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Lesson 28 Part 1: Introduction CCLS 7.SP.3 Using Mean and Mean Absolute Deviation to Compare Data In this lesson, you will compare two distributions by looking at their centers and variabilities. Take a look at this problem. Marcus is the basketball reporter at Chesapeake State University. He wants to write a story comparing the heights of players on the men’s team to the heights of players on the women’s team. He made two dot plots to help him compare heights. Players’ Heights Players’ Heights 72 74 76 78 80 Men (in inches) 82 84 66 68 70 72 74 Women (in inches) 76 78 Explore It Use the math you already know to solve this problem. What are some heights that you see only on the men’s team? What are some heights you see only on the women’s team? What are some heights that you see on both the men’s team and the women’s team? Estimate the typical height for the men’s team. Mark it on the graph. Do the same for the women’s team. How does the typical height for the men’s team compare to the typical height for the women’s team? How does the spread for the distribution of heights on the men’s team compare to that of the women’s team? 264 L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. Part 1: Introduction Lesson 28 Find Out More For data sets like the ones on the previous page, it can be helpful to compute the measures of center and compare them against your estimates. One measure of center you can use for each data set is the mean. For the men’s basketball team, the mean height is 77.2 inches (rounded to the nearest tenth). For the women’s basketball team, the mean height is 71.5 inches (rounded to the nearest tenth). You can also compare estimates of spread against a formal measure. One measure of spread is the mean absolute deviation (MAD). The MAD describes the average distance between the data values in a distribution and the mean of the distribution. To compute the MAD for the men’s team, start by subtracting the value of each data point from the mean. Height Difference from Mean 78 76 73 76 80 77 75 20.8 1.2 4.2 1.2 22.8 0.2 2.2 83 79 79 25.8 21.8 21.8 75 73 80 2.2 4.2 22.8 Then, you can average the absolute values of all of those results to get the MAD: 1.2 1 2.8 1 0.2 1 2.2 1 5.8 1 1.8 1 1.8 1 2.2 1 4.2 1 2.8 0.8 1 1.2 1 4.2 1 5 2.4 13 ························································· You can use the same procedure to compute the women’s team’s MAD, which in this case would equal 2.3 (rounded to the nearest tenth). The difference between the mean heights of the teams is about 6 (77.2 2 71.5). The MAD for the two distributions are about the same (between 2.3 and 2.4). 6 is a little more than two times 2.4, so the difference between the means is a little more than twice the value of the MAD for each distribution. If the difference between the means was three or more times as great as the MAD for each distribution, there would be an even greater difference between the mean heights. A dot plot would show fewer heights in common between the men’s and women’s teams. Reflect 1 How would the distributions change if the difference between the means was the same as the MAD for each distribution? L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. 265 Part 2: Modeled Instruction Lesson 28 Read the problem below. Then explore ways to compare data sets that have similar variabilities but different centers. A lot of the basketball players at Chesapeake State had graduated from Central Middle School. How can Marcus compare the heights of players on the boys’ team at Central Middle School to those on the men’s team at Chesapeake State? Model It You can compare the dot plots for each distribution. The mean height of the men’s team is 77.2. Using the data for the boy’s team, you can find the mean: 62 1 62 1 63 1 63 1 64 1 65 1 66 1 66 1 67 1 67 1 67 1 69 60 1 60 1 61 1 5 64.1 15 ···························································· You can use vertical segments to mark the mean for each data set. Players’ Heights 68 70 72 74 76 78 Men (in inches) 80 82 84 72 74 76 Players’ Heights 60 266 62 64 66 68 70 Boys (in inches) L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. Part 2: Guided Instruction Lesson 28 Connect It Now you will use these representations to compare the data sets. 2 What is the difference between the mean height for men on the Chesapeake State team and the mean height for boys on the middle school team? 3 Compute the mean absolute deviation for boys’ height, to the nearest tenth. How does it compare to 2.4, the mean absolute deviation for men’s height? 4 By what number would you have to multiply the MAD to get the difference between the mean heights you found in problem 1? Round your answer to the nearest tenth. 5 What would the dot plot look like if you combined the data from boys at Central Middle School with the data from men at Chesapeake State to make one big dot plot? 6 You would have to multiply the MAD by more than 3 to get the difference between the means in these distributions. When the means of distributions are more than 3 MADs apart, do you expect them to have a lot of values in common? Why or or why not? Try It Use what you just learned about comparing distributions to solve this problem. 