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Mean, Median, Mode and Range and the relationship between Median and Mean

Lesson Plan

Mean, Median, Mode and Range and the relationship between Median and Mean

Grade Levels

6th Grade

Course, Subject

Vocabulary

box and whisker plot--(also called a box plot) a histogram-like way of displaying data.  One can create a box and whisker plot by drawing a line at the median.  The ends of the box extend to the point to include all numbers that are not outliers.  Outliers are placed on the "whisker" portion of the plot that extends from the box that groups the concentration of data points together. 

mean--a value that is intermediate between other values, such as an average.

median--the middle value in a set of statistical values that are arranged in ascending or descending order.

mode--the number of numbers that occur most often in a data set. 

range--tells you how spread out the data is

minimum--the least, or smallest value, in a data set

maximum--the greatest, or largest value, in a data set

Objectives

Students will be able to correctly calculate the mean and median when given a data set.

Students will be able to demonstrate how the values of a data set impact the mean and median. 

Students will be able to recognize and explain the relationship between the median and mean. 

Lesson Essential Question(s)

How can we use the mean and median to describe a data set?

Why do we need multiple measures of central tendency?

Duration

1 -- 60 minute lesson

Materials

http://illuminations.nctm.org/ActivityDetail.aspx?ID=160  "Mean and Median."  Illuminations, an affiliate site of Thinkfinity.org. Web. 17 March 2010. http://www.thinkfinity.org 

http://illuminations.nctm.org/LessonDetail.aspx?ID=L452#inst1 "Comparing Properties of the Mean and the Median through the use of Technology."  Illuminations, an affiliate site of Thinkfinity.org.  Web. 17 March 2010. http://www.thinkfinity.org

 

Interpreting a Data Set.doc--Primary Student Worksheet

Additional Data Set Examples and Answers.doc--Provides additional data set examples for teachers to use in the lesson along with correct answers.

Developing a Data Set Number Line.doc--A 0-30 Number Line that can be used for supplemental

 

Suggested Instructional Strategies

Instructional Procedures

By sixth grade, students should be comfortable with the terms mode, mean, median, and range.  This lesson is designed to provide students with a review of these concepts and then an in-depth look at how the overall composition of the data set can impact the calculation of both the mean and median.

1.)  For a warm-up, provide students with simple facts that review their addition, subtraction, multiplication and division.

2.) Lead the students in a discussion about "Why it is important to know how to analyze and interpret data?"  and "How can we do this?"  Answer soliciting student responses, explain to students today that we are going to reviewing several sets of data and identifying different ways that we can extract meaning from the data that is given.

3.)  Distribute the worksheet, Investigating a Data Set.  Use this worksheet to guide student inquiry of mean, median, mode, range. 

4.)  Use additional data set example worksheet to provide students with additional practice as student demonstrate need for practice of concepts.  Students can use the worksheet, Investigating a Data Set as a step by step tool of how to calculate each analysis method.

5.)  After several examples, transition students to begin thinking about how the measures of central tendency (mean, median and mode) are related to each other.  Is there times in a data set when these numbers are very similar and other times when these numbers are quite different?

6.)  Using the activity listed in the Explorations section of the Mean and Median webpage from Illuminations, as a class design the three data sets according to the provided specifications.  As you drag each data point onto the number line, the mean and median are calculated so we can see how the addition of each data point affects these figures and how the overall placement of the point on the number line can have similar affects. 

      The copied text from Mean and Median from Illuminations, for reference:

      Can you create three data sets, all of which have 6 data points, a mean of 50, a median of 50, and meet the following criteria?

  • Set A: Every data point is between 35 and 65.
  • Set B: Every data point is either less than 25 or greater than 75.
  • Set C: The difference between every pair of two consecutive data points is the same.

How are these sets different from one another? How are they alike? Are there other data sets with 6 points, a mean of 50, and a median of 50 that look different from the three you've created?

7.) The last question of the activity, "Are there other sets with 6 points, a mean of 50, and a median of 50 that look different from the three you've created?" is a great way for students to put the class theories that were discussed with the first three sets to the test.  You can clear out the data sets that you created in the applet on the Mean and Median page by clicking the "deactivate" button and then design additional samples that meet these criteria.

8.)  For additional practice in understanding the relationship between mean and median and how the values in a data set impact these measure, use "Comparing Properties of the Mean and the Median through the use of Technology" website found through Illuminations.  This applet work similarly to the "Mean and Median" applet used above, but this time students are also looking at the relationship between the mean and median, in addition the overall composition of the data set. 

