Lessons

A percentage change is meaningless without the base it changes.

A single number cannot capture a lopsided distribution.

Individual observations can be messy, skewed, or spiky, yet averages of many independent observations often settle into a predictable bell-shaped pattern.

Large-scale claims are often estimates, so the honest answer includes assumptions and uncertainty.

Past results don't change the odds of an independent event.

The number of possible pairs grows roughly with the square of the group size.

Filling in the last few items of a collection costs far more than the early ones, because duplicates pile up near the end.

Repeated doubling stays small, then explodes past anything you expected.

Record-setting and rock-bottom results usually move toward the average next time, with no cause behind it.

Each extra zero is a ten-fold jump our gut quietly ignores.

Many real-world quantities spread over several orders of magnitude, so smaller first digits appear more often.

How a number is shown steers what people conclude from it.

Average many independent guesses and their random errors offset, leaving an estimate closer to the truth than most individuals.

A metric can be a useful signal until people are rewarded for moving it; then the signal often gets gamed.

Conditional probability is about updating the sample space after you learn something.

More data helps, but random error shrinks slowly: to halve the noise in an average, you usually need four times as many observations.

Compare dangers by a common unit, not by how vivid they are.

Real randomness clumps; an even, tidy spread is the surprise.

Even strong evidence can mislead when the thing you are testing for is very uncommon.

A hidden common cause can make unrelated things move together.

A small p-value says the data would be surprising under a null model, not that the claim is probably true or important.

A small sample can land far from the truth by luck alone, while a larger one averages that luck away. This lesson shows how spread shrinks with the square root of the sample size and why the smallest groups dominate the extremes of any ranking.

Means, variances, correlations, and regression lines can match while the underlying data tell completely different stories.

Expected value ignores how badly the rare outcome would hurt.

An arbitrary starting number drags every estimate toward it.

Your conclusion can be wrong when the data you see excludes important missing cases.

A measured number is usually a best estimate with sampling wobble around it, so honest comparisons need ranges as well as point estimates.