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Measures of Variation Lesson 10 Measures of Variation SURVEYS Jamie asked her classmates how many Glasses of Water Consumed Main Idea glasses of water they drink on a typical day. × × Find the measures of 1. What is the median of the data set? × × × × × × × × variation of a set of 2. Organize the data into two groups: the top × × × × × × data. half and the bottom half. How many data 012345 values are in each group? New Vocabulary 3. measures of variation What is the median of each group? range 4. Find the difference between the two numbers from Exercise 3. quartile lower quartile upper quartile Measures of variation are used to describe the distribution of the data. interquartile range The range is the difference between the greatest and least data values. outlier Quartiles are values that divide the data set into four equal parts. Math Online LQ median UQ glencoe.com 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5, 5, 5 ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ The median of the lower The median of the upper 6.SP.3, 6.SP.5c half of a set of data is half of a set of data is the lower quartile or the upper quartile or LQ; in this case, 1. UQ; in this case, 3.5. One fourth of the data lie below the lower quartile and one fourth of the data lie above the upper quartile. So, one half of the data lie between the lower quartile and upper quartile. This is called the interquartile range. Measures of Variation Upper and Lower Quartiles The upper and lower quartiles are the medians of the upper half and lower half of a set of data, respectively. 14, 18, 19, 20, 24, 29, 31 lower quartile median upper quartile Interquartile Range The range of the middle half of the data. It is the difference between the upper quartile and the lower quartile; in this case, 29 - 18, or 11. The interquartile range is 29 - 18, or 11. Range The difference between the greatest and least data values; in this case, 31 - 14, or 17. Lesson 10 Measures of Variation 39 Find Measures of Variation SPEED Find the measures Animal Speeds of variation for the data. Animal Speed (mph) Range 70 - 1 or 69 mph cheetah 70 lion 50 Quartiles cat 30 elephant 25 Order the numbers from mouse 8 least to greatest. spider 1 lower half median upper half ⎩ ⎨ ⎧ ⎩ ⎨ ⎧ 1 8 25 30 50 70 ⎩ ⎧ ⎨ 25 + 30 LQ _ = 27.5 UQ 2 Interquartile Range 50 - 8 or 42 UQ - LQ The range is 69, the median is 27.5, the lower quartile is 8, the Interquartile Range If the upper quartile is 50, and the interquartile range is 42. interquartile range is low, the middle data are Basketball Scores grouped closely together. a. SPORTS Determine the measures 64 61 67 59 60 of variation for the data in the table. 58 57 71 56 62 An outlier is a data value that is either much greater or much less than the median. If a data value is more than 1.5 times the value of the interquartile range beyond the quartiles, it is an outlier. Find Outliers ELECTIONS The ages of candidates in an election are 23, 48, 49, 55, 57, 63, and 72. Name any outliers in the data. Find the interquartile range. 63 - 48 = 15 Multiply the interquartile range by 1.5. 15 × 1.5 = 22.5 Subtract 22.5 from the lower quartile and add 22.5 to the upper quartile. 48 - 22.5 = 25.5 63 + 22.5 = 85.5 The limits for the outliers are between 25.5 and 85.5. The only age beyond this is 23. So, it is the only outlier. b. BRIDGES The lengths, in feet, of various bridges are 88, 251, 275, 354, and 1,121. Name any outliers in the data set. Analyze Data SCIENCE The table shows a set of scores on a Room Room science test in two different classrooms. Compare A B and contrast their measures of variation. 72 63 Find the measures of variation for both rooms. 100 93 67 79 Room A Room B 84 83 Range 100 - 65 = 35 98 - 63 = 35 65 98 Median 80 81 78 87 87 + 92 87 + 93 UQ _ = 89.5 _ = 90 92 73 2 2 87 81 67 + 72 65 + 73 LQ _ = 69.5 _ = 69 80 65 2 2 Interquartile 89.5 - 69.5 = 20 90 - 69 = 21 Range Both classrooms have a range of 35, but Room B has an interquartile range of 21 while Room A’s interquartile range is 20. There are slight differences in the medians as well as the upper and lower quartiles. Antelope, Augusta, Month MT ME c. WEATHER Temperatures for the January 21 28 first half of the year are given for Antelope, Montana, and February 30 32 Augusta, Maine. Compare and March 42 41 contrast the measures of April 58 53 variation of the two cities. May 70 66 June 79 75 Examples 1 and 2 1. WIND SPEED The average wind Wind Speed speeds for several cities in Speed Pennsylvania are given in the table. Pennsylvania City (mph) a. Find the range of the data. Allentown 8.9 b. Find the median and the lower and Erie 11.0 upper quartiles. Harrisburg 7. 