Transparencies and CD Kit. This Really Good Stuff Activity Guide. Mean, Median, Mode, and Range Transparencies and CD Kit

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1 Mean, Median, Mode, and Range Transparencies and CD Kit Congratulations on your purchase of this Really Good Stuff Mean, Median, Mode, and Range Transparencies and CD Kit a series of sequential lessons for introducing the data-analysis concepts of mean, median, mode, range, and outliers. This Really Good Stuff product includes: Finding the Mean Overhead Transparency Finding the Median Overhead Transparency Finding the Mode Overhead Transparency Finding the Range and Outliers Overhead Transparency Mixed Review Overhead Transparency Mean, Median, Mode, and Range Transparencies CD This Really Good Stuff Activity Guide Displaying the Mean, Median, Mode, and Range Transparencies and CD Kit Before utilizing the Mean, Median, Mode and Range Transparencies and CD Kit, make copies of this Really Good Stuff Activity Guide and file the pages for future use. This Activity Guide addresses the use of the Transparencies; however, for all of the activities that follow, you may choose to use the CD and project the images on your classroom whiteboard. Introducing the Finding the Mean Overhead Transparency Explain that finding the mean is one way to find the average. Display the Finding the Mean Overhead Transparency, and obscure the division problem at the bottom of the Transparency with a piece of paper so that students can only see the definition, problem, table, and question. Read through the problem together. Review the content in the table, noting how it reflects the information in the problem. Using a wet erase marker, demonstrate how distributing the pencils across the six columns, or six students, will visually show the average number of pencils that the students in this group have in their desks. Reiterate that when you distribute the total number of pencils evenly across the six columns, the amount of pencils in each column is two. Reveal the algorithm at the bottom of the Transparency. Explain that the steps you just performed on the table are in actuality a division problem. Tell students to look at the numerical data provided in the problem, and ask them how many pencils there are in total. Explain that to find the mean, they will need to find the total, which in this case is 12; and then divide by the number of pieces of data, which in this case is 6. By dividing the 12 (pencils) into 6 equal groups (students/columns) they will find that 2 is the average number of pencils found in the desks of the students in this group thus matching the results of the earlier work done on the table. To review, read the definition at the top of the Transparency together. Introducing the Finding the Median Overhead Transparency Explain to students that another way to analyze a set of data is to find the median. Explain that the median is sometimes used, instead of the mean, to represent a set of data when the data includes All activity guides can be found online: Helping Teachers Make A Difference 2011 Really Good Stuff Made in Guangzhou, China #159225

2 Mean, Median, Mode, and Range Transparencies and CD Kit that the mode of a set of numerical data is the number (or numbers) that occur most often. Indicate that it is often preferable to use the mode when working with a set of categorical data. For example, if you ask a group of students what they would like to have for lunch, and their responses are pizza, a hamburger, pizza, noodles, peanut butter and jelly, and pizza, then pizza would be the mode for this set of data because it is the most commonly occurring response. Explain that people use the mode for categorical data, because there is not a way to find the mean or median. Display the Finding the Mode Overhead Transparency. Explain that when finding the mode for a larger set of data, creating a tally table can help. Show students how they can create a twocolumn table, and record the first piece of data in Column One and a tally mark in Column Two. They should then cross out the first piece of data in the original table to keep track. Explain to students that an alternative way to find the mode is to use the same strategy that they used to find the median. Indicate that they may put the numbers in order to see quickly what number occurs most often. Redisplay the already completed Finding the Mode Overhead Transparency, and use the previous work to model finding the mode for these two problems. (Modes= 51 and 2 snow days) Record the modes on the Transparency, and lead a discussion comparing the newly found modes, with the previously found medians. Note that if there is not a number that occurs more than any other, we conclude that there is no mode for the data. Explain that it is also possible to have more than one mode for a set of data. To review, read the definition at the top of the Finding the Mode Overhead Transparency together. Introducing the Finding the Range and Outliers Overhead Transparency Display the Finding the Range and Outliers Overhead Transparency, obscuring the bottom half of the Transparency. Explain to students that calculating the range of a set of data will tell them how spread out the data is. Tell students that the range is the difference between the lowest and highest values. Read the problem on the Range section of the Transparency together and ask students to identify the least and greatest number in the set of data. Circle the 2 and the 6. Explain that subtracting the 2 from the 6 gives the range of 4 for this set of data. To review, read the definition at the top of the Finding the Range and Outliers Overhead Transparency together. Reveal the Outliers section of the Transparency. Explain to students that when analyzing data, they should look for outliers, which are the unusual numbers that are distant from the other numbers in the set. Explain that outliers can affect the mean, the median, and the range. Read the problem on the Transparency with your class. Draw students attention to the fact that the data in this problem isn t neatly listed for them, and urge them to look carefully for all the data contained within the text. Helping Teachers Make A Difference 2011 Really Good Stuff Made in Guangzhou, China #159225

