lucid.rodeo · incerto · mean

The lying average

Draw a million samples from a fat-tailed world and the running average keeps promising a number it cannot yet deliver. A Gaussian with the same true mean settles almost at once.

Exact Real draws from the real distributions — seeded exact sampling (inverse transform for the Pareto, Box–Muller for the Gaussian); the