7 The Central Middle School girls’ basketball team has a mean height of 62.3 inches, and a MAD of 2.3 (the same MAD as the Chesapeake University women’s team). How many MADs greater is the mean height for the Chesapeake University women’s team? (Remember: the mean height for the women was 71.5 inches). L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. 267 Part 3: Guided Practice Lesson 28 Read the situation described below. Then solve problems 8–10. Student Model The student divided the difference in the means by the mean absolute deviation to solve the problem. Sara was curious about how many text messages students with cell phones send each day. She surveyed a random sample of students who own cell phones at the middle school and another at the high school. For the middle school data set, the mean number of texts per day was 60. For the high school data set, the mean number was 76. The MADs for the data sets were the same. Both MADs were 5. How does the difference of the means compare to the mean absolute deviations? Look at how you can use the information in the problem to answer the question. Pair/Share Why is it good to compare data sets using MADs along with looking at the differences between the means of the data sets? What does it mean when the MADs of two sets of data are about the same? The difference in the means is 16. 16 4 5 5 3.2 The difference of the means is a little more than 3 times Solution: the mean absolute deviations. 8 How would the dot plots of the data described in the student model differ? How might they be the same? Explain. Show your work. Pair/Share What are some kinds of data sets that you might want to compare that would NOT have similar MADs? 268 Solution: L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. Part 3: Guided Practice 9 Do you think that the two data sets described on the previous page have a lot of values in common? Why or why not? Show your work. Lesson 28 What information tells me whether the data sets have a lot of values in common? Solution: 10 Which pair of data sets is most likely to have the greatest number of values in common? A Data set 1: mean 5 7 Data set 2: mean 5 15; MAD for both data sets is 8 Pair/Share Why do data sets usually have a lot of shared values when their means are one MAD or less apart? What values should I be comparing? B Data set 1: mean 5 7 Data set 2: mean 5 15 MAD for both data sets is 4 C Data set 1: mean 5 10 Data set 2: mean 5 18 MAD for both data sets is 2 D Data set 1: mean 5 10 Data set 2: mean 5 15 MAD for both data sets is 1 Bryce chose D as the correct answer. How did he get that answer? L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. Pair/Share How would you help Bryce understand his error? 269 Part 4: Common Core Practice Lesson 28 Answer Form 1 B C D 2 B C D 3 B C D Solve the problems. Mark your answers to problems 1–3 on the Answer Form to the right. Be sure to show your work. 1 2 Number Correct 3 Two data distributions with similar MADs and many values in common often have which of the following? A small MADs B large MADs C means that are many MADs apart from one another D means that are not many MADs apart from one another Choose the data set with the greatest MAD. A mean = 81.9389 72 74 76 78 80 Height (in inches) 82 84 mean = 76.8109 B 72 74 76 78 80 Height (in inches) 82 84 mean = 76.9897 C 72 74 76 78 80 Height (in inches) 82 84 mean = 77.3007 D 72 270 74 76 78 80 Height (in inches) 82 84 L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. Part 4: Common Core Practice 3 4 Lesson 28 The mean absolute deviation is best described as: A The value obtained when you add all the values in a data set and divide by the number of values. B The value obtained when averaging the distances between each data point in a distribution and the mean. C The difference between the means of two similar distributions. D An approximation of the center of a statistical distribution. Make up two data sets. List all the values in each data set and write a story to describe where they may have originated. The data sets should meet the following conditions: • The means should be different. • The MADs should be similar. • The means should be more than one MAD apart. Be sure to show work to demonstrate that your data sets meet the above conditions. Self Check Go back and see what you can check off on the Self Check on page 247. L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. 271 Lesson 28 (Student Book pages 264–271) Using Mean and Mean Absolute Deviation to Compare Data Lesson Objectives The Learning Progression •Use visual representations, such as dot plots, to compare two real-world numerical data sets with similar and differing variabilities. In Grade 7, students continue their work with data display and analysis. In previous grades, they have described the shape of a data set and calculated mean, median, mode, and range. They have worked with dot plots and box plots. In Grade 7, students learn measures of variability and use them to compare data sets. They assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. •Compare data sets and measure the difference between the centers. •Represent the difference between centers of data sets by using the mean. •Step through the calculations necessary to find the mean absolute