0732_hsm07a1_se_ny_1002.qxd 6/13/07 3:49 PM Page 732 Quartiles and Box-and-Whisker Plots NY-2 Learning Standards for Mathematics GO for Help Check Skills You’ll Need A.S.11 Find the percentile rank of an item in a data set and identify the point values for ﬁrst, second, and third quartiles. Find the range and the median. 1. 3 8 12 15 20 3. 0 2 7 10 –1 –4 –11 A.S.5, A.S.9, A.S.6 Construct, analyze, and interpret a box-andwhisker plot using the ﬁve number summary. Lesson 1-6 2. 2.1 3.3 –5.4 0.8 3.5 4. 64 16 23 57 14 22 New Vocabulary • ﬁrst quartile • third quartile • ﬁve number summary • minimum • maximum • percentiles • percentile rank • quartile • box-and-whisker plot • interquartile range 1 2 Part Graphing and y ≠ ax Percentiles 1 1 Quartiles In Lesson 1-6, you used measures of central tendency to describe data sets. In this lesson, you will use quartiles to separate data in four equal parts. The median separates the data into upper and lower halves. The ﬁrst quartile (Q1) is the median of the lower half of the data. The third quartile (Q3) is the median of the upper half of the data. minimum 1 EXAMPLE Q1 Q2 Q3 first quartile median of lower half median third quartile median of upper half maximum Finding Median, First, and Third Quartiles For the data below, ﬁnd the median (Q2), ﬁrst quartile (Q1), and third quartile (Q3). 39 27.5 26 31 42 26 37 30 22 Step 1 Arrange the data in order from least to greatest. Find the median. 22 26 26 27.5 30 31 37 39 42 The median (Q2) is the middle value 30. Step 2 Find the ﬁrst quartile and third quartile, which are the medians of the lower and upper halves. 22 26 | 26 27.5 30 31 37 | 39 42 first quartile (Q1) = 26 1 26 = 26 2 Quick Check NY 732 third quartile (Q3) = 37 1 39 = 38 2 1 For the following data, ﬁnd the median, the ﬁrst quartile, and the third quartile. 480 500 600 250 660 570 490 610 530 Chapter NY New York Additional Topics 0732_hsm07a1_se_ny_1002.qxd 6/13/07 3:49 PM Page 733 The ﬁve number summary of a set of data includes the minimum, the ﬁrst quartile, the median, the third quartile, and the maximum. The minimum of a data set is the least value. The maximum is the greatest value. 2 EXAMPLE Finding the Five Number Summary Find the ﬁve number summary for the data below. 125 80 140 135 126 140 350 Step 1 Arrange the data in order from least to greatest. 80 125 126 135 140 140 350 Step 2 Find the minimum, maximum, and median. 80 125 126 135 140 140 350 The minimum is 80. The maximum is 350. The median is 135. Step 3 Find the ﬁrst quartile and third quartile. ﬁrst quartile (Q1) = 125 1 126 = 125.5 2 third quartile (Q3) = 140 1 140 = 140 2 The ﬁve number summary is minimum = 80, Q1 = 125.5, median = 135, Q3 = 140, and maximum = 350. Quick Check 2 Find the ﬁve number summary for each set of data. a. 95 85 75 85 65 60 100 105 75 85 75 b. 11 19 7 5 21 53 Data sets can be separated into smaller equal parts. Percentiles separate data sets into 100 equal parts. The percentile rank of a score is the percentage of scores that are less than or equal to that score. 3 EXAMPLE Finding a Percentile Rank Of 25 test scores, eight are less than or equal to 75. What is the percentile rank of a test score of 75? Write a ratio of the number of scores less than or equal to 75 compared to the total number of test scores. 