A growing library of statistical illusions & everyday data mistakes
Data that tells the truth. Conclusions that don't.
Each illusion here is a way for flawless arithmetic to walk you confidently to the wrong answer — explained slowly, with figures you can poke. Alongside them sit hands-on tools and longer articles.
28 ILLUSIONS · 6 INTERACTIVE TOOLS · 6 MECHANISM FAMILIES
How to read the library
Same trap, different costumes
Statistical illusions are not exotic. They decide drug approvals, court verdicts, league tables, hiring rounds and headlines — and they survive because each is built from numbers that are individually, checkably true. The lie lives in the step from number to conclusion.
The library is organised by mechanism rather than by field, because the same trap resurfaces everywhere wearing different clothes: an illusion of aggregation looks the same in a hospital and a ballpark. Each illusion opens with an interactive figure where it happens in front of you — one toggle, two honest readings. Throughout, the colour claret marks the misleading reading, and each one closes with the question that would have saved you, collected at the bottom of this page as a pocket checklist. New illusions are added as they're written; the dashed cards are next in the queue.
The tools
Play with the deceptions yourself
Five hands-on instruments for committing — and catching — statistical sleights of hand. No reading required; just poke.
The Benford Detector
Paste any numbers and see whether their leading digits obey Benford's Law — the fraud-detection rule, and how it gets misused to cry fraud.
INTERACTIVE TOOL↳ playThe P-Hacking Sandbox
Torture a noise dataset through defensible analysis choices until it turns “significant” — then watch the specification curve expose the painted target.
INTERACTIVE TOOL↳ playThe Base-Rate Explorer
Set a disease's rarity and a test's accuracy, then watch 1,000 people sort into true and false positives — and see why a “positive” usually isn't.
INTERACTIVE TOOL↳ playLie With This Chart
Drag a bar chart's y-axis off zero and watch a 3% change become a cliff. Commit the deception yourself — then reveal the honest version.
INTERACTIVE TOOL↳ playThe Spurious Correlation Machine
Generate two completely unrelated random datasets and watch them correlate anyway — a working proof that correlation is cheap, and trends are the trap.
INTERACTIVE TOOL↳ playThe Simpson's Machine
Plot two groups whose trends both point one way, then slide a lurking variable until the single line through all the data points the other way — Simpson's paradox, built and broken by hand.
The library
Ways to be honestly wrong
The full set of illusions, grouped by the mechanism that powers them. Each is a self-contained explainer with an interactive figure.
No illusions match that filter — try a broader word like “average”, “risk” or “bias”.
Simpson's Paradox
A trend that holds in every subgroup reverses the moment the groups are pooled. Berkeley admissions, Ken Ross's ballpark, vaccine tables.
№ 2Lord's Paradox
Two statisticians analyse the same before-and-after data. One finds an effect, one finds none. Both are right — they're asking different questions.
№ 8The Will Rogers Phenomenon
Move one patient between two groups and the average improves in both. Stage migration, and medical progress that exists only on paper.
№ 9The Ecological Fallacy
Rich states vote one way; rich people the other. What's true of groups can be false of every individual inside them.
№ 18Volatility Drag & the Ergodicity Trap
A coin flip with a +5% expected value that bankrupts almost everyone who plays it. When money compounds, the average is a rumour about a luckier you.
№ 20Mean vs Median
When a few values are enormous, the “average” stops describing anyone. Most people earn below average — and the median is the honest typical.
Survivorship Bias
The bombers that came home, the funds that still exist, the founders who get interviewed. The missing data is the data.
№ 4Berkson's Paradox
Filter on two unrelated virtues and they become rivals inside the filtered group. Hospitals, casting calls, your dating shortlist.
№ 12The Winner's Curse & Publication Bias
The significant studies you read overestimate the effects they report — the modest results and quiet replications are still in the drawer.
№ 19The Behaviour Gap
A fund can post a glittering return while its investors lose money — they bought after the rises and sold after the falls. The advertised return is real; nobody earned it.
№ 22Self-Selection Bias
Online polls, star ratings and surveys hear only from the people who chose to answer — the loud, the angry, the delighted. A self-selected sample misrepresents everyone who stayed silent, and a bigger one (Literary Digest, 1936) only looks more authoritative.
The Inspection Paradox
Why your bus is always late, your classes are oversized, and your friends have more friends than you do. Big things catch you.
№ 27Lead-Time & Length-Time Bias
Screening finds cancer earlier — and that alone makes five-year survival soar, even when death comes on the very same day. How the survival clock and the slow-tumour skew let statistics rise while nobody lives longer.
The Base-Rate Fallacy
The test is 99% accurate, the disease is rare, and your positive result is probably a false alarm. Count people, not percentages.
№ 7The Texas Sharpshooter
Test twenty hypotheses and chance hands you a “discovery”. Featuring a dead salmon with opinions about photographs.
№ 16The Monty Hall Problem
Three doors, one car. You pick; the host reveals a goat; should you switch? Yes — and almost everyone's intuition screams no.
№ 10Regression to the Mean
Punishment seems to work and praise seems to backfire — because extreme performances drift back toward average all by themselves.
№ 11The Gambler's Fallacy & the Hot Hand
The roulette wheel owes you nothing; the shooter might owe you something. Telling apart the processes that remember from the ones that can't.
№ 21The Prosecutor's Fallacy
“Only one in a million would match” is not “one in a million chance he’s innocent” — the transposed conditional that helped jail Sally Clark, and how a rare match still leaves a crowd of suspects.
№ 24The Birthday Paradox
In a room of just 23 people, two probably share a birthday — not because the room is special, but because 23 people make 253 pairs. Why coincidences are arithmetic, not miracles.
№ 25Confounding
Ice cream sales and drownings rise together — yet ice cream drowns no one; summer drives both. How a lurking third variable fakes a real, strong correlation with no cause behind it.
№ 26The Law of Small Numbers
The counties with the lowest cancer rates are small and rural — and so are the ones with the highest. Small samples swing to extremes, so the top and bottom of any ranking fill up with the tiniest groups.
№ 28The Conjunction Fallacy
Told that Linda is an activist philosopher, most people rank “bank teller and feminist” above “bank teller” — impossible, since every feminist bank teller is a bank teller. Why a vivid detail feels likelier but can only be rarer.
Goodhart's Law & the McNamara Fallacy
Make a measure a target and it stops measuring; count only the countable and the rest of reality quietly disappears.
№ 23Algorithmic Bias
An algorithm trained on the past acts on it, writing the next record — so bias compounds until a model invents a hotspot out of nothing and calls it objective. Predictive policing, and Amazon’s scrapped hiring AI.
Absolute vs. Relative Risk
“Doubles your risk!” — of what, exactly? A relative change is meaningless until you know the baseline it quietly hides.
№ 14Anscombe's Quartet
Four datasets with identical means, variances, correlations and regression lines — and four completely different stories. The case for plotting.
№ 15The Truncated Axis
A 2% difference drawn as a cliff. Where an axis starts — and where it secretly doesn't — decides what your eye concludes.
The blog
Illusions caught in the wild
Longer articles tracing these traps through real headlines, studies and markets — from bacon scares to fund backtests.
The pocket checklist
The questions that would have saved you
Each illusion is disarmed by a single question, asked before believing the number. The checklist grows with the library — memorise it and you can skip the rest of the site, though you'd miss the salmon.