When the average is a trap
2026-08-05 · Self-organisation and criticality
You can compute a country's average height, and you can compute the average energy released by an earthquake. The first number means something. The second does not merely mislead — it does not exist. Not for want of data. It does not exist.
You are doing something entirely ordinary: taking an average. How many copies a book sells. How many households a blackout affects. How much revenue a new user brings in. Pull the history, divide, get a number, plan around it.
For heights, weights and commute times this is fine. For those three it walks you into a ditch — and into the particular kind of ditch where trying harder makes it worse: collect another year of data and the number will not settle down, it may double. Collect another year and it may go back.
The problem is not sample size. The problem is that "take an average" quietly assumes something nobody ever told you — that a typical value exists. For a large class of systems it does not, and that class happens to contain the things you most want to predict: crashes, breakout hits, accidents, epidemics, fortunes.
The issues in this phase each ask a different question about criticality. Topic 16 asked why microscopic detail stops mattering at the critical point. Topic 17 asked how connectivity suddenly spans a system. Topic 18 asked who tuned the system to criticality. This one is not about a mechanism but about the output — what the statistical signature these systems emit actually looks like, and which of your habitual moves it cancels.
Look at the thing before naming it.
Below are two sets of one hundred numbers whose totals are identical. Think of them as two ways of splitting the same sum of money, the same pool of traffic, the same audience. Both are sorted largest-first and drawn as bars against the same vertical scale.
On the left, "the average is 6.7" is a sentence with content. Nearly every bar sits close to it and the tallest is only 1.6 times as high. Tell someone the average and the picture in their head is roughly right.
On the right the same sentence carries almost nothing. Seventy-one of the hundred numbers fall below the mean. The median is 3.1, less than half the mean, while the largest is twelve times the mean and the top ten together take 44% of everything. "Average 6.7" describes no actual bar: most are far below it, a few are far above, and the region around it is the emptiest part of the distribution.
The shape on the right is a power law. Its definition can be stated plainly: the number of things exceeding size x is proportional to x raised to some negative power. Written out, P(exceeding x) ∝ x−β, and β is the tail index. Smaller β means a heavier tail and more monsters. → ref · Identifying and Misidentifying Power Laws
That formula comes with an immediately usable reading: put both axes on a scale where each step multiplies by ten — log-log axes — and a power law is a straight line whose slope is −β. A bell-shaped normal distribution on the same paper curves down and hits the floor: the number of events it predicts beyond five times the mean is essentially zero.
The reason for the two shapes lies in how they are generated. A normal distribution comes from many small factors added together — height is hundreds of genes and nutritional conditions summed, none of them decisive alone. Power laws typically come from multiplication or contagion: a book that already sells well is seen by more people and so sells better still; a grid node that is already connected carries more load and so fails more readily. Addition grinds differences flat. Multiplication amplifies them.
One overused claim deserves handling here. The "80/20 rule" — twenty percent of the items hold eighty percent of the total — is not the definition of a power law. It is what one particular tail index happens to produce: β ≈ 1.16 gives exactly 80/20. Change β and the split changes completely (smaller β might give 90/10, larger β only 60/20). Treating 80/20 as a universal law is treating a parameter as a principle.
Given any set of "size" data — sales, blast radius of an incident, customer value, loss per accident — the first move is not to average it. It is to sort it and compute what fraction of the total the largest 1% holds. If that fraction exceeds 10%, drop the mean from your reporting and report the median together with the top-1% share instead. This step needs no statistics and takes five minutes.
Saying the average "describes nobody" is only a complaint about description. This section makes a harder claim: you may not be able to measure that average at all.
You probably half-believe a rule: more data means more stability. Flip a coin a hundred times and the fraction of heads jumps around; flip it a million times and it sits on 50%. That rule is the law of large numbers, and it underwrites nearly every piece of advice that begins "collect more data".
It has preconditions, and in the heavy-tailed world — the broad class of distributions whose extreme events decay more slowly than exponentially, of which the power law is the cleanest member — those preconditions can fail. The figure below was actually run: draw a hundred thousand random numbers from each of two distributions, recompute the running average after every single draw, and watch where it goes. The vertical axis is the running average divided by the theoretical mean, so 1.0 means converged.