9.) Using this information provided under the task heading on the webpage, guide student discussion and interaction with the data set to answer the listed questions, including:

  • Can you find ways to move the data points that keep the median the same but change the mean?


  • Can you find ways to move the data points that keep the mean the same but change the median?


  • How do the mean and median change when you keep the points in the same order but just change their positions on the number line?


  • What happens if you pull some of the data values way off to one extreme or the other extreme?


  • By moving data points, can you construct data sets in which the mean seems to be a typical value but the median is not? Vice versa? For what types of data sets, if any, is the mean not very representative? When is the median not very representative?

10.)  To wrap up this activity, students should be able to answer the questions, "What sorts of changes in a data set make the mean change?" and "What sorts of changes in a data set make the median change?"

11.)  As a culminating activity, have students recall what mean, median, mode, and range are and how they are calculated.  Ask students to think of examples of when they have seen on television or in the newspaper that included references to the mean, median, mode or range of a data set.  If possible, provide examples on hand to show students how understanding how these terms mean can assist them in being a more informed citizens about a variety of topics. 

12.)  Assign homework or additional practice as needed. 

WHERETO

W - How will you help your students to know where they are headed, why they are going there, and what ways they will be evaluated along the way?

Students will understand where they are headed and why they are going there, through a discussion at the beginning of the lesson the direction of the lesson through the questions, "Why it is important to know how to analyze and interpret data?"  and "How can we do this?"  Students will understand how they are assessed throughout the lesson through teacher explanation at different points throughout the lesson.

H - How will you hook and hold students’ interest and enthusiasm through thought-provoking experiences at the beginning of each instructional episode?

Students will be actively engaged throughout the lesson through the motivation provided at the beginning of the lesson, the inquiry based worksheet activity to introduce the concepts and through the technology applets that will engage the class in participatory discussion and manipulation of data points.

E - What experiences will you provide to help students make their understandings real and equip all learners for success throughout your course or unit?

Students will experience how central tendency changes as the data set changes and recognize the importance of interpreting data sets in a variety of ways and recognize how data can be found in many real world situations.

R - How will you cause students to reflect, revisit, revise, and rethink?

The second half of this lesson is designed to force students to reflect upon what they know about mean and median and make predictions about their relationship.  Students will then test their predictions and their understanding and revise and rethink to develop sound conclusions about the role of mean and median.

E - How will students express their understandings and engage in meaningful self-evaluation?

Students will engage in self-evaluation when they complete guided practice problems after the inquiry based worksheet is completed and at the conclusion of the applet activities when students are asked "What sorts of changes in a data set make the mean change?" and "What sorts of changes in a data set make the median change?"

T - How will you tailor (differentiate) your instruction to address the unique strengths and needs of every learner?

Practice problems can be adapted to meet student needs.  Worksheets can be adapted to provide additional support to students if needed.  The number line worksheet and small objects such as counters could be used for the more tactile learners while students could also physically get up and move and create their number line that way, responding to the needs of the kinesthetic learner.  For students who demonstrate a high understanding of these concepts, challenge them to investigate the relationship between mode and mean.

O - How will you organize learning experiences so that students move from teacher-guided and concrete activities to independent applications that emphasize growing conceptual understandings as opposed to superficial coverage?

The lesson is structured so that students are introduced to a concept by the teacher and provided with the needed support to develop understanding and then students are provided with the opportunity to work independently or in small groups to further master content and skills.  The teacher should ensure that students demonstrate mastery of one skill before advancing to another topic of more in-depth nature. 

Formative Assessment

1.  The teacher will assess the student's understanding of the concepts and vocabulary throughout the lesson through effective questioning and monitoring of the student work.  This will be extremely important when students are given additional examples to work through independently to demonstrate their mastery prior to advancing to the more in-depth, higher level analysis portion of the lesson. 

2.  The teacher will evaluate student understanding of the relationship between mean and median through their responses to questions and their development of data sets as a class through the simulation or with paper and pencil. 

3. If provided to students, the teacher will also evaluate student work from additional examples and/or a related homework assignment. 

Related Materials & Resources

http://illuminations.nctm.org/LessonDetail.aspx?ID=L297  "Dealing with Data in the Elementary School." Illuminations, an affliate site of Thinkfinity.org. Web. 17 March 2010 http://www.thinkfinity.org

http://illuminations.nctm.org/LessonDetail.aspx?ID=L356 "Eat Your Veggies."  Illuminations, an affliate site of Thinkfinity.org.  Web. 17 March 2010. http://www.thinkfinity.org/

 

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Date Published

March 03, 2010
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