Irving Lubliner's (used with permission) Mean Median Mode

The “Mean, Median, Mode, and Range Challenges”

Grade level: 5-12

Core and Content Standard:

Fifth Grade: MA.05:SP:01: Compare two related sets of data using measures of center (mean, median, and mode) and spread (range).

Seventh Grade: MA.07.SP.01: Find, use, and interpret measures of center and spread, including mean and interquartile range for given or derived data.

CIM (High School): MA.CM.SP.01: Estimate from a graph or a set of data the mean and standard deviation of a normal distribution and draw conclusions about the distribution of data using measures of center and spread (e.g., analyze a variety of summary statistics and graphical displays).

Is this activity your original idea? Yes, it is.

Reference any other source(s) you used: none

Submitted by: Irving Lubliner

Email: irving.lubliner@sou.edu

Address: Home:

1351 Ponderosa Drive

Ashland, OR 97520

School:

Mathematics Department

Southern Oregon University

1250 Siskiyou Blvd.

Ashland, OR 97520

Phone: Home:

(541) 488-8225

School:

(541) 552-6142

Lesson Goal:

Students will use and master the concepts related to measures of central tendency.

Preparation:

• • Classroom time needed to complete: This activity can easily be broken up into “bite-size” pieces to serve as warm-ups or end-of-class challenges. It can also serve as a whole-class lesson or a “Problem of the Week.”

  • Materials needed: paper and pencil

Launch:

• • What background or skill is needed? a basic understanding of mean, median, mode, and range

  • How should the class be organized? Students may work individually, in pairs, or in small groups.

  • Describe the lesson/activity: Consider the set of numbers {2, 3, 4, 8, 8}. For this particular set, we see that the mean is 5, the median is 4, the mode is 8, and the range is 7. Thus, we can write the following statement: the median < the mean < the range < the mode. For each of the following statements, can you find a set of numbers for which the statement is true? As an additional challenge, can you do this in such a way that the mean, median, mode, and range are all whole numbers?

• 1. the median < the mean < the range < the mode (this was shown above)

  2. the median < the mean < the mode < the range

  3. the median < the range < the mean < the mode

  4. the median < the range < the mode < the mean

  5. the median < the mode < the mean < the range

  6. the median < the mode < the range < the mean

  7. the mean < the median < the mode < the range

  8. the mean < the median < the range < the mode

  9. the mean < the mode < the median < the range

  10. the mean < the mode < the range < the median

  11. the mean < the range < the median < the mode

  12. the mean < the range < the mode < the median

  13. the range < the median < the mean < the mode

  14. the range < the median < the mode < the mean

  15. the range < the mean < the median < the mode

  16. the range < the mean < the mode < the median

  17. the range < the mode < the median < the mean

  18. the range < the mode < the mean < the median

  19. the mode < the median < the mean < the range

  20. the mode < the median < the range < the mean

  21. the mode < the mean < the median < the range

  22. the mode < the mean < the range < the median

  23. the mode < the range < the median < the mean

  24. the mode < the range < the mean < the median

Explore:

• • What are good questions to prompt, probe, and encourage? What do mean, median, mode, and range tell us about a set of data? Why do we use more than one measure of central tendency? Can you make up a data set for which the mean, median, mode, and range are all equal? Can you make up a data set for which the mean, median, mode, and range are all different? Do you think that each of the twenty-four “challenges” is possible?

  • How should students record and report their work? For each of the “challenges,” students should write down an appropriate data set, as well as showing the processes used to determine the mean, median, mode, and range.

Summarize:

• • What questions can be used to connect, extend, and generalize the math concept(s)?

• 1. Did you find that all of the challenges were possible? If not, which ones proved to be impossible?

  2. What did you find most difficult as you tried to determine a data set that met certain conditions?

  3. For each of the twenty-four challenges, consider this question: What is the fewest number of elements that can be in a data set that meets the given criteria?

  4. Can you make up a data set for which the mean, median, mode, and range form an arithmetic sequence?

  5. Can you make up a data set for which the mean, median, mode, and range are all consecutive integers?

  6. Is it possible for the mean, median, mode, range, and standard deviation to all be equal?

  7. What if we include standard deviation as one of the descriptors of our data set? For example, can you find a data set that satisfies these conditions?

the mean < the median < the mode < the standard deviation < the range

• 1. If this set of challenges were being prepared for next year’s students, what additional challenges could you add to the list?

• • How will the ideas be applied after the summary? Students will use the various measures of central tendency to describe given data sets.