8 25 8 = 0.32 25 = 32% Number of test scores less than or equal to 75 Total number of test scores 8 Rewrite 25 as a percent. The percentile rank of 75 is 32. Quick Check 3 a. Of the 25 test scores, 15 scores are less than or equal to 85. What is the percentile rank of 85? b. Critical Thinking Is it possible to have a percentile rank of 0? Explain. Lesson NY-2 Quartiles and Box-and-Whisker Plots NY 733 0732_hsm07a1_se_ny_1002.qxd 2 1 6/13/07 3:49 PM Page 734 Box-and-Whisker Plots A box-and-whisker plot is a graph that summarizes a data set along a number line. The box represents the ﬁrst quartile, the median, and the third quartile of the data. The whiskers on the left and right sides represent the minimum and maximum values. Use the ﬁve number summary to construct a box-and-whisker plot. Box-and-Whisker Plot whisker minimum 4 EXAMPLE whisker box first quartile third maximum quartile median Real-World Problem Solving Agriculture The table below shows United States crops harvested from 1992 to 2004. Find the ﬁve number summary and construct a box-and-whisker plot of the data. Step 1 Arrange the data in order from least to greatest. Find the minimum, maximum, and the median. 308 314 316 317 321 321 321 324 325 326 326 327 332 Step 2 Find the ﬁrst quartile and third quartile. first quartile = 316 1 317 = 633 = 316.5 2 2 third quartile = 326 1 326 = 652 = 326 2 2 Crops Harvested Acres Acres Year (millions) Year (millions) 1992 1993 1994 1995 1996 1997 1998 317 308 321 314 326 332 326 1999 2000 2001 2002 2003 2004 327 325 321 316 324 321 SOURCE: Statistical Abstract of the United States. Step 3 Draw a number line. Construct the box-and-whisker plot. Mark the values of the minimum, ﬁrst quartile, median, third quartile, and maximum. Crops Harvested (millions of acres) 290 300 310 320 330 340 350 Quick Check 4 Construct a box-and-whisker plot for the following distances of bird migrations in thousands of miles. 5 2.5 6 8 9 2 1 4 6.2 18 NY 734 Chapter NY New York Additional Topics 0732_hsm07a1_se_ny_1002.qxd 6/13/07 3:49 PM 5 Page 735 EXAMPLE Using a Graphing Calculator Use a graphing calculator to make a box-and-whisker plot of the data below. Find the ﬁve number summary. 10 26 18 35 14 11 17 29 31 25 27 20 19 12 13 26 L1 L2 L3 1 25 27 20 19 12 13 26 1:L1 Plot1 Plot2 Plot3 On Off Type: Xlist: L1 Freq: 1 L1(16) = 26 Med=19.5 Step 1 Press 1: Edit . . . . Enter data under L1. (To clear L1, move the cursor over L1 and press , .) Step 2 Y= for Press STAT PLOTS. Enter 1. Highlight ON and also Type: box-and-whisker plot. Press . Step 3 Press and enter 9. Press and move the cursor across the box-and-whisker plot to ﬁnd the ﬁve number summary. The ﬁve number summary is minimum = 10, Q1 = 13.5, median = 19.5, Q3 = 26.5, maximum = 35. Quick Check 5 Use a graphing calculator to ﬁnd the ﬁve number summary for the following set of student test grades shown below. 98 92 76 84 93 82 74 68 85 91 77 83 94 97 72 88 70 84 87 82 The interquartile range is the difference between the ﬁrst and third quartiles. In a box-and-whisker plot, this is the width of the box. You can use a parallel box-and-whisker plot to compare two sets of data. 6 EXAMPLE Comparing Data Sets The box-and-whisker plot below shows average monthly rainfall for Miami and New Orleans. Which city shows the greater range in average monthly rainfall? Average Monthly Rainfall (millimeters) 20 40 60 80 100 120 140 160 180 200 220 Miami, Fla. New Orleans, La. Miami has the longer box-and-whisker plot, so it has the greater range. Quick Check 6 What do the interquartile ranges tell you about the average monthly rainfall for Miami and New Orleans? Lesson NY-2 Quartiles and Box-and-Whisker Plots NY 735 0732_hsm07a1_se_ny_1002.qxd 6/22/07 12:59 AM Page 736 EXERCISES For more exercises, see Extra Skill and Word Problem Practice. Practice and Problem Solving A Practice by Example Example 1 GO for Help (page NY 732) Example 2 (page NY 733) Example 3 (page NY 733) Example 4 (page NY 734) Example 5 (page NY 735) Find the median, the ﬁrst quartile, and the third quartile. 1. 12 10 11 7 9 10 5 2. 4.5 3.2 6.3 5.2 5 4.8 6 3.9 12 3. 