Lesson 14 — Mean, Median, Mode, and Range: Choosing the Best Measure

Summary

The unit capstone: learners compute the mean, median, mode, and range of a data set, then choose the measure that best answers a question. They see how an outlier pulls the mean but not the median or mode, and revisit ratios, signed numbers, equations, graphs, and geometry as they summarize real data and defend a choice.

Objectives

• Summarize a data set with mean, median, mode, and range, and choose the measure that best answers a question. (D05.S4.06.02)
• Recognize the center, spread, and shape of a data set and explain how an outlier affects each. (D05.S4.06.01)

Connection

A class records how many books each learner read this month: most read 2, 3, or 4, but one fast reader finished 20. What is “typical”? The mean (about 4.08) gets pulled up by that single 20 and overstates most learners; the median (3) and mode (3) tell the truer story of “what most did.” Choosing the measure that fits your question is real judgment — a nurse reading a growth chart, a market trader reading prices, a planner reading rainfall — and it is the honest way to let data answer, rather than to make data say what you want.

Materials

• Data summary recording page
• Data set cards
• Counters

Preparation

• Copy the data summary recording page and data set cards (include at least one set with an outlier).
• Have a worked example ready: books read 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 20 → mean ≈ 4.08, median 3, mode 3, range 1 to 20.

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) the mean is the balance point (sum ÷ count); the median is the middle value in order; the mode is the most common value; the range is the spread (largest − smallest); and (2) choosing the best measure depends on the question — the median or mode resists outliers, the mean uses every value. Because this is a capstone of foundational skills, use explicit instruction and worked examples before independent work (Kirschner, Sweller & Clark, 2006). The classic slip is finding the mean of a data set with an outlier and reporting it as “typical” without noticing the pull. Retrieval: this lesson is a spiral review — have learners recall the unit’s ratios, signed numbers, one-step equations, tables and graphs, and area/volume, and notice each one hiding in the data work (ordering the data uses signed-number order; the mean is a kind of “fair share” ratio; spread is range). The egalitarian lens is sharp (docs/philosophy.md §4): a single wealthy person can distort a country’s average income, which is why fair summaries often report the median — choosing the measure is a question of honesty and fairness, not just arithmetic. Note the global point: “averages” and middle values appear in ancient record-keeping from Babylon to India; the modern care with outliers is a refinement of that old habit. Distinguish evidence (the computed measures) from values (which measure is “fairest” for a purpose).

Procedure

1. Gather (5 min). You have spent this unit learning to reason with numbers. Today you put a whole pile of numbers into one honest summary — and choose which number to trust.
2. Meet the four measures (10 min). The mean is the balance point: add all the values and divide by how many. The median is the middle value when they are in order. The mode is the value that appears most. The range is the spread: largest minus smallest.
3. Worked example — compute (10 min). For the books-read data (1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 20): sum = 53, count = 13, so mean = 53 ÷ 13 ≈ 4.08. Put them in order; the 7th value is median = 3. The mode = 3 (appears most). The range = 20 − 1 = 19.
4. Worked example — choose (10 min). The single 20 pulls the mean up to 4.08, but most learners read 3. If your question is “what did a typical learner read?”, the median or mode (3) is the better answer. If your question is “how much reading did the class do in total?”, the mean helps. The best measure depends on the question.
5. Practice with a partner (10 min). Take your data set cards. For each, compute all four measures, then say which measure best answers the question and why. Check each other’s arithmetic and reasoning.
6. Close (10 min). Summarize the unit aloud: ratios kept shares fair; signed numbers placed things above and below zero; variables and equations found unknowns; tables and graphs showed relationships; area, volume, and nets measured shape; and today’s measures summarize data honestly. Choose the measure that answers your question — that is mathematics in service of truth.

Differentiation

• Support: Use small data sets (5–7 values) with no outlier first; find the mean by physically sharing counters evenly and the median by lining up objects. Read each step aloud so learners with low vision can follow by listening and touch.
• Extension: Given a data set with an outlier, compute all four measures, explain exactly how the outlier affects each, and write a one-paragraph summary defending the best measure for a stated purpose.

Assessment

• Formative/Summative (self/peer): Can the learner compute mean, median, mode, and range and choose the measure that best answers a question, with a reason?
• Self-check: The learner asks, “Did I compute all four measures correctly, and did I choose the measure that fits my question — not just the one I like — and say why?”

Home connection

Collect a small data set at home (daily steps, coins saved, hours of sleep over a week) and find its mean, median, mode, and range; tell someone which measure best describes “typical” and why.

Resources

• Mean, median, mode, and range are standard in middle-grades statistics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019). The observation that a single outlier distorts the mean but not the median is a standard statistical fact; the claim that fair summaries often report the median is an egalitarian value with real policy import (docs/philosophy.md §4). The note that averages and middle values appear in ancient Babylonian and Indian record-keeping is documented history.