3 Mean, Median, Mode, and Range Transparencies and CD Kit Explain to students that to find any outliers, they should put their data in order from smallest to largest, crossing out numbers from the problem as they write them down so that they don t repeat any or forget to include any numbers. Indicate that the number 10 is an outlier, because the range of the other numbers in the data set is only 3 (3 0 = 3), and the largest number of 10 is 7 away from the highest number in the data set. Ask students what might explain the 10. Among others, one possible response is that the person who bought 10 candy bars was a family member of Cesar s. To review, read the outliers definition in blue next to the word Outliers together. the data, because it is not representing goals scored for one of the games, but instead represents the number of games, or number of pieces of data. Therefore, the seven is used as the divisor when figuring out the mean. Use the results of the data analysis to lead a discussion in comparing and contrasting the results for the mean, median, mode, range, and outliers. Were the mean, median, and mode very different? Which number best represents the data set? How does the range affect the conclusions that can be drawn from the mean, median, and mode? Introducing the Mixed Review Overhead Transparency Display the Mixed Review Overhead Transparency and review all of the skills learned. Note that the data is not highlighted in red in this problem. Read the first problem, and ask students which numbers in the text should be considered. Indicate to students that the seven shouldn t be considered as part of Allow students the opportunity to collect real data, and use the results for practicing the concepts of mean, median, mode, range, and outliers. Helping Teachers Make A Difference 2011 Really Good Stuff Made in Guangzhou, China #159225

4 Mean, Median, Mode, and Range Transparencies and CD Kit extremely high or extremely low values that might affect the overall mean. Ask students to think about the word median and ask where they ve heard this word before. Students may respond that they know there are sometimes medians on the road. Ask students to describe that kind of median, noting the fact that medians are in the middle of the road. Students may also respond that the word median sounds like medium. Ask students what it would sound like if you were to put the volume of a TV on medium. Would it be very loud? very soft? Reiterate the fact that it would be somewhere in the middle. Explain that when finding the median, they are finding the number in the middle of the data. Display the Finding the Median Overhead Transparency, obscuring the bottom half of the Transparency. Read the first problem together with students. Explain that in order to find the median height of this group, the pieces of data need to be placed in numerical order. Model how you find the smallest number in the problem, cross it out, and write it on the first line. Emphasize that it is important to cross out the piece of data in the problem as it is put in order so that no numbers are missed or used more than once. After all of the heights have been recorded on the lines, indicate that a way to ensure that the middle number is found is to cross out a number from each end, until the middle number is reached. Explain that it is easy to find the median of an odd set of data. Then indicate to students that sometimes they will have to find the median of an even set of numbers. Reveal the second problem on the Finding the Median Overhead Transparency and guide students through the problem. Since this problem contains additional numbers in the text, use this problem to highlight the fact students must carefully read each problem to ensure that they are working with the correct numbers. Circle both of the numeral 10s in the problem; then reread these parts of the question aloud, and indicate that one must be careful not to include numbers that are not actually part of the data collected. Once you ve identified the correct pieces of data, cross out the numbers in the problem as you write them on the lines. Find the median by modeling again how students should cross out the numbers from each end of the set of data until they reach the middle number. Indicate that since the number of pieces of data from this problem are even, they will look at the middle two numbers, and with your wet erase marker, circle the middle two 2s. Explain that since the middle two numbers are the same value, that the number 2 is the median for this set of data. Elaborate further, explaining that if the two middle numbers in an even amount of ordered numbers are different, that they must find the number that would fall in between those two numbers. Explain that finding the mean for these two middle numbers will also give them the median. To review, read the definition at the top of the Transparency together Introducing the Finding the Mode Overhead Transparency Explain to students that mathematicians often look at the mode when analyzing a set of data. Explain Helping Teachers Make A Difference 2011 Really Good Stuff Made in Guangzhou, China #159225