deviation for each of two data sets. •Describe the variation in data sets. Prerequisite Skills •Find the mean, median, mode, and range of a data set. •Read and create data displays, including dot plots. In this lesson, students will use dot plots to compare two real-world numerical data sets with similar and differing variabilities, measure the difference between centers, and represent the difference in the means. Students will step through the calculations necessary to find the mean absolute deviation for and describe the variation in each data set. •Find the mean absolute deviation of a data set. Toolbox Vocabulary Teacher-Toolbox.com Prerequisite Skills mean: the average of the numbers; the sum of the values divided by the number of values Ready Lessons mean absolute deviation (MAD): the average distance of each data point from the mean Tools for Instruction Interactive Tutorials ✓ ✓ 7.SP.3 ✓ ✓ ✓ CCLS Focus 7.SP.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. STANDARDS FOR MATHEMATICAL PRACTICE: SMP 1–7 (see page A9 for full text) L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. 271 Part 1: Introduction Lesson 28 At a Glance Students explore a problem and are guided to compare two distributions by looking at centers and variabilities. Lesson 28 Step By Step 72 •Ask the pairs to measure each other’s heights to the nearest inch. •Record the measurements for each student on the board, sorting the measures of boys from girls. •Have the pairs create one dot plot for girls and one for boys. •Ask pairs to compare the shapes of the graphs, the averages of the typical heights, and the spreads. •Discuss how the students’ data compares with the data of the players’ heights. 272 76 78 80 Men (in inches) 82 84 66 68 70 72 74 Women (in inches) 76 78 use the math you already know to solve this problem. What are some heights that you see only on the men’s team? What are some heights you see only on the women’s team? men’s team: 78, 79, 80, and 83; women’s team: 67, 69, 70, 71, and 72 What are some heights that you see on both the men’s team and the women’s team? Possible answers: some heights that you see on both teams are 73, 75, and 77. Estimate the typical height for the men’s team. Mark it on the graph. Do the same for the women’s team. Possible answer: men’s team about 77; women’s team about 71 How does the typical height for the men’s team compare to the typical height for the women’s team? the typical height for men is about 6 inches taller than the ELL Support Have students work in pairs and save their work for the Hands-On Activity on page 275. 74 explore it •Ask student pairs or groups to answer the last question and share their explanation with the class. Materials: yard sticks, rulers Players’ Heights Players’ Heights •Give students time to make observations based on the dot plots, prompting them as necessary to describe clumps, gaps, and spreads. Encourage students to debate and justify their observations. Make dot plots to compare height data. 7.sP.3 Marcus is the basketball reporter at Chesapeake State University. He wants to write a story comparing the heights of players on the men’s team to the heights of players on the women’s team. He made two dot plots to help him compare heights. •Work through Explore It as a class. Hands-On Activity CCLs in this lesson, you will compare two distributions by looking at their centers and variabilities. take a look at this problem. •Tell students that this page guides them in understanding how to compare two distributions by looking at their centers and variabilities. Have students read the problem at the top of the page and examine the dot plots. On this page, some everyday words are used in a math context: typical, center, spread, distribution. Make a chart with the words, their everyday meanings, and their meanings as math terms. Part 1: introduction using Mean and Mean absolute Deviation to Compare Data typical height for women. How does the spread for the distribution of heights on the men’s team compare to that of the women’s team? Possible answer the spreads are pretty similar because the tallest height for the men is 10 inches greater than the shortest height. the tallest height for the women is also 10 inches greater than the shortest height. 264 L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse •How did you estimate the typical heights of the men and the women? On what did you base your estimates? Students may have different ways to estimate the “typical” height. Follow up by asking whether others used a different method. Listen for ideas about “most” heights. •How would the shape of the dot plots change if you added data from this class? Student observations may include a change in the mean, a greater range, a greater spread distribution, and changes in the shape of the dot plot. •Describe a scenario in which information from the problem may be of value. Knowing the typical heights of chosen players. L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. Part 1: Introduction Lesson 28 At a Glance Students examine how to determine the mean absolute deviations of two data sets to assess the degree of overlap. Step By Step •Read Find Out More as a class. •Review briefly what the mean is and how it is calculated. Explain that for this