Measures of Spread and Their Effects Grade Seven


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1 Ohio Standards Connection Data Analysis and Probability Benchmark F Determine and use the range, mean, median and mode to analyze and compare data, and explain what each indicates about the data. Indicator 3 Analyze a set of data by using and comparing combinations of measures of center (mean, mode, median) and measures of spread (range, quartile, interquartile range), and describe how the inclusion or exclusion of outliers affects those measures. Mathematical Processes Benchmarks J. Communicate mathematical thinking to others and analyze the mathematical thinking of others. K. Recognize and use mathematical language and symbols when reading, writing and conversing with others. Lesson Summary: In this lesson, students learn about statistical measures using the heights of the students in the class. They measure and order the heights to find the range, first quartile and third quartile. They learn to calculate the interquartile range by finding the difference of the first quartile and third quartile. Students practice determining the statistics of sets of data and compare the statistics. Estimated Duration: One and onehalf to two hours Commentary: Students have some conception of measures of spread and exploring range from experiences with data in the elementary grades. Students have experience in describing, organizing, representing and analyzing data. This experience progresses over the years from informal descriptions in earlier grades to using technology to analyze and represent in the middle grades. Students begin to use statistical terms in the analysis of the data sets. PreAssessment: Students create concept maps to demonstrate their knowledge and understanding of the concepts related to measures of center (mean, median and mode), measures of spread (range, quartiles, interquartile range, extremes) and outliers. The preassessment also assesses students knowledge of ways to display data (boxandwhisker plot, stemandleaf plot and frequency tables). The second part of this preassessment requires students to discuss their ideas in small groups and develop a group concept map that will scaffold their thinking in preparation for the lesson. Have each student cut out the vocabulary cards from Measures of Spread, Attachment A. Instruct students to create a concept map by grouping cards in ways that seem to go together best and make the most sense. A sample organizer is included on Attachment B. Instructional Tip: A concept map is a special form of diagram for showing connections among concepts. Concept mapping can be used for several purposes: to generate ideas, communicate complex ideas, aid learning by explicitly integrating new and old knowledge and assess understanding or diagnose misunderstanding. Have students glue the cards on paper and give each group a title that best describes its characteristics. As an option, have students complete the activity on paper. 1
2 Create groups of three or four students. Have each student discuss the concept map with other group members. Have students create a small group concept map that reflects the group s thinking and record the group s ideas on chart paper. Have each group present its concept map to the class for discussion. Develop informal definitions for the vocabulary words. Collect individual student concept maps to check for gaps in understanding and misconceptions. Instructional Tips: As the students thinking develops, encourage them to make changes to their group map in another color. Ask students to explain the changes they made to their maps to assess progress. If students show understanding of the relationships and use the vocabulary accurately in their discussion, skip Part One of this lesson. Scoring Guidelines: This is an informal assessment intended to assess students readiness for the lesson by checking their level of knowledge of measures of center, measures of spread and data displays. Check that they have grouped together mean, median and mode as measures of center. This idea is basic to the lesson. Intervention is necessary if they do not understand this idea. Assess students level of knowledge and understanding by checking if they have grouped together range, quartile, lower quartile, upper quartile, interquartile range under measures of spread. Assess students awareness of the different graphical displays that can more clearly represent measures of center and spread. Lack of awareness does not necessarily indicate intervention. Students should provide explanations (verbal or written) for their groupings. Refer to PreAssessment Answer Key, Attachment B, for one type of example. The students concept maps may be less hierarchical and more weblike. PostAssessment: This is a formal paperandpencil assessment that measures students ability to arrange data into a graphical display, identify the median, ranges, quartiles and outliers, and interpret the data. Distribute calculators and Measures of Spread PostAssessment, Attachment E, to each student. Have them complete the assessment individually or in pairs. Collect the postassessment after students have completed it. Evaluate the assessment using Measures of Spread, Attachment E, and Measures of Spread PostAssessment Answer Key, Attachment F. Scoring Guidelines: Evaluate students on the degree to which they demonstrate understanding of the following concepts: Create an appropriate graphical display that best reflects the data. Identify median, range, interquartile range, lower quartile, upper quartile and outliers. Describe the attendance at the movies using the data. 2