5 Mean, Median, Mode, and Range Poster Congratulations on your purchase of this Really Good Stuff Mean, Median, Mode, and Range Poster a colorful reference poster to help students define and practice calculating mean, median, mode, and range problems, as well as evaluate and defend which of the above best represents a number set. This Really Good Stuff product includes: Mean, Median, Mode, and Range Poster, laminated This Really Good Stuff Activity Guide Introducing the Mean, Median, Mode, and Range Poster Display the poster in a visible area. Begin by pointing out to students that although all of the terms defined on the poster are related, they each have their own specific meaning and use. Point out that what these terms have in common is that they are helpful in finding patterns in number sets, and therefore make the number sets easier to understand. Activity 1: All About Mean Read the definition of mean on the poster and look at the example showing how to calculate the mean number (of a number set). Note that the words mean and average are synonyms. The process of finding the mean of a number set is often used to calculate grades in school because it is a single number that can help show how well a student is doing in a subject. 1. Write the definition of mean: _ 2. Another name for the average of a number set is the. 3. The steps in the process of finding the mean of a number set are: a. all the numbers in the group together. b. Divide the sum by how many numbers were together to get the sum. The resulting number is the, or average. 4. For example, to find the mean of 4, 9, 7, 8, and 10: a. Add the numbers in the group: The sum of the numbers is. b. Divide the sum by 5 because there are numbers in the number set. The resulting answer is, which is the mean, or average, of the group. Practice Problems Show your work, using number 4 above as a model. 1. At the end of the marking period, a student s test scores are 95, 70, 86, 83, and 90. Calculate the mean. 2. A coach needs to find the mean of each player s points per game. Player A scored 6 points in game 1, 15 points in game 2, and 18 points in game 3. Calculate the mean points per game for Player A. 3. Each member of a family has their own collection of CDs. One person has 20, the second has 35, the third has 100, and the fourth has 61. What is the mean, or average, number of CDs in the family? 4. A bookcase has five shelves with 20, 24, 24, 18, and 24 books on them. What is the mean number of books on a shelf in this bookcase? 5. You are shopping with three friends who spend $20, $28, and $15. If you spend $16, what is the mean spending for all of you? Helping Teachers Make A Difference 2007 Really Good Stuff #155874