data set, we’ll use the mean as the “typical” height. •Read aloud the third paragraph. Ask three or four students to explain in their own words what the mean absolute deviation (MAD) is and what it tells about a data set. Ask others to agree or add on to the explanations. •Direct students’ attention to the final steps in determining the MAD. Here, all the differences are combined and then divided by the total number of absolute values. •Discuss how to find the difference between the mean heights and how to compare it to the MADs for the distributions. Part 1: introduction Lesson 28 Find out More For data sets like the ones on the previous page, it can be helpful to compute the measures of center and compare them against your estimates. One measure of center you can use for each data set is the mean. For the men’s basketball team, the mean height is 77.2 inches (rounded to the nearest tenth). For the women’s basketball team, the mean height is 71.5 inches (rounded to the nearest tenth). You can also compare estimates of spread against a formal measure. One measure of spread is the mean absolute deviation (MaD). The MAD describes the average distance between the data values in a distribution and the mean of the distribution. To compute the MAD for the men’s team, start by subtracting the value of each data point from the mean. height Difference from Mean 78 76 73 76 80 77 75 20.8 1.2 4.2 1.2 22.8 0.2 2.2 83 79 SMP Tip: Students reason abstractly when they consider how the distributions or dot plots would change if there were different MAD values (SMP 2). •Have pairs or groups do Reflect and share their answers with the class. ELL Support To help students understand the abstract meanings of such words as mean absolute deviation, compare the words to their everyday uses: mean (not to be confused with cruel): average, in the middle absolute: unlimited; a dictator has absolute power; an absolute number ignores its positive or negative signs deviate, deviation: go away from L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. 75 73 80 2.2 4.2 22.8 Then, you can average the absolute values of all of those results to get the MAD: 0.8 1 1.2 1 4.2 1 1.2 1 2.8 1 0.2 1 2.2 1 5.8 1 1.8 1 1.8 1 2.2 1 4.2 1 2.8 5 2.4 13 ························································· You can use the same procedure to compute the women’s team’s MAD, which in this case would equal 2.3 (rounded to the nearest tenth). The difference between the mean heights of the teams is about 6 (77.2 2 71.5). The MAD for the two distributions are about the same (between 2.3 and 2.4). 6 is a little more than two times 2.4, so the difference between the means is a little more than twice the value of the MAD for each distribution. If the difference between the means was three or more times as great as the MAD for each distribution, there would be an even greater difference between the mean heights. A dot plot would show fewer heights in common between the men’s and women’s teams. reflect 1 How would the distributions change if the difference between the means was the same as the MAD for each distribution? Possible answer: there would be a lot more overlap between the distributions. you would see more heights in common between the men’s team and the women’s team. L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. •Emphasize that a data set with a smaller MAD has data values that are closer to the mean than a data set with a greater MAD. Create a dot plot on the board with combined data to support this reasoning. 79 25.8 21.8 21.8 265 Real-World Connection Ask students to think of situations in which it would be helpful to know the mean absolute deviations of two data sets. If they have trouble with this, suggest this situation: Suppose a teacher writes two versions of a test covering the same subject. To be a good test, students should score about the same on both tests. How would knowing the mean absolute deviations of the scores help? [The MADs of the scores on the two tests would show whether students as a group were scoring roughly the same on both tests.] 273 Part 2: Modeled Instruction Lesson 28 At a Glance Part 2: Modeled instruction Students will use dot plots to compare the mean average deviation of two data sets. read the problem below. then explore ways to compare data sets that have similar variabilities but different centers. Step By Step A lot of the basketball players at Chesapeake State had graduated from Central Middle School. How can Marcus compare the heights of players on the boys’ team at Central Middle School to those on the men’s team at Chesapeake State? •Read the problem at the top of the page as a class. Model it •Direct students to Model It. You will discuss with students what they can determine from casual observation of the dot plots. you can compare the dot plots for each distribution. The mean height of the men’s team is 77.2. Using the data for the boy’s team, you can find the mean: 60 1 60 1 61 1 62 1 62 1 63 1 63 1 64 1 65 1 66 1 66 1 67 1 67 1 67 1 69 5 64.1 15 ···························································· •Ask, Are the height data of the boys tightly clustered or generally spread out? [The boys’ height data are spread out, ranging from 60 to 69 inches.] You can use vertical segments to mark