Note the detail that matters about the orange line: it does not wobble, it steps. For twenty thousand draws it sits near 0.7, and had you stopped there you would have believed you had measured it. Then one number arrives and lifts the whole average to 2.9 times the truth in a single step. That number is not an error, not dirty data, not something to strip out — it is what this distribution is supposed to do. And more such jumps are coming, just ever more rarely.
So "collect more data" fails here completely: your average does not creep toward the truth, it sits low for long stretches and is occasionally over-corrected by a single jump. No individual snapshot is trustworthy, and you have no way of knowing where between two jumps you currently stand.
Behind this is a clean mathematical fact worth memorising, because it turns "can this quantity be averaged" into something you can look up. For a power law with tail index β:
The mean exists only if β exceeds 1; the variance — that is, the very concept of "typical fluctuation" — only if β exceeds 2. "Does not exist" does not mean "is very large". It means the integral diverges and there is no limiting value: the sample average you compute is a random number wandering with sample size, converging to nothing.
(A convention trap lurks here: many papers quote the exponent α of the probability density instead, related by α = β + 1, so the same fact is written there as "the mean exists if α > 2, the variance if α > 3". Before using any quoted number, check which convention it uses, or you will be off by exactly one.)
This is not a mathematician's toy. Earthquakes live in the leftmost band. The Gutenberg-Richter law says each unit of magnitude cuts the count roughly tenfold; converted into released energy, the tail index is about 2/3 — less than 1. Which means the question "how much energy does an average earthquake release" has no answer within the model. The only reason a number can be computed in practice is that the Earth is finite and fault lengths are bounded, which chops the tail off. The average you compute is set by where the truncation falls, not by the physics.
Subject any average you plan to decide on to a running-mean test: add the samples one at a time in order, recompute the average after each, and plot the curve. If it lies flat, the mean is usable. If it is still stepping, your average is currently determined by the few largest samples — report the median and quantiles instead. Ten lines of code, and it should come before any formal statistical test.
"Scale-free" is often used as an adjective, as though it were a literary way of saying "big and very uneven". It is in fact a specific, checkable statement.
Start with its opposite. Suppose adult male height is bell-shaped with a mean of 1.75 m and a standard deviation of 7 cm. Then 24% clear 1.80 m, 1.6% clear 1.90 m, 0.018% clear 2.00 m and 0.0000287% clear 2.10 m. Each extra 10 cm cuts the population to 1/15, then 1/90, then 1/619 of what it was. The divisor itself grows fast: harder as you go up, and the difficulty accelerates.
A power law does nothing of the kind. The probability of doubling is independent of where you currently are. Whether a book has sold ten thousand copies or a million, its chance of doubling again is the same number: 2−β. At β = 1.16 that is 45%, always 45%.
That is what scale-free means: no position on the distribution is special. There is no normal range, therefore nothing outside the normal range, therefore no typical scale to anchor on. Switch the horizontal units from copies to tens of thousands of copies and the picture is identical.
This property has a direct and thoroughly counterintuitive consequence. Under a power law, already large is evidence of larger still. Precisely: if you know a quantity has already exceeded x, the expectation of its final value is x times the fixed factor β/(β−1) — the expected value is proportional to the current value. At β = 1.16 that factor is 7.25: a book already at 100,000 copies has an expected final figure of 725,000; one at a million expects 7.25 million. It never tops out.
A normal distribution does the opposite. Given that a man clears 1.90 m, his expected height is 1.925 m; given that he clears 2.00 m, 2.017 m. The conditional expectation hugs the threshold — raise the threshold and the expectation barely moves. This is where regression to the mean lives: the distribution has a centre, and an observation far from it is usually followed by one closer to it.
So "it has run up so far, it is due for a pullback" is not general wisdom. It is a claim with a precondition, and the precondition is that the quantity has a centre. For scale-free quantities it fails, and it fails in the opposite direction.