3 Instructional Procedures: Part One 1. Divide students into groups of four. Have students measure the heights of group members in centimeters and record each height on an index card. Have students post their index cards on the board in order from shortest to tallest. 2. Have the students determine the range of the heights in this set of data for the whole class. Observe methods students use to find the difference of the shortest and tallest. 3. Tell the students that range is one measure used to analyze and describe the spread of data. Explain that they will explore and identify other measures to describe and analyze the spread of data. 4. Direct the students to find the median height of the students in the class. (This is the middle value of all the heights in the data set. It divides the data set into two halves.) Draw a line on the board to represent the median of the data. If the number of entries is an odd number, then the median is one of the statistics. If the number of entries is an even number, the median will be the average of two statistics. 5. Have the students analyze the lower half of the ordered list of heights to find the median of the lower half. Ask, What is the median of this lower set of numbers? (Answers will depend on the class data. This is the lower quartile.) On the board draw the appropriate mark that separates the lower half into two equal parts known as the first and second quartiles. 6. Direct the students to analyze the upper half of the ordered list of heights to find the median of the upper half. Ask students to identify the median of this upper set of numbers and explain that this median represents the upper quartile. On the board, draw the appropriate mark that separates the upper half into two equal parts known as the third and fourth quartiles. Ask students what part of the data one quartile represents (about 25 percent or a quarter of the data). 7. Direct the students to find the difference between the lower and upper quartiles. (This is the interquartile range.) What percentage of the data falls between the upper and lower quartiles? (about 50 percent of the data). 8. Discuss with students the number of data points (heights) in each quartile. Assist students in realizing that about 50 percent of the data falls within the interquartile range, and about 25 percent of the data is at or below the lower quartile and that about 25 percent is at or above the upper quartile 9. Clarify the statistical measures that describe the spread of the data. Have students write definitions and examples of the measures in their journal. Play a quiz game to help students recall definitions of the words, by asking questions such as, I represent the middle 50 percent of the data. Part Two 10. Distribute Olympia Diving Team Problem, Attachment C, to each student. Pair the students and provide time for them to work on the task. If needed, review the vocabulary and the definitions of the measures from Part One. An answer key is provided on Olympia Diving Team Problem Answer Key, Attachment D. 11. To close the lesson, have students use the information from the Olympia Diving Team Problem, Attachment C, to formulate an argument based on their data. Ask, If you had to choose either Larry or Leslie to be on your team for a competition, whom would you choose? 3
4 Why? Have the students base their responses on the data as given to support their arguments. Differentiated Instructional Support: Instruction is differentiated according to learner needs, to help all learners either meet the intent of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified indicator(s). Allow students to present and support conjectures orally. Have students make multiple representations of the data in the postassessment and then compare the information or statistics revealed by each of the graphs. Provide opportunities to use calculators during the activities to remove computational challenges that may hinder student understanding of the standard. Expect students demonstrating evidence of exceeding expectations to create problems using realworld data. Have them formulate and defend conjectures about the problems. Extensions: Access realworld data from the Internet, newspapers, almanacs and other appropriate resources. Challenge students to create problems using this data and calculate the measures of spread and center and represent the data using appropriate graphs such as a stemandleaf plot or a boxandwhisker plot. Explore outliers in data. Develop an informal definition of an outlier and make some generalizations about an outlier s effect on the statistical measures. Home Connection: Record data on television viewing for two weeks, noting the times of viewing, program(s) viewed and number of commercials, etc. Have students compare their data with the data collected by other classmates. Students should compute measures of center, measures of spread, etc., and decide which data points would be considered outliers. Students can also make data displays choosing appropriate representations. Materials and Resources: The inclusion of a specific resource in any lesson formulated by the Ohio Department of Education should not be interpreted as an endorsement of that particular resource, or any of its contents, by the Ohio Department of Education. The Ohio Department of Education does not endorse any particular resource. The Web addresses listed are for a given site s main page, therefore, it may be necessary to search within that site to find the specific information required for a given lesson. Please note that information published on the Internet changes over time, therefore the links provided may no longer contain the specific information related to a given lesson. Teachers are advised to preview all sites before using them with students. For the teacher: Board, overhead calculator (if available), measuring tape or yardsticks with metric markings (one per group) For the student: Calculators (graphing if available), measuring tape or yardsticks with metric markings (one per group) 4