5 Middletown 7. 7 c. Find the interquartile range. Philadelphia 9.5 d. Identify any outliers in the data. Pittsburgh 9.0 Williamsport 7. 6 Example 3 2. TREES The heights of several types of palm trees, in feet, are 40, 25, 15, 22, 50, and 30. The heights of several types of pine trees, in feet, are 60, 75, 45, 80, 75, and 70. Compare and contrast the measures of variation of both kinds of trees. Examples 1 and 2 3. GOLF COURSES The table shows the number Number of Golf Courses of golf courses in various states. California 1,117 a. Find the range of the data. Florida 1,465 b. Find the median and the lower and upper Georgia 513 quartiles. Iowa 437 Michigan 1,038 c. Find the interquartile range. New York 954 d. Name any outliers in the data. North Carolina 650 Ohio 893 South Carolina 456 Texas 1,018 4. INTERNET The table shows the countries Millions of Internet Users with the most Internet users. China 99.8 a. Find the range of the data. Germany 41.88 b. Find the median and the lower and upper India 36.97 quartiles. Japan 78.05 c. Find the interquartile range. South Korea 31.67 d. Name any outliers in the data. United Kingdom 33.11 United States 185.55 Example 3 5. EXERCISE The table shows the number Minutes of Exercise of minutes of exercise for each person. Week 1 Week 2 Compare and contrast the measures of Tanika 45 30 variation for both weeks. Tasha 40 55 Tyrone 45 35 Uniqua 55 60 Videl 60 45 Wesley 90 75 6. FOOTBALL The table Penalties By NFL Teams shows the top teams NFC AFC in the National Football Dallas Cowboys 104 New England Patriots 78 Conference (NFC) and Arizona Cardinals 137 Indianapolis Colts 67 the American Football Green Bay Packers 113 Jacksonville Jaguars 76 Conference (AFC). New Orleans Saints 68 San Diego Chargers 94 a. Which conference had New York Giants 77 Cleveland Browns 114 a greater range of Seattle Seahawks 59 Pittsburgh Steelers 80 penalties? Minnesota Vikings 86 Houston Texans 82 b. Find the measures of variation for each conference. c. Compare and contrast the measures of variation for each conference. For each data set, find the median, the lower and upper quartiles, and the interquartile range. 7. daily attendance at the water park: 346, 250, 433, 369, 422, 298 8. texts per day: 24, 53, 38, 12, 31, 19, 26 9. cost of admission: $13.95, $24.59, $19.99, $29.98, $23.95, $28.99 B 10. SCIENCE The table shows the number of Known Moons of Planets known moons for each planet in our solar Mercury 0 Jupiter 63 system. Use the measures of variation to Venus 0 Saturn 34 describe the data. Earth 1 Uranus 27 Mars 2 Neptune 13 11. EXERCISE Lucy and Dena are training for a bike race and recorded their mileage for a week. Find the measures of variation of each person’s mileage. Which measures of variation show the girls’ similarities in their training? the differences? Explain. Monday Tuesday Wednesday Thursday Friday Saturday Sunday Lucy 7 mi 3 mi 5 mi 8 mi 6 mi 10 mi 9 mi Dena 6 mi 4 mi 6 mi 8 mi 11 mi 9 mi 7 mi C 12. FIND THE ERROR Hiroshi was finding the measures of variation of the following set of data: 89, 93, 99, 110, 128, 135, 144, 152, and 159. Find his mistake and correct it. median = 128 lower quartile = 99 upper quartile = 144 interquartile range = 45 range = 70 13. OPEN ENDED Create a list of data with at least six numbers that has an interquartile range of 15 and two outliers. 14. WRITE MATH Explain why the median is not affected by very high or very low values in the data. Lesson 10 Measures of Variation 43 TestT Practice 15. The number of games won by 10 chess 17. The normal monthly rainfall in inches players is given. for a city are given in the table. 13, 15, 2, 7, 5, 9, 11, 10, 12, 11 Jan Feb Mar Apr May June 0.65 1.39 0.63 2.16 2.82 4.21 Which of the following statements is July Aug Sept Oct Nov Dec NOT supported by these data? 3.22 1.20 9.31 11.25 0.70 0.80 A.
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