Concepts

A relative change ('50% more', 'doubles your risk') hides how big the underlying risk actually is. A large relative increase on a tiny base rate is still tiny. Always ask for the absolute numbers — how many in 100 before and after — not just the percentage change.

An initial number — even an arbitrary or irrelevant one — pulls later estimates toward it. People adjust away from the anchor too little, so the starting point quietly shapes the final judgment.

The mean adds everything up and divides, so a few extreme values can drag it far from the typical case. In skewed data the median — the middle value — better represents what is normal. Ask about the shape of the data, not just its average.

The background frequency of something before you consider new evidence. A strong signal can still produce many false alarms when the thing being tested for is rare, because there are many more non-cases than cases. Good reasoning combines the new signal with the base rate.

The central limit theorem says that averages of many independent, similar-sized random influences tend to form a bell-shaped sampling distribution, even when the original data are not bell-shaped. It explains why averages are often more predictable than individual observations.

Collider bias appears when you only look at cases selected by a shared outcome, such as hospital patients or admitted applicants. Conditioning on that shared outcome can create a relationship between causes that were not related in the full population.

Putting different dangers on one common scale — like the micromort, a one-in-a-million chance of death — lets you compare activities honestly, instead of ranking them by how frightening they feel.

The probability of something after new information changes what is still possible. The important move is to update the denominator: cases ruled out by the new information should no longer be counted. Many mistakes happen when people keep using the original odds after the situation has narrowed.

A confidence interval is a range produced by a method that captures the true value a stated share of the time across repeated samples. It describes uncertainty in the estimating method, not a personal guarantee that one already-computed interval contains the truth.

Two things moving together does not mean one causes the other. A third variable (a confounder) can drive both, or the link can be coincidence. Establishing causation needs more than correlation — usually a controlled comparison.

When many independent estimates each carry random error in different directions, averaging them cancels much of the error and leaves the shared signal. Some guesses land too high and some too low, so the over- and under-shoots offset one another in the average. This is why a crowd average — or a repeated measurement — can beat most individual guesses.

The long-run average outcome: each result weighted by how likely it is. It tells you whether a bet is favourable on average, but it ignores how badly a single rare outcome would hurt — which is why people rationally pay to avoid losses they could not absorb.

Quantities that grow by a constant percentage multiply rather than add, so each step is bigger than the last. The total stays deceptively flat for a long time, then erupts. Doubling is the clearest case: every step is as large as everything that came before it combined.

The same true numbers can give very different impressions depending on how they are presented: axis ranges, baselines, absolute versus relative, or wording. Framing changes perception without changing the facts, so check how a number is shown, not just what it is.

Goodhart's law says that when a measure becomes a target, it tends to stop being a good measure. People optimize what is rewarded, so the metric can improve while the thing it was meant to represent gets worse.

Independent events do not influence each other: the probability of the next outcome is the same no matter what came before. A coin, a die, and a roulette wheel have no memory of their past results.

As a sample grows, its average gets closer to the true average and random swings shrink, roughly in proportion to 1 over the square root of the sample size. Small samples can land far from the truth purely by chance, which is why a handful of observations is so easy to misread.

A way of measuring multiplicative growth, where equal distances mean equal ratios instead of equal differences. Moving from 1 to 10 is the same size jump as moving from 10 to 100 because both are times ten. Log scales are useful when quantities span several orders of magnitude.

A margin of error is the plus-or-minus range around a survey estimate caused by sampling uncertainty. If two estimates have overlapping uncertainty ranges, the apparent leader may not be truly ahead.

A factor-of-ten step. We handle small additive differences well but badly underestimate how far apart numbers are when they span many tens — which is why a million and a billion feel similar even though one is a thousand times the other.

A p-value is the probability of seeing data at least this extreme if a specific null model were true. It is not the probability that the claim is false, not the chance the result was random, and not a measure of how large or important the effect is.

When every item can be compared with every other item, the number of possible pairs grows much faster than the number of items. A group of n items has n x (n - 1) / 2 pairs, so each new item connects to everything already there. This is why coincidences become common before any single comparison feels likely.

Publication bias happens when studies with exciting or statistically significant results are more likely to be published, shared, or noticed than studies with null results. The visible literature can then exaggerate how strong an effect really is.

Truly random events clump and streak far more than intuition expects. Runs and clusters are a normal feature of randomness, not proof of a pattern, a cause, or a hot streak. An even, tidy spread is the unlikely outcome.

An extreme measurement is usually a mix of a stable, repeatable part and a one-off lucky part. Because the lucky part rarely shows up the same way twice, the next measurement of the same thing tends to land closer to the average. This is not a force that pulls things toward the middle. It is simply what selecting on luck looks like once you measure again.

A distorted result caused by looking at data that does not represent the population you care about. More data does not fix the problem if the same kinds of cases are still missing. Before trusting a conclusion, ask who had a chance to appear in the sample and who was left out.

A way of sampling where an item's chance of being picked grows with its size or its number of connections, so larger or more-connected items are over-represented. It is why a name occurrence pooled from every friend list looks more popular than a random person: well-connected people appear in many lists, so they get drawn more often. The same effect makes classes feel bigger when you ask students than when you ask the registrar, because more students sit in the large classes.

Standard deviation describes how spread out individual observations are around their mean. It is about variation in the data itself. Standard error is different: it describes how much an estimate such as the sample mean would vary across repeated samples.

Standard error is the typical sampling wobble of an estimate such as a mean. For independent observations, the standard error of a mean shrinks with the square root of the sample size, so cutting the error in half usually takes four times as much data.

Statistical power is the chance that a study will detect an effect of a given size if that effect is real. Low-powered studies can miss real effects, while very large studies can detect effects too small to matter in practice.

Summary statistics compress a dataset into numbers such as the mean, variance, correlation, or regression line. They are useful, but they can hide shape, outliers, clusters, and curved relationships that a graph would reveal immediately.

The honest range left after measurement limits, model choices, and missing information. A single number can be useful as a midpoint, but decisions usually need plausible low and high values too. Good estimates state the assumptions that would move the answer.