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The Three M's of Statistics: Mode, Median, Mean
Lesson Question:How can students use the Visual Thesaurus to learn more about mode, median, and mean, and to apply their knowledge in solving some basic statistical math problems?
Lesson Overview:In this lesson, students use the Visual Thesaurus to define the statistical concepts of mode, median, and mean, and then analyze a set of class data using the three statistical concepts.
Length of Lesson:One hour to one hour and a half
- use the Visual Thesaurus to define mode, median, and mean in a statistical context
- use mnemonic devices to memorize the concepts of mode, median, and mean
- analyze statistical mode, median, and mean problems in a large group setting
- independently solve mode, median, and mean problems
- student notebooks
- white board
- computers with Internet access
Analyzing a statistical mode problem:
- Distribute blank index cards to the class and ask each student to write his or her shoe size on a card.
- Have a volunteer student collect the index cards and sort them into piles of alike sizes (i.e. , a pile of 2's, 3's, 4's, 5's etc. ).
- Count aloud the number of cards in each pile and draw a simple chart on the board to record the different shoe sizes and how many students wear shoes of each of the sizes.
- Ask students to imagine that their sizes are representative of the general population of kids. How could this data help a shoe store owner decide how to stock their shelves?
- Elicit students' responses, and establish that a store owner would need to order more shoes of the most popular size and fewer shoes of the other sizes.
Introducing the statistical concept of mode:
Display the Visual Thesaurus word map for mode, pointing out the definition "the most frequent value..." and explaining that the most popular shoe size of the class would be called the mode.
One way to remember that the mode is the "most popular" value is that mode and most both begin with the letter combination "m-o. " Or, you could point out another meaning of mode relates to fashion (as in "a la mode") since students may associate both meanings as they relate to the concept of "popular. "
Present students with the following problem to solve:
- Q: How can Theo use the concept of "mode" to prove that he is a valuable member of his basketball team? Here are Theo's point totals from the nine games in which he played: 6, 8, 14, 12, 11, 14, 4, 6, 14.
- [A: Since mode is the most popular or frequently occurring value among a set of numbers, Theo can use the modal value of 14 points to show off his basketball scoring record. ]
Introducing the statistical concept of median:
Display the Visual Thesaurus word map for median, pointing out the definition "the value below which 50% of the cases fall" and explaining that the median value in a set of numbers is the number in the middle.
One way to remember that the median value is the number in the middle of a set of numbers is to have students picture the median strip of land (or island) that is located in the middle of a large road or highway, dividing the lanes of opposing traffic.
Collectively solve a median problem:
- Choose something that each student in the class can individually count—such as a number of buttons they are wearing, number of pencils or markers in their desks, number of pages in a book they are reading, etc.
- Direct students to line up in order of least to greatest, depending on the number they counted.
- Explain to students that in order to find the median value in the group, the pairs of students at the beginning and end of the line will have to keep sitting down until only one or two students remain standing (depending on whether or not there was originally an odd or even number of students).
- If there is one student remaining, he or she represents the median value. If there are two students remaining, add their two number values and divide by two to determine the median.
Introducing the statistical concept of mean:
Display the Visual Thesaurus word map for mean, pointing out the definition "an average of n numbers..." and explaining that most people use the verb form of "to average" to describe how to find the mean value (i.e., total all the numbers and then divide by how many numbers there are).
One way to remember that the mean is the average of a set of numbers is to associate the two words in a sentence, such as "It's mean to call someone 'average'. "
Present students with the following problem to solve:
- Q: A teacher told the class that he would use the mean value of students' test scores to determine their final grades. If Mary scored 90, 85, 80, 85, and 100 on her exams, what will be her final grade?
- [A: Adding the test scores totals 440. 440 divided by 5 equals 88—Mary's final grade. ]
Solving mode, median, and mean problems independently:
- For homework or as an in-class review, have students review the concepts of mode, median, and mean independently.
- Provide students with one set of data (a set of numbers) and have them determine the three values (i.e., mode, median, and mean) for the same set of numbers independently.
- Invite three students to the front of the classroom to share the mode, median, and mean values through oral explanations that include their problem-solving thought processes.
Extending the Lesson:
- To further challenge students, you could ask students to think of "real life" situations where mode, median, and mean values are helpful. Students could also interview adults outside the classroom to see how they use these statistical concepts.
- Check whether or not students accurately completely the three mode, median, and mean problems in a group setting.
- Assess students' comprehension of the three concepts by checking their independent "wrap-up" work. Make sure that students have shown their work in calculating the three values accurately.
List of Benchmarks for Mathematics
Standard 6. Understands and applies basic and advanced concepts of statistics and data analysis
Level II (Grades 3-5)
1. Understands that data represent specific pieces of information about real-world objects or activities
2. Understands that spreading data out on a number line helps to see what the extremes are, where the data points pile up, and where the gaps are
3. Understands that a summary of data should include where the middle is and how much spread there is around it
4. Organizes and displays data in simple bar graphs, pie charts, and line graphs
5. Reads and interprets simple bar graphs, pie charts, and line graphs
6. Understands that data come in many different forms and that collecting, organizing, and displaying data can be done in many ways
7. Understands the basic concept of a sample (e.g. , a large sample leads to more reliable information; a small part of something may have unique characteristics but not be an accurate representation of the whole)
Level III (Grades 6-8)
1. Understands basic characteristics of measures of central tendency (i.e. , mean, mode, median)
2. Understands basic characteristics of frequency and distribution (e.g. , range, varying rates of change, gaps, clusters)
3. Understands the basic concepts of center and dispersion of data
4. Reads and interprets data in charts, tables, and plots (e.g. , stem-and-leaf, box-and-whiskers, scatter)
5. Uses data and statistical measures for a variety of purposes (e.g. , formulating hypotheses, making predictions, testing conjectures)
6. Organizes and displays data using tables, graphs (e.g. , line, circle, bar), frequency distributions, and plots (e.g. , stem-and-leaf, box-and-whiskers, scatter)
7. Understands faulty arguments, common errors, and misleading presentations of data
8. Understands that the same set of data can be represented using a variety of tables, graphs, and symbols and that different modes of representation often convey different messages (e.g. , variation in scale can alter a visual message)
9. Understands the basic concept of outliers
10. Understands basic concepts about how samples are chosen (e.g. , random samples, bias in sampling procedures, limited samples, sampling error)