2. Box-and-Whisker Plots describe a data set? How can you use a box-and-whisker plot to ACTIVITY: Drawing a Box-and-Whisker Plot Work with a partner. The numbers of first cousins of the students in an eighth-grade class are shown. A box-and-whisker plot uses a number line to represent the data visually. 9 2 62 Numbers of First Cousins 0 9 45 24 8 0 6 8 2 8 0 5 4 a. Order the data set and write it on a strip of grid paper with 24 equally spaced boxes. Fold the paper in half to find the median. b. Fold the paper in half again to divide the data into four groups. Because there are 24 numbers in the data set, each group should have six numbers. least first median third greatest COMMON CORE Box-and-Whisker Plots In this lesson, you will make and interpret box-and-whisker plots. find inter ranges of data sets. compare box-and-whisker plots. Learning Standards S.ID. S.ID.2 S.ID. c. Draw a number line that includes the least and the greatest in the data set. Graph the five numbers that you found in part (b). 0 5 0 5 20 25 0 5 40 45 50 d. Explain how the box-and-whisker plot shown below represents the data set. least first median third greatest 0 4 0 7 45 0 5 0 5 20 25 0 5 40 45 50 Number of First Cousins 68 Chapter 2 Data Analysis and Displays
2 ACTIVITY: Conducting a Survey Math Practice View as Components How do the different components of a box-and-whisker plot help you interpret the s? Conduct a survey in your class. Ask each student to write the number of his or her first cousins on a piece of paper. Collect the pieces of paper and write the data on the chalkboard. Now, work with a partner to draw a box-and-whisker plot of the data. Two people are first cousins if they share at least one grandparent, but do not share a parent. ACTIVITY: Reading a Box-and-Whisker Plot First Cousins Work with a partner. The box-and-whisker plots show the test score distributions of two eighth-grade standardized tests. The tests were taken by the same group of students. One test was taken in the fall and the other was taken in the spring. a. Compare the test results. b. Decide which box-and-whisker plot represents the results of each test. How did you make your decision? 50 6 72 82 00 2 50 60 7 00 0 0 20 0 40 50 60 70 80 90 00 0 Test Score 4. IN YOUR OWN WORDS How can you use a box-and-whisker plot to describe a data set? 5. Describe who might be interested in test score distributions like those shown in Activity. Explain why it is important for these people to analyze test score distributions. Use what you learned about box-and-whisker plots to complete Exercise 4 on page 62. Section 2. Box-and-Whisker Plots 69
2. Lesson Lesson Tutorials Key Vocabulary box-and-whisker plot, p. 620, p. 620 five-number summary, p. 620 inter range, p. 62 Box-and-Whisker Plot A box-and-whisker plot displays a data set along a number line using medians. Quartiles divide the data set into four equal parts. The median (second ) divides the data set into two halves. The median of the lower half is the first. The median of the upper half is the third. least first whisker box median third whisker greatest The five numbers that make up the box-and-whisker plot are called the five-number summary of the data set. EXAMPLE Making a Box-and-Whisker Plot Make a box-and-whisker plot for the ages of the members of the U.S. women s wheelchair basketball team. 24, 0, 0, 22, 25, 22, 8, 25, 28, 0, 25, 27 Step : Order the data. Find the median and the s. least lower half upper half 8 22 22 24 25 25 25 27 28 0 0 0 first, 2 third, 29 median, 25 greatest Step 2: Draw a number line that includes the least and greatest s. Graph points above the number line for the five-number summary. Study Tip Step : Draw a box using the s. Draw a line through the median. Draw whiskers from the box to the least and greatest s. A box-and-whisker plot shows the variability of a data set. 8 9 20 2 22 2 24 25 26 27 28 29 0 Age Exercises 5 7. A basketball player scores 4, 6, 20, 5, 22, 0, 6, and 28 points during a tournament. Make a box-and-whisker plot for the points scored by the player. 620 Chapter 2 Data Analysis and Displays
The figure shows how data are distributed in a box-and-whisker plot. of the data are 4 in each whisker. of the data are 2 in the box. of the data are 4 in each whisker. first third Another measure of dispersion for a data set is the inter range, which is the difference of the third and the first. It represents the range of the middle half of the data. EXAMPLE 2 Interpreting a Box-and-Whisker Plot The box-and-whisker plot represents the lengths of songs (in seconds) played by a rock band at a concert. 40 60 80 200 220 240 260 280 00 20 Song Length (seconds) a. Find and interpret the range of the data. The least is 60. The greatest is 00. So, the range is 00 60 = 40 seconds. This means that the song lengths vary by no more than 40 seconds. b. Describe the distribution of the data. 25% of the song lengths are between 60 and 220 seconds. 50% of the song lengths are between 220 and 280 seconds. 25% of the song lengths are between 280 and 00 seconds. c. Find and interpret the inter range of the data. inter range = third first = 280 220 = 60 So, the inter range is 60 seconds. This means that the middle half of the song lengths vary by no more than 60 seconds. Exercises 0 and Use the box-and-whisker plot in Example. 2. Find and interpret the range and inter range of the data.. Describe the distribution of the data. Section 2. Box-and-Whisker Plots 62