the mean for each data set. Players’ Heights 68 • Ask, How do the height data for the boys compare to the height data for the men? [The boys’ height data range from 60 to 69 inches and appear to have a mean of about 64 inches. The men’s height data range from 73 to 83 inches and appear to have a mean of about 77 inches.] • Ask, Do the boys’ height data overlap with the men’s height data? How do you know? [No, the data sets do not overlap. The greatest height for the boys is 69 inches; the least height for the men is 73 inches. Caution students not to be fooled by the apparent overlap caused by the dot plots being so close to each other.] SMP Tip: When students describe the centers and spread of data without using a formula, they are reasoning inductively and making reasonable conjectures (SMP 2). Periodically ask students to describe and make conjectures before doing formal calculations. 274 Lesson 28 70 72 74 76 78 Men (in inches) 80 82 84 72 74 76 Players’ Heights 60 266 62 64 66 68 70 Boys (in inches) L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse •What predictions can you make about the dot plots? Explain and justify this prediction. Listen for explanations which include no overlapping of values when all of the data are combined on a single dot plot. •If you made a dot plot of players’ heights for a men’s team from another university, how would it compare to the dot plot for the players’ heights from Chesapeake State? Students will likely say that the data values for both universities will be similar. Some say that the data would overlap if graphed on a single dot plot. L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. Part 2: Guided Instruction Lesson 28 At a Glance Students revisit the problem on page 266 to compare data sets by assessing the degree of overlap by using the mean absolute deviation. Part 2: guided instruction Lesson 28 Connect it now you will use these representations to compare the data sets. 2 What is the difference between the mean height for men on the Chesapeake State team Step By Step •Read Connect It as a class. Point out that the questions refer to the problem on page 266 and also the team data from the beginning of the lesson. Point out that the data points for the men’s team have not changed and still have a mean of 77.2. •As you work through problems 2–4 with the class, have students describe how to solve the problems and share their answers. •For problems 5 and 6, use the dot plot on page 266 to help students measure the distance between the mean heights. Ask, How can the number of MAD segments between the two means be determined? [find the difference between the two means and then divide by the MAD] •Read Try It as a class. Have students explain how they got their answers to the class. and the mean height for boys on the middle school team? compare to 2.4, the mean absolute deviation for men’s height? 4.1 1 4.1 1 3.1 1 2.1 1 2.1 1 1.1 1 1.1 1 0.1 1 0.9 1 1.9 1 1.9 1 2.9 1 2.9 1 2.9 1 4.9 5 2.4; 15 ····································································· it is about the same as the MaD for the men. 4 By what number would you have to multiply the MAD to get the difference between the mean heights you found in problem 1? Round your answer to the nearest tenth. 13.1 4 2.4 < 5.458; about 5.5 inches 5 What would the dot plot look like if you combined the data from boys at Central Middle School with the data from men at Chesapeake State to make one big dot plot? Possible answer: there wouldn’t be any overlap. 6 You would have to multiply the MAD by more than 3 to get the difference between the means in these distributions. When the means of distributions are more than 3 MADs apart, do you expect them to have a lot of values in common? Why or or why not? Possible answer: no, i would not expect many common values. as data sets are more MaDs apart, there is less overlap. try it use what you just learned about comparing distributions to solve this problem. 7 The Central Middle School girls’ basketball team has a mean height of 62.3 inches, and a MAD of 2.3 (the same MAD as the Chesapeake University women’s team). How many MADs greater is the mean height for the Chesapeake University women’s team? (Remember: the mean height for the women was 71.5 inches). Possible answer: 71.5 2 62.3 5 9.2 inches difference between the means. if you divide 9.2 by 2.3, you get 4. so, the mean heights are 4 MaDs apart. L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. Hands-On Activity Calculate means and MADs to compare data. Materials: dot plots from Hands-On Activity page 272 •Have students work in the same pairs they did earlier. •Ask the pairs to determine the mean of each data set, the MAD of each data set, and the difference of the means using MAD. 77.2 2 64.1 5 13.1 3 Compute the mean absolute deviation for boys’ height, to the nearest tenth. How does it 267 Try It Solution 7 Solution: 4 MADs apart; Students may determine the difference of the mean heights by calculating; 71.5 2 62.3 5 9.2 inches. Divide 9.2 by 2.3 5 4. So, the mean heights are 4 MADS apart. ERROR ALERT: Students who wrote 27.1 MADs did not determine the difference between the mean heights. •Have the pairs describe in writing the meaning of the differences in the means using MAD. •Have students add data points to the data sets to change the outcome of the differences in the means using the MAD. •Discuss everyone’s results and analyses as a class. L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. 