERATURE The table shows record high temperatures for Kentucky in July. Make a line plot of the data. (Lesson 8-1) See margin. Find the simple interest earned to the nearest cent for each principal, interest rate, and time. (Lesson 7-8) 34. $1,250, 3.5%, 2 years $87.50 35. $569, 5.5%, 4 months $10.43 July Temperatures 99 98 96 98 98 97 100 103 103 103 100 95 100 103 105 36. FOOD The United States produced almost 11 Uses of Apples in the United States billion pounds of apples in a recent year. Use the information in the graph to find how many pounds of apples were used to make juice and cider. (Lesson 7-1) about 2.53 billion 37. Name the property shown by the statement 4 6 = 6 4. (Lesson 1-8) Commutative () Juice and Cider 23% Canned Fruit 11% Other 7% Source: U.S. Apple Association Fresh Fruit 59% PREREQUISITE SKILL Name the place value of the underlined digit. (p. 669) 38. 581 hundreds 39. 6,295 ones 40. 4,369 thousands 41. 2.84 tenths 408 Chapter 8 Statistics: Analyzing Data