Classify the quantity before choosing your intuition. Thin-tailed quantities — height, reaction time, daily commute — obey regression to the mean, so expect extremes to be followed by a fallback. Heavy-tailed quantities — sales, reach, loss per incident, revenue per customer — do the reverse: size reached so far is the best available predictor of further growth. Concretely: in your post-mortem template, replace the question "is this an outlier we should exclude" with "which class is this quantity in". The first question guarantees that you throw away your most informative data whenever the tail is heavy.
Power laws are the most abused tool in complexity science, for a simple reason: they look like a straight line, and straight lines make people relax. Four limits before use.
First, "it looks straight" is not evidence. Log-log axes are an extremely forgiving way to draw: they compress the vertical range across several orders of magnitude, and the eye's sensitivity to curvature collapses along with it. A log-normal distribution — what you get when many small factors multiply — plotted on the same paper is nearly inseparable from a power law across two or three decades. In 2009 Clauset, Shalizi and Newman re-examined a batch of widely cited power-law data sets under one uniform statistical procedure; several failed outright, and among those that passed, many could not be distinguished from a log-normal. → ref · Identifying and Misidentifying Power Laws
Second, real tails are always truncated. No book sells to more people than exist; no earthquake exceeds the crust. So a strictly non-existent mean never shows up in real data as infinity. It shows up as the curve in section 02: computable, non-converging, and with a value effectively set by where the truncation falls. That does not weaken the conclusion, it makes it operational — if you must report a mean, report alongside it how long your observation window was and what the largest observed value is.
Third, and most important: the same straight line can be emitted by entirely different mechanisms.
This directly bounds how far the previous issue's conclusion travels. Seeing a power law entitles you to say "some mechanism in this system amplifies across scales". It does not entitle you to conclude "therefore this system is in a self-organised critical state", because proportional growth, preferential attachment and extremal aggregation all emit the same line and none of them involve criticality. Inferring mechanism from shape is inverting a many-to-one map, and that inverse is not unique.
Fourth, heavy-tailed is not the same as power law. This issue uses "heavy tail" for the broad class in which extreme events decay more slowly than exponentially; the power law is only its cleanest member. Log-normal and stretched-exponential distributions are also in the class, yet all their moments exist. So the section-02 results about non-existent means apply only to the power-law branch and cannot be carried wholesale onto "heavy tails". Saying a quantity is heavy-tailed and saying it follows a power law are claims of very different strength.
Never treat "fits a power law" as evidence for a mechanism. If you want to claim that some specific mechanism is at work — rich-get-richer, self-organised criticality, contagion — you must produce another testable consequence of that mechanism: an ordering in time it predicts, an intervention effect it predicts, the shape of some other quantity it predicts. With the distribution shape as your only evidence, the defensible statement is "there is amplification across scales", and you stop there.
Because many questions are about totals, and totals are necessarily set by the tail. An insurer owes the sum of all claims, not the typical claim; a server must absorb the total wait across all requests. On such questions the median is safe nonsense — stable, but answering a different question. The fix is not one substitute number but a set: the median for the typical case, quantiles for the shape of the tail, and the largest observation to record how long you have been watching.
Look for an intrinsic ceiling mechanism. Height is bounded by skeleton and metabolism, exam scores by the maximum mark; such quantities have a centre and regress toward it. Sales, reach and wealth have no such mechanism and do have positive feedback — selling well means being seen means selling better — so the tail stays open. A crude but serviceable test: could this quantity go up tenfold without hitting any physical or institutional wall? If yes, do not assume regression.
There is more to do than in the thin-tailed world, not less. If outcomes are decided by a few draws, then increasing the number of attempts, keeping the attempts genuinely different from one another, and bounding the downside of each becomes more effective than raising the per-attempt success rate. That is why venture funding, drug screening and creative work all take the shape of portfolios — not gambling, but design fitted to the shape of the distribution.
It does, and that often matters more than its level. Markets have different tail shapes in calm and in crisis; a social network's spread distribution changes when the recommender is rewritten. The trouble is that estimating a tail index requires many tail samples, and tail samples are inherently scarce — so "the tail index has shifted" is usually measurable only long after the shift. That is a genuine weakness of this issue's toolkit; do not expect real-time warning from it.