Measures of variation discuss 2.1
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This document provides information about measures of variation, including range, median, quartiles, and interquartile range. It defines these terms and shows examples of calculating them for various data sets. The examples include finding the measures of variation for data on water consumption, speeds, golf courses, internet users, exercise times, moons of planets, and more. Students are provided practice problems to calculate these measures of variation for additional data sets.
Measures of variation discuss 2.1
- 1. Lesson 10 glencoe.com Math Online 6.SP.3, 6.SP.5c Lesson 10 Measures of Variation 39 Measures of Variation SURVEYS Jamie asked her classmates how many 0 1 2 3 4 5 Glasses of Water Consumed × × × × × × × × × × × × × × × × glasses of water they drink on a typical day. 1. What is the median of the data set? 2. Organize the data into two groups: the top half and the bottom half. How many data values are in each group? 3. What is the median of each group? 4. Find the difference between the two numbers from Exercise 3. Measures of variation are used to describe the distribution of the data. The range is the difference between the greatest and least data values. Quartiles are values that divide the data set into four equal parts. LQ median UQ 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5, 5, 5 ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ The median of the lower half of a set of data is the lower quartile or LQ; in this case, 1. The median of the upper half of a set of data is the upper quartile or 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. Main Idea Find the measures of variation of a set of data. New Vocabulary measures of variation range quartile lower quartile upper quartile interquartile range outlier
- 2. Find Measures of Variation SPEED Find the measures of variation for the data. Range 70 - 1 or 69 mph Quartiles Order the numbers from least to greatest. lower half median upper half ⎧ ⎨ ⎩ 1 8 25 30 50 70 ⎧ ⎨ ⎩ LQ 25 + 30_ 2 = 27.5 UQ Interquartile Range 50 - 8 or 42 UQ - LQ The range is 69, the median is 27.5, the lower quartile is 8, the upper quartile is 50, and the interquartile range is 42. a. SPORTS Determine the measures Basketball Scores 64 61 67 59 60 58 57 71 56 62of variation for the data in the table. 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. Animal Speeds 50 30 25 8 1 70 elephant mouse spider cheetah lion cat Speed (mph)Animal ⎧ ⎨ ⎩ Interquartile RangeInterquartile Range If the interquartile range is low, the middle data are grouped closely together.
- 3. Analyze Data SCIENCE The table shows a set of scores on a Room A Room B 72 63 100 93 67 79 84 83 65 98 78 87 92 73 87 81 80 65 science test in two different classrooms. Compare and contrast their measures of variation. Find the measures of variation for both rooms. Room A Room B Range 100 - 65 = 35 98 - 63 = 35 Median 80 81 UQ 87 + 92_ 2 = 89.5 87 + 93_ 2 = 90 LQ 67 + 72_ 2 = 69.5 65 + 73_ 2 = 69 Interquartile Range 89.5 - 69.5 = 20 90 - 69 = 21 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. c. WEATHER Temperatures for the Month Antelope, MT Augusta, ME January 21 28 February 30 32 March 42 41 April 58 53 May 70 66 June 79 75 first half of the year are given for Antelope, Montana, and Augusta, Maine. Compare and contrast the measures of variation of the two cities. Examples 1 and 2 1. WIND SPEED The average wind speeds for several cities in Pennsylvania are given in the table. a. Find the range of the data. b. Find the median and the lower and upper quartiles. c. Find the interquartile range. d. Identify any outliers in the data. 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. Wind Speed Pennsylvania City Speed (mph) Allentown Erie Harrisburg Middletown Philadelphia Pittsburgh Williamsport 8.9 11.0 7.5 7.7 9.5 9.0 7.6
- 4. Examples 1 and 2 3. GOLF COURSES The table shows the number Number of Golf Courses California 1,117 Florida 1,465 Georgia 513 Iowa 437 Michigan 1,038 New York 954 North Carolina 650 Ohio 893 South Carolina 456 Texas 1,018 of golf courses in various states. a. Find the range of the data. b. Find the median and the lower and upper quartiles. c. Find the interquartile range. d. Name any outliers in the data. 4. INTERNET The table shows the countries Millions of Internet Users China 99.8 Germany 41.88 India 36.97 Japan 78.05 South Korea 31.67 United Kingdom 33.11 United States 185.55 with the most Internet users. a. Find the range of the data. b. Find the median and the lower and upper quartiles. c. Find the interquartile range. d. Name any outliers in the data. Example 3 5. EXERCISE The table shows the number Minutes of Exercise Week 1 Week 2 Tanika 45 30 Tasha 40 55 Tyrone 45 35 Uniqua 55 60 Videl 60 45 Wesley 90 75 of minutes of exercise for each person. Compare and contrast the measures of variation for both weeks. 6. FOOTBALL The table Penalties By NFL Teams NFC AFC Dallas Cowboys 104 New England Patriots 78 Arizona Cardinals 137 Indianapolis Colts 67 Green Bay Packers 113 Jacksonville Jaguars 76 New Orleans Saints 68 San Diego Chargers 94 New York Giants 77 Cleveland Browns 114 Seattle Seahawks 59 Pittsburgh Steelers 80 Minnesota Vikings 86 Houston Texans 82 shows the top teams in the National Football Conference (NFC) and the American Football Conference (AFC). a. Which conference had a greater range of penalties? b. Find the measures of variation for each conference. c. Compare and contrast the measures of variation for each conference.
- 5. Lesson 10 Measures of Variation 43 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 Mercury 0 Jupiter 63 Venus 0 Saturn 34 Earth 1 Uranus 27 Mars 2 Neptune 13 known moons for each planet in our solar system. Use the measures of variation to describe the data. 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.
- 6. Test PracticeT 44 Domain: Statistics and Probability 15. The number of games won by 10 chess players is given. 13, 15, 2, 7, 5, 9, 11, 10, 12, 11 Which of the following statements is NOT supported by these data? A. Half of the players won more than 10.5 games and half won less than 10.5 games. B. The range of the data is 13 games. C. There are no outliers. D. One fourth of the players won more than 7 games. 16. SHORT RESPONSE The ages in months of dogs enrolled in obedience class are: 8, 12, 20, 10, 6, 15, 12, 9, and 10. Find the range, median, lower and upper quartiles, and interquartile range of the dogs’ ages. 17. The normal monthly rainfall in inches for a city are given in the table. Jan Feb Mar Apr May June 0.65 1.39 0.63 2.16 2.82 4.21 July Aug Sept Oct Nov Dec 3.22 1.20 9.31 11.25 0.70 0.80 What values, if any, are outliers? F. 9.31 only G. 11.25 only H. both 9.31 and 11.25 I. There are no outliers. 18. Which of the following sets of data has an interquartile range of 10? A. 3, 4, 9, 16, 17, 24, 31 B. 41, 43, 49, 49, 50, 53, 55 C. 12, 14, 17, 19, 19, 20, 21 D. 55, 56, 56, 57, 58, 59, 62