5 Vocabulary: boxandwhisker plot interquartile range lower extreme lower quartile measures of spread outlier quartile stemandleaf plot upper extreme upper quartile Measures of Spread and Their Effects Grade Seven Technology Connections: Data can be entered into a graphing calculator and boxandwhisker plots can be constructed to examine the data. Access data from the Internet. Make graphical representations using statistical software to understand concepts. Use spreadsheets to develop these concepts. Research Connections: Burke, Jim. Tools for Thought: Graphic Organizers for Your Classroom. Portsmouth, N.H.: Heinemann, Jonassen, D.H., Beissner, K., & Yacci, M.A. Structural knowledge: Techniques for conveying, assessing, and acquiring structural knowledge. Hillsdale, N.J.: Lawrence Erlbaum Associates, Marzano, Robert J., Jane E. Pollock and Debra Pickering. Classroom Instruction that Works: ResearchBased Strategies for Increasing Student Achievement. Alexandria, Va: Association for Supervision and Curriculum Development, General Tips: Have calculators readily available for student use. Attachments: Attachment A, Measures of Spread: PreAssessment Attachment B, Measures of Spread: PreAssessment Answer Key Attachment C, Olympia Diving Team Problem Attachment D, Olympia Diving Team Problem  Answer Key Attachment E, Measures of Spread: PostAssessment Attachment F, Measures of Spread: PostAssessment  Answer Key 5
6 Attachment A Measures of Spread: PreAssessment Name Date Directions: Cut out the cards below. Arrange them into groups based on a commonality. Name each group. Describe why you grouped the words the way you did. Record your ideas on paper. Boxandwhisker plot Interquartile range Lower quartile Mean Measures of center Measures of spread Median Mode Range Upper quartile Stemandleaf plot Frequency table Lower extreme Upper extreme 6
7 Attachment B Measures of Spread: PreAssessment Answer Key 7
8 Attachment C Olympia Diving Team Problem Name Date Directions: Analyze each diver s performance using the data given. Larry and Leslie are on the Olympia Middle School diving team. In a recent competition, they were awarded points by three judges on twelve separate dives. Their total scores for each dive are as follows: Dive Larry s Leslie s Score Score Mean Median Range Lower Quartile Upper Quartile Interquartile Range 1. Calculate the mean, median, range, upper and lower quartiles, and interquartile range for each of their sets of scores. 2. Write a description about the diving performances of Larry and Leslie based on the data above.
9 Attachment C (continued) Olympia Diving Team Problem Name Date 3. a. Who is the more consistent diver? b. Why? c. How could you show that Leslie is more consistent? 4. Organize your data into a graphical display (boxandwhisker plot, stemandleaf plot, frequency table, etc.) that compares the performances of the two divers.
10 Attachment D Olympia Diving Team Problem Answer Key Dive Larry s Leslie s Score Score Mean 22 ~23.4 Median Range Lower Quartile Upper Quartile Interquartile Range A sample response: Leslie s performance is better but less consistent. Leslie s mean is slightly higher and her median (middle) is 3.5 points higher than Larry s. Twentyfive percent of her dives scored at or above 27 while Larry s highest 25 percent were only at or above However, Leslie s scores have a wider overall range, making her less consistent. 3. Sample response: Larry is more consistent because the range of his scores is 16 while Leslie s range is 22. c. If the lower scores for each are removed, Leslie is more consistent. 4.
11 Attachment E Measures of Spread PostAssessment Name Date Directions: Answer each of the questions based on the data given. You may use a calculator if needed. Ages of people at the movies last Saturday afternoon Organize the data into a graphical display (boxandwhisker plot, stemandleaf plot, frequency table, etc.) that best reflects the data. Attach to a separate sheet. 2. What is the range of the data? Show how you found it. 3. Find the lower quartile, upper quartile, median and Interquartile range. Lower quartile: Median: Upper quartile: Interquartile range: 4. Write a description about the attendance at the movie for last Saturday afternoon.
12 Attachment F Measures of Spread PostAssessment Answer Key Ages of people at the movies last Saturday afternoon Organize the data into a graphical display of some kind (boxandwhisker plot, stemandleaf plot, frequency table, etc.) that best reflects the data. Explain why boxandwhisker plot is best here or ask students to do boxandwhisker plot only What is the range of the data? Show how you found it = Find the lower quartile, upper quartile, median and Interquartile range. Lower quartile: 11 Upper quartile: 15.5 Median: 14 Interquartile range: Are there any outlier(s). If so, what are they? Yes; 47 is an outlier 5. Answers will vary. Ideas may include: the 47yearold person is the owner of the movie theater; the older teenagers probably work there; the movie is probably a movie that middle school students would like.
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