55 53 67 52 50 49 51 52 52 4. 101 100 100 105 101 102 104 5. Find the ﬁve number summary for the set of data. 1 1 3 3 4 5 7 7 8 9 9 32 31 29 35 30 32 11 6. Of 20 scores, 19 are less than or equal to 10. Find the percentile rank of 10. 7. Gerry ranks 48th in a class of 120 students. What is his percentile rank? 8 Construct a box-and-whisker plot for the ages of the members of the art club. 13 18 16 14 15 17 15 16 16 15 16 14 14 17 19 9. Use a graphing calculator to draw a box-and-whisker plot for the data below. Then ﬁnd the interquartile range. 16 18 59 75 30 34 25 49 27 16 21 58 71 19 50 Example 6 10. Compare the box-and-whisker plots below. What can you conclude? (page NY 735) Ages of U.S. Olympic Soccer Team Players 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Men Women B Apply Your Skills For Exercises 11–16, use the data below. Find each measure. 1.8 2.5 3.9 4.6 4.7 4.8 4.8 4.9 11. median 12. ﬁrst quartile 13. minimum 14. maximum 15. third quartile 16. percentile rank of 4.6 17. Use the box-and-whisker plot below. What can you conclude about the areas of state parks? Areas of State Parks (acres) 100 200 300 400 500 600 For Exercises 18–21, use the graph below showing box-and-whisker plots for two sets of data, A and B. 66 68 70 72 74 76 78 Data Set A Data Set B 18. Which set of data has the greater range? 19. Which set of data has the lesser median? NY 736 Chapter NY New York Additional Topics 80 82 84 86 0732_hsm07a1_se_ny_1002.qxd 6/13/07 3:49 PM Page 737 20. Which set of data has the greater interquartile range? 21. Which set of data has the lesser minimum? For Exercises 22–24, use the ﬁve number summary given below. Minimum = 10 Q1 = 20 Median = 30 22. Find the interquartile range. Q3 = 42 Maximum = 75 23. Find the range of the data. 24. Use the data to construct a box-and-whisker plot. 25. Social Studies The plots below compare the percents of the voting-age population who said they registered to vote in U.S. elections to the percents who said they voted. Which conclusion best reﬂects the data collected? Percents of Population Who Registered and Voted, 1990–2000 40 50 60 70 Registered Voted A. The percent who voted was about 15% less than the percent who registered. B. The percent who voted was about half the percent who registered. C. The percent who voted was equal to the percent who registered. D. The percent who voted was about 15% more than the percent who registered. 26. Error Analysis In a class of 250 students, Emily had the tenth highest grade average. She computed her percentile rank as 4. What was her error? C Challenge 27. Reasoning Can you ﬁnd the mean, median, and mode of a set of data by looking at a box-and-whisker plot? Explain. REGENTS Test Prep Multiple Choice 28. Find the ﬁrst quartile and third quartile in the following data set. 17 20 30 19 20 18 25 28 31 23 17 29 31 33 28 A. 19 and 30 B. 17 and 25 C. 19.5 and 30.5 D. 17 and 33 29. Of 20 test scores, sixteen are less than or equal to 80. What is the percentile rank of a test score of 80? F. 16th Short Response G. 85th H. 25th J. 80th 30. Describe how you could ﬁnd the 75th percentile score in a set of 20 scores. Mixed Review Lesson 1-1 Deﬁne variables and write an equation to model each situation. 31. The total cost of the number of CDs times $5.00. 32. The perimeter of a square equals 4 times the length of the side. Lesson 1-3 Decide whether each statement is true or false. If the statement is false, give the counterexample. 33. All whole numbers are rational numbers. 34. The square root of a number is always smaller than the number. Lesson NY-2 Quartiles and Box-and-Whisker Plots NY 737

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