275 Part 3: Guided Practice Part 3: guided Practice Lesson 28 Lesson 28 read the situation described below. then solve problems 8–10. StudentModel The student divided the difference in the means by the mean absolute deviation to solve the problem. Sara was curious about how many text messages students with cell phones send each day. She surveyed a random sample of students who own cell phones at the middle school and another at the high school. For the middle school data set, the mean number of texts per day was 60. For the high school data set, the mean number was 76. The MADs for the data sets were the same. Both MADs were 5. How does the difference of the means compare to the mean absolute deviations? Look at how you can use the information in the problem to answer the question. Pair/share Why is it good to compare data sets using MADs along with looking at the differences between the means of the data sets? What does it mean when the MADs of two sets of data are about the same? the difference in the means is 16. Part 3: guided Practice 9 Do you think that the two data sets described on the previous page have a lot of values in common? Why or why not? Show your work. values in common. When data sets are more than 3 MaDs apart, they don’t usually overlap much. 10 Which pair of data sets is most likely to have the greatest number of values in common? a Data set 1: mean 5 7 Data set 2: mean 5 15; MAD for both data sets is 8 b Data set 1: mean 5 7 Data set 2: mean 5 15 MAD for both data sets is 4 C Data set 1: mean 5 10 Data set 2: mean 5 18 MAD for both data sets is 2 D Data set 1: mean 5 10 Data set 2: mean 5 15 MAD for both data sets is 1 the mean absolute deviations. 8 How would the dot plots of the data described in the student model differ? How might they be the same? Explain. Show your work. What information tells me whether the data sets have a lot of values in common? Solution: Possible answer: no, i don’t think there are a lot of 16 4 5 5 3.2 Solution: the difference of the means is a little more than 3 times Lesson 28 Pair/share Why do data sets usually have a lot of shared values when their means are one MAD or less apart? What values should I be comparing? Bryce chose D as the correct answer. How did he get that answer? bryce probably chose D because the difference between the means is the lowest. but bryce probably didn’t see that the means for choice D are 5 MaDs apart. the means for choice a are Pair/share What are some kinds of data sets that you might want to compare that would NOT have similar MADs? 268 Solution: one dot plot would center around 60, the other would center around 76. since the MaDs are the same, the only 1 MaD apart, so there is a better chance the two data sets will have more overlap. spread of the data would be similar. L28: Using Mean and Mean Absolute Deviation to Compare Data Pair/share How would you help Bryce understand his error? L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. ©Curriculum Associates, LLC Copying is not permitted. 269 At a Glance Solutions Students will compare data sets by assessing the overlap of two numerical data distributions. Students will also measure the difference between the centers of data sets by expressing it as a multiple of a measure of variability. Ex Solution: Find the difference of the means and divide that difference by the MAD. Step By Step 9 Solution: No; When data sets are more than 3 MADs apart, the overlap is minimal. •Ask students to solve the problems individually by using mean absolute deviation to compare data sets. •When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group. 276 8 Solution: One dot plot would center around 60, the other around 76; The MADs are the same, so the spread of the data would be similar. 10 Solution A; The difference between the means is the same as the MAD for each distribution, so there are most likely many data values in common. Explain to students why the other two answer choices are not correct: B is not correct because MADs of length 4 would fit into the difference of the means twice, which is not as good as the overlap in A. C is not correct because MADs of length 2 would fit into this difference of the means 4 times. Data points would not. L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. Part 4: Common Core Practice Part 4: Common Core Practice Lesson 28 Lesson 28 Part 4: Common Core Practice Lesson 28 answer Form 1 A B C D 2 A B C D 3 A B C D Solve the problems. Mark your answers to problems 1–3 on the Answer Form to the right. Be sure to show your work. 1 2 number Correct 3 3 The mean absolute deviation is best described as: A The value obtained when you add all the values in a data set and divide by the number of values. Two data distributions with similar MADs and many values in common often have which of the following? B The value obtained when averaging the distances between each data point in a distribution and the mean. A small MADs C The difference between the means of two similar distributions. B large MADs D An approximation of the center of a statistical distribution. C means that are many MADs apart from one another D means that are not many MADs apart from one another 4 Choose the data set with the greatest MAD. A mean = 81.9389 72 74 76 78 80 Height (in inches) 82 Make up two data sets. List all the values in each data set and write a story to describe where they may have originated. The data sets should meet the following conditions: • The means should be different. • The MADs should be similar. • The means should be more than one MAD apart. Be sure to show work to demonstrate that your data sets meet the above conditions. 