A box-and-whisker plot shows the shape of a distribution. Study Tip If you can draw a line through the median of a box-and-whisker plot, and each side is a mirror image of the other, then the distribution is symmetric. Shapes of Box-and-Whisker Plots Skewed left Symmetric Skewed right Left whisker longer than right whisker Most data on the right Whiskers about same length Median in the middle of the data Right whisker longer than left whisker Most data on the left EXAMPLE Comparing Box-and-Whisker Plots The double box-and-whisker plot represents the test scores for your class and your friend s class. Your Class Friend s Class 55 60 65 70 75 80 85 90 95 00 a. Identify the shape of each distribution. For your class, the left whisker is longer than the right whisker, and most of the data are on the right side of the display. So, the distribution is skewed left. Test Score For your friend s class, the whisker lengths are equal. The median is in the middle of the data. The data appear to be evenly distributed on both sides of the median. So, the distribution is symmetric. b. Which test scores are more spread out? The range of the test scores in your friend s class is greater than the range in your class. Also, because the box for your friend s class is longer than the box for your class, the inter range is also greater. So, the test scores in your friend s class are more spread out. Exercise 20 4. The double box-and-whisker plot represents the surfboard prices at Shop A and Shop B. Identify the shape of each distribution. Which shop s prices are more spread out? Explain. Shop A Shop B 0 00 200 00 400 500 600 700 800 900 Surfboard Price (dollars) 622 Chapter 2 Data Analysis and Displays
2. Exercises Help with Homework. VOCABULARY In a box-and-whisker plot, what percent of the data is represented by each whisker? by the box? 2. WRITING Describe how to find the first of a data set.. NUMBER SENSE What does the length of a box-and-whisker plot tell you about the data? 9+(-6)= +(-)= 4+(-9)= 9+(-)= 4. The box-and-whisker plots show the monthly car sales for a year for two sales representatives. Compare the sales for the two representatives. Sales Rep A Sales Rep B 0 4 8 2 6 20 24 28 Cars Sold Make a box-and-whisker plot for the data. 5. Hours of television 6. Lengths (in inches) of 7. Elevations (in feet): watched: 0,, 4, 5, cats: 6, 8, 20, 25, 7, 2, 0, 5, 4,,, 2,, 4, 6, 5 22, 2, 2 0, 2,, 6, 5, 9, 4, 6, 7,, 4, 7, 8,, 5, 8 8. ERROR ANALYSIS Describe and correct the error in making a box-and-whisker plot for the data. 2 4 5 6 7 8 9 0 9. FISH The lengths (in inches) of the fish caught on a fishing trip are 9, 0, 2, 8,, 0, 2, 4, 7, 4, 8, and 4. a. What is the median of the data set? b. What are the first and third s of the data set? c. Make a box-and-whisker plot for the data. Section 2. Box-and-Whisker Plots 62
2 0. INCHWORM The table shows the lengths of 2 inchworms. a. Make a box-and-whisker plot for the data. b. Find and interpret the range of the data. c. Describe the distribution of the data. d. Find and interpret the inter range of the data. Length (cm) 2.5 2.4 2. 2.5 2.7 2. 2 2.8 28 2.6 26 2. 2 2.6 26 2.9 29 2.0 Entrée Prices (dollars) 4.00 7.00 2.50 0.00.00 8.25 9.00 8.50 4.75 5.00 4.00 2.00. ENTRÉE The table shows the prices of entrées at a restaurant. a. Make a box-and-whisker plot for the data. b. Find and interpret the inter range of the data. c. Describe the distribution of the data. d. Find the standard deviation. Interpret your result. 2. WRITING Given the numbers 6 and 2, identify which number is the range, and which number is the inter range, of a set of data. Explain your reasoning. Determine whether the shape of the box-and-whisker plot is symmetric, skewed left, or skewed right. Explain.. 4. 5. 6. 7. ERROR ANALYSIS Describe and correct the error in describing the box-and-whisker plot. The shape of the distribution is skewed right. So, there are more data s to the right of the median than to the left of the median. 8. LOGIC Give examples of real-life data that are symmetric and real-life data that are not symmetric. Justify your answer. 624 Chapter 2 Data Analysis and Displays
9. CALORIES The table shows the numbers of calories burned per hour for nine activities. a. Make a box-and-whisker plot for the data. b. Identify the outlier. c. Make another box-and-whisker plot without the outlier. d. WRITING Describe how the outlier affects the whiskers, the box, and the s of the box-and-whisker plot. Calories Burned per Hour Fishing 207 Mowing the lawn 25 Canoeing 26 Bowling 77 Hunting 295 Fencing 54 Bike racing 944 Horseback riding 26 Dancing 266 20. CELL PHONES The double box-and-whisker plot compares the battery lives (in hours) of two brands of cell phones. Brand A a. Identify the shape of each distribution. b. What is the range of the upper 75% of each brand? c. Compare the inter ranges of the two data sets. d. Which brand do you think has the greater standard deviation? Explain. Brand B 2 2.5.5 4 4.5 5 5.5 6 6.5 7 Battery Life (hours) Modeling Create a set of data s for the box-and-whisker plot that has the given characteristic(s). 2. The least, greatest, s, and median are all equally spaced. 22. Both whiskers are the same length as the box. 2. The box between the median and the first is three times as long as the box between the median and the third. 24. There is no right whisker. Write an equation of the line that passes through the points. (Section 2.6) 25. ( 4, 0), (2, 8) 26. (, ), (0, ) 27. ( 4, ), (4, ) 28. (6, 7), (8, 8) 29. MULTIPLE CHOICE What is the quotient of (2z 2 z + 2) and (z )? (Section.5) A 2z + 7 B 2z 7 C z + 6 D z 7 Section 2. Box-and-Whisker Plots 625