Lecture Notes — Lecture 28: Summaries That Don't Mislead

Companion to the lecture plan and the worked examples. Read the Definitions before class, the Explanation after it; the practice questions and the lecture quiz close the loop.

Definitions

• Mean — the total divided by the count; sensitive to every value, especially extremes
• Median — the middle value of the sorted data; robust to extremes
• Skew — asymmetry in the data's shape; the mean-median gap is its alarm
• Percentage points — the arithmetic difference of two percentages
• Relative change — the difference divided by the starting value
• Honest summary — a claim plus its denominator, its n, and the summary choice it depends on

Explanation

A summary is an argument, and it can mislead without one false digit. Mean versus median: the mean answers 'total cost per person', the median answers 'the middle experience' - and in skewed data (salaries, waiting times) they diverge enormously. The gap itself is the skew alarm; report both or report the median. Percentages carry two units that get swapped in the wild: percentage points (10% to 8% is -2 points) versus relative change (which is -20%). Compounded changes multiply on shifting bases: up 20% then down 20% lands at 96, not 100. The honest summary bundles the number with its denominator, its n, and the alternative summary that would tell a different story. The discipline: before trusting any chart, ask what would make this look better - then look for that choice in the fine print.

Examples

The lecture's worked examples live in examples.md; the case studies this lecture builds on are worked in class from the projector. The case sequence with difficulties:

• cs-109 (Beginner)
• cs-110 (Foundational)
• cs-111 (Intermediate)
• cs-112 (Advanced)

Common misconceptions

• Quoting the mean of skewed data as 'the average' without the median beside it.
• Swapping percentage points and percent - churn 'down 2 points' is 'down 20 percent', not the same claim.
• Adding percentage changes across shifting bases - up 20% then down 20% is a net 4% loss.
• Believing a chart cannot lie if the numbers are true - truncated axes and chosen baselines mislead.
• Reporting an n alongside a percentage is optional - without n, the reader cannot judge stability.

Summary and key takeaways

1. Mean answers 'total per person'; median answers 'the middle experience' - report both under skew.
2. The mean-median gap is the skew alarm; the tail fraction is the experience of the unlucky.
3. Percentage points and percent are different units; name the base every time.
4. Compounding multiplies on shifting bases - +20% then -20% is a 4% loss.
5. An honest summary names its denominator, its n, and the choice it depends on.

Practice questions

• Compute mean and median for 5 salaries with one extreme; write the one-sentence honest summary.
• Restate 'churn fell from 10% to 8%' in both units; which makes the team look better, and to whom?
• Compound: price +15% then -15%; then -15% then +15%. Explain why order does not matter here.
• Enrichment: find a real-feeling dataset where median and mean tell opposite stories; write both headlines.
• Enrichment: critique a truncated-axis bar chart (drawn by hand) and redraw it honestly.

Where this leads

Next lecture: Simulation as a Way of Knowing. The quiz below checks this lecture's essentials before we build on them.