84 Check to see that students’ work meets each of the conditions described above. mean = 76.8109 B 72 74 76 78 80 Height (in inches) 82 84 mean = 76.9897 C 72 74 76 78 80 Height (in inches) 82 84 mean = 77.3007 D 72 74 76 78 80 Height (in inches) 82 84 self Check Go back and see what you can check off on the Self Check on page 247. 270 L28: Using Mean and Mean Absolute Deviation to Compare Data L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. ©Curriculum Associates, LLC Copying is not permitted. 271 At a Glance Solutions Students solve problems that might appear on a mathematics test. They compare data sets by assessing the degree of overlap of two numerical data distributions with similar variabilities, and measure the difference between the centers by expressing it as a multiple of a measure of variability. 1 Solution: D; A lot of values in common means the overlapping of data values and mean absolute deviations will have little difference. Step By Step •First tell students that they will solve problems about data sets and mean absolute deviation. Then have students read the directions and answer the questions independently. Remind students to fill in the correct answer choices on the Answer Form. 2 Solution: D; Students may recognize the spread of the data horizontally. 3 Solution: B; Students identify the definition of mean absolute deviation. 4 Solution: Check to see that students’ work meets each of the conditions described. •After students have completed the Common Core Practice problems, review and discuss correct answers. Have students record the number of correct answers in the box provided. L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted. 277 Differentiated Instruction Lesson 28 Assessment and Remediation •Ask students to compare these data sets to determine whether the distributions overlap. Have them show their work and write a few sentences to explain. Round answers to the nearest whole number. Set A: 58, 38, 54, 48, 26, 36 Set B: 58, 42, 64, 62, 70, 40 [Means: 43, 56; MADs: 10, 10. The MADs are similar, which means the spread of data is similar. The difference of the means is 13, and the MADs indicate a good chance that the two data sets will have some overlap.] •For students who are struggling, use the chart below to guide remediation. •After providing remediation, check students’ understanding. Ask students to compare data sets by assessing the degree of overlap of these data sets: Set A: 12, 11, 9, 9, 7 Set B: 10, 11, 8, 8, 6 •If a student is still having difficulty, use Ready Instruction, Level 6, Lesson 27. If the error is . . . Students may . . . To remediate . . . MADs of 43 and 56 have confused MADs and the means of the data sets. Review the process of determining MADs. Student will also benefit from using a dot plot to see the locations of MADs and the means. Ask the student to write out the steps for comparing data sets using the mean absolute deviations. MADs of –2 and 0 not understand that they need to take the absolute value of the differences and then find the average. Work with students to build two tables showing the MADs as the average distance from the mean. overlapping of data have confused the MAD of the data sets with the distance between the means. Create dot plots of the data. Focus on the number of MADs between the means as the determining factor in overlapping data. Hands-On Activity Challenge Activity Describe variations and overlap in two data sets. Create and compare data sets. Give students the following data sets: Set A: 88, 116, 94, 114, 112, 124 Set B: 100, 127, 105, 120, 130, 135 Have students create a dot plot to represent each set of data. Tell them to calculate the mean of each set and the MADs to compare the sets. Tell students to round their answers to the nearest whole number. Then have students write a few sentences to describe and compare the data sets using mean absolute deviation. Tell students to create two sets of data that satisfy the following conditions: •The mean absolute deviation of the first set is less than the mean absolute deviation of the second set. •The mean of the first set is greater than the mean of the second set. Have students write a few sentences to describe and compare the data sets. Have students exchange their data sets with another student who will determine if the data sets meet the criteria. Means: 108, 120; MADs: 11, 11. The MADs are similar, which means the spread of the data is similar. The difference of the means is 12. The MADs indicate a good chance that the two data sets will have some overlap. 278 L28: Using Mean and Mean Absolute Deviation to Compare Data ©Curriculum Associates, LLC Copying is not permitted.