Data Analysis — Mean, Median, and Mode
Warm-up
Write a basketball player's last seven game scores: 12, 15, 11, 14, 38, 13, 12. Ask: "What's a fair single number for how this player usually scores?" Compute the mean together (16.4) and watch the room object: she scored below 16 in six of seven games!
The outlier (38) dragged the mean. Enter the median (13) and mode (12) — three different "centres," each honest about different things. Choosing among them IS the lesson.
Explore
Measure-hunting lab: pairs compute mean, median, and mode for four datasets engineered to disagree — the basketball scores (outlier), shoe sizes (mode matters — a store restocks the most common size, not the average), house prices on a street with one mansion (median is the market truth), and a symmetric quiz-score set (all three agree — the boring case that explains why the disagreement cases matter).
Each dataset closes with the forced choice: "which ONE number would you report, and to whom?" The debate is the deliverable.
Formalize
Formalize the three measures and the outlier's differential impact:
Sensitivity summary: the mean feels every value (outliers included); the median only feels the middle's position (outlier-resistant); the mode only counts repetition. Even-length lists average the two middle values for the median. Range (max − min) rides along as the simplest spread measure.
Practice
Practice: compute all three for two datasets; one even-length median; one "a new value arrives — which measures move?" thought experiment; one measure-choice defence for a landlord's rent report vs a tenant's.
Exit ticket: dataset 3, 5, 5, 8, 29 — all three measures, plus one sentence on which best represents a "typical" value and why. (Mean 10, median 5, mode 5; the 29 makes the mean unrepresentative.)
Exit ticket
Practice: compute all three for two datasets; one even-length median; one "a new value arrives — which measures move?" thought experiment; one measure-choice defence for a landlord's rent report vs a tenant's.
Exit ticket: dataset 3, 5, 5, 8, 29 — all three measures, plus one sentence on which best represents a "typical" value and why. (Mean 10, median 5, mode 5; the 29 makes the mean unrepresentative.)
The data: a small company's salaries (thousands): 38, 42, 45, 45, 48, 52, 160 (the owner).
Step 1: Mean: k. Median: sorted middle = 45k. Mode: 45k. Range: 122k.
Step 2: The rhetoric demonstration: recruiting ad — "average salary \$61k!" (mean, inflated by the owner). Union rep — "typical worker earns \$45k" (median). Both cite the same data truthfully.
Step 3: Which is honest? For "what will I likely earn," the median — six of seven employees earn 52k or less; 61k describes nobody.
Step 4: The transferable defence: whenever a single "average" is quoted about skewed data (incomes, house prices, injury rates), ask for the median and the range before forming an opinion. Today's lesson is consumer protection wearing a math costume.
The situation: after four tests, Dana's mean is 82. What score on test five raises the mean to 85?
Step 1: Convert means to totals (the move that unlocks everything): four tests at mean 82 → total .
Step 2: Five tests at mean 85 needs total .
Step 3: The fifth score: .
Step 4: Reality check: 97 is steep but possible — and the structure explains why: hauling a mean up 3 points across five tests costs points above the old average, and ✓ (a second route to the same answer, revealing the mean as a balance: one test must supply everyone's uplift).
Step 5: The generalization students can carry: means hide totals; totals obey arithmetic. Un-hiding the total turns every mean puzzle into addition.
Two classes' quiz results (out of 10): Class A: 7, 7, 7, 7, 7. Class B: 3, 5, 7, 9, 10… wait — sum check: 34 vs 35. Use B: 3, 5, 7, 9, 11 is out of range… B: 4, 5, 7, 9, 10 → sum 35 ✓.
Step 1: Means: both . Identical single-number summaries.
Step 2: Ranges: A — ; B — . The classes are nothing alike: A is uniform; B spans strugglers to stars.
Step 3: The teaching decisions differ completely: A gets one lesson for all; B needs differentiation. A single centre-number ERASED the difference; the spread restored it.
Step 4: The habit: never report a centre without a spread (mean with range, for now; fancier spreads come in Grade 9+). A summary is a compression, and compression loses data — report enough numbers to keep the story honest.