LESSON PLAN

Central Tendency — Mean, Median, and Mode

Apothem Team

Grade 8 · Data & Probability

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

The teacher "accidentally" reports the class's last quiz: "Average: 71%. So most of you got about 71." Hand out the actual (anonymized) distribution: a cluster in the 60s, a cluster in the 80s, almost nobody near 71. The average described NOBODY.

Grade 8 central tendency is measure-choice as a critical-thinking skill: mean, median, mode — computed fluently, then CHOSEN and DEFENDED, with outliers and skew as the deciding factors.

Explore

The salary-negotiation redux with new muscles: a company's wages (one owner outlier) analyzed three ways, then the follow-up: what happens to each measure when the owner's pay DOUBLES? (Mean leaps; median doesn't twitch; mode sleeps.) Sensitivity, quantified by experiment.

Then the weighted-mean workshop — the Grade 8 upgrade: your course grade when tests count 50%, assignments 30%, participation 20%: $0.5(78) + 0.3(85) + 0.2(95) = 83.5$ . Weights are how the world actually averages (GPAs, indexes, ratings) — equal weighting is just the special case $w_i = \frac{1}{n}$ .

Formalize

Formalize the weighted mean alongside the familiar three:

$\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i} \qquad \text{mean } | \text{ median } | \text{ mode}$

The choice heuristics, posted: skewed data or outliers → median tells the typical story; categorical data → mode is all you have; symmetric numeric data → mean uses every value's information; unequal importance → weighted mean. "Which centre?" is a modelling decision that changes conclusions — treat it like one.

Practice

Practice: compute all three for one skewed dataset and recommend; one weighted GPA computation; one reverse problem (what final-exam score forces the weighted grade to 80?); one media clipping where "average" is doing rhetorical work — identify which measure and whose interest it serves.

Exit ticket: house prices — 300k, 320k, 340k, 360k, 2.1M. Which centre for a buyer's guide, and why? (Median 340k; the mansion drags the mean to ~684k, describing no purchasable house.)

Exit ticket

Exit ticket: house prices — 300k, 320k, 340k, 360k, 2.1M. Which centre for a buyer's guide, and why? (Median 340k; the mansion drags the mean to ~684k, describing no purchasable house.)

TIP The reverse-weighted-mean problem ("what do I need on the final?") is the highest-engagement computation in the entire strand. Use it early and often; it's self-interest doing pedagogy's job.

WORKED EXAMPLES

Example 1 — The grade forecast: weighted mean run backwards

The syllabus: tests 40%, project 25%, homework 15%, final exam 20%. Standing so far: tests 74, project 88, homework 95. What final-exam score yields an 80 course grade?

Step 1: Bank the known contributions: $0.40(74) + 0.25(88) + 0.15(95) = 29.6 + 22 + 14.25 = 65.85$ .

Step 2: The target: $80$ . The final must supply: $80 - 65.85 = 14.15$ points through its 20% weight.

Step 3: Solve: $0.20 f = 14.15 \to f = 70.75$ — a 71% on the final secures the 80.

Step 4: Fully worth asking: what final score is needed for 90? $0.2f = 24.15 \to f = 120.75$ — impossible. The weighted structure CAPS the achievable grade at $65.85 + 20 = 85.85$ ; knowing the ceiling changes how a student spends the last week. Algebra as academic triage.

Example 2 — Two data stories, one number line: mean vs median under skew

The dataset: minutes students spent on last night's homework — 10, 15, 15, 20, 20, 25, 25, 30, 120.

Step 1: Mean: $\frac{280}{9} \approx 31$ minutes. Median: the 5th sorted value = 20. Mode: 15, 20, 25 (tri-modal — barely useful).

Step 2: The story test: "a typical student spent about ___ last night." Thirty-one describes nobody (eight of nine students finished under half an hour); twenty describes the room.

Step 3: Diagnose the culprit: one student's 120 (stuck? distracted? overachieving?) dragged the mean 11 points. The mean's sensitivity is a FEATURE when totals matter (the teacher's total marking load DOES include the 120) and a BUG when typicality matters.

Step 4: Report like a pro: "median 20 min (one student at 120 raised the mean to 31)" — both numbers, the outlier flagged, no lies. Good statistics is full disclosure in one sentence.

Example 3 — The mode's day in court: when it's the ONLY sensible centre

The dataset: shoe sizes sold at a store last week — 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 11.

Step 1: All three centres: mean ≈ 8.3, median 8, mode 8. They agree — boring? No: ask the ordering question.

Step 2: The store restocks ONE size heavily. Mean 8.3 is not a shoe. Median 8 happens to be a shoe, luckily. Mode 8 is THE shoe — the most-sold size, the direct answer to "what will customers most likely ask for?"

Step 3: Now break the agreement: suppose sales were 6, 6, 6, 6, 10, 10, 11, 11 — mode 6, median 8 (not even a sold size pattern-wise), mean 8.25. The mode alone respects that demand is BIMODAL-ish and clustered at 6.

Step 4: The classification insight: for categorical-like decisions (which size, which flavour, which bus time), frequency is the question and the mode is its answer. Centres aren't rivals — they answer different questions, and the question picks its measure. That sentence is the entire strand.

MATERIALS

•Distribution handouts

•Weighted-grade calculators (paper)

•Media clippings

•Practice set (PDF)

WATCH FOR

!Weights ignored (straight-averaging the category scores). The syllabus IS a weight vector; averaging against it changes grades.

!Median believed to require odd counts. Even lists average the middle two.

!"The average person" language accepted uncritically — averages describe datasets, not persons; the 71% quiz demo is the vaccine.