6 Mean, Median, Mode, and Range Poster Activity 2: All About Median Another way to describe a number set using only one number is to find the median, or the middle number, of a set. Carefully read the definition of median on the poster. The median answers the question, What number is the halfway point of the data? To find the median of a set of numbers: Arrange the numbers in order from smallest to largest. Count the number of numbers in the set. When there is an ODD number of numbers in the set, pick the middle number. When there is an EVEN number of numbers, average the two middle numbers. 1. Write the definition of median: 2. When finding the median, first arrange the numbers in. 3. The median of the set of numbers: 2, 3, 5, 6, 7, 8, 9 is because it is the number in the (there are three numbers in front of it and three numbers behind it). 4. The median of the set of numbers 2, 3, 4, 5, 6, 7, 8, 9 is because it is the of the two middle numbers. Practice Problems 1. Find the median for each set of numbers below. Show your work. 33, 23, 43, 44, 63, 42, 29 92, 84, 87, 86, 77, 2 17, 42, 62, 27, 25, 12 22, 35, 34, 36, 40, 36, 23, Find the mean of each set of numbers above. Show your work on the back of this paper. 3. Is the mean and median the same for each set of numbers? 4. Which problem above has the greatest difference between its mean and median? What might have caused the mean and median to be so different in this number set? Think About It Both the mean and median are attempts to describe a data set using only one number. At times the mean and median are very close. However, when one of the numbers in a set is very different from all the other numbers, the mean is greatly affected; therefore, the median may be a more useful number to describe the data. This very different number is called an outlier. Which number sets in problem 1 above has an outlier? Write another example of a number set with an outlier. Helping Teachers Make A Difference 2007 Really Good Stuff #155874

7 Mean, Median, Mode, and Range Poster Activity 3: All About Mode and Range Mode is another way that can be used to represent a number set with one number. Read the definition of mode on the poster and look at the example showing how to find the mode of a number set. 1. Write the definition of mode: 2. To find the mode of a number set, find the number that occurs most. 3. In the number set 4, 8, 6, 2, 8, 3, the number is the mode because it occurs at least two times or the most often in the number set. 4. Some number sets may have more than one mode. Write a number set with more than one mode: 5. Write a general statement that describes any number set with more than one mode: 6. Some numbers sets have no mode. Write a number set with no mode: 7. Write a general statement that describes any numbers sets with no mode: Range is the number that shows the spread of the numbers in a set. While mean, median, and mode help describe the middle of a data set with a single number, sometimes it is important to tell how spread out the numbers are in a set. Read the definition of range on the poster and look at the examples for finding the range of a number set. 1. Write the definition of range: 2. The steps to find the range are: a. Put numbers in order from smallest to largest b. the smallest number from the largest number 3. Suppose you were keeping track of the number of hours you slept in a week and your data was 8 hours, 6 hours, 7 hours, 8 hours, 8 hours, 5 hours, and 8 hours. a. Write your data set in order b. Subtract the smallest number from the largest number. The number is the range for the number set. A small range number indicates all the numbers in the number set are. Helping Teachers Make A Difference 2007 Really Good Stuff #155874

8 Mean, Median, Mode, and Range Poster Activity 4: Finding the Mean, Median, Mode, and Range 1. The chart below shows the data on the number of hits on a Web site per week. Week Number of Hits a. Calculate the mean, median, mode, and range of the data in the chart above. b. Not all of these numbers are equally helpful in accurately representing a number set. Tell whether the mean, median, mode, or range would be most helpful in predicting the number of hits in the next four weeks. Explain why: c. Tell whether the mean, median, mode, or range would be least helpful in predicting the number of hits in the next four weeks. Explain why: 2. Emerson s math test scores are 87, 93, 92, 25, 96, 88, and 96. a. Calculate the mean, medium, mode, and range for these scores. Show all your work. Use the back of the paper if necessary. b. Which term (mean, median, mode, or range) is the BEST representation of Emerson s math scores? c. Defend your answer: 3. The hourly wage for a group of employees is $5.50, $6.20, $5.50, $5.50, $6.40, $8.00, $16.00, and $6.00. a. Calculate the mean, median, mode, and range of the salaries. b. After you have calculated the values, take EACH of values above and write an explanation of why you think it is or is not a good descriptor of the number set: 4. Challenge yourself. a. Your test scores are 90, 85, and 88. What score do you need on the next test to raise your average to 90? b. Which of the terms (mean, median, mode, or range) could be used to show a pattern in a group of non-numeric data? Give an example of a non-numeric data set. Helping Teachers Make A Difference 2007 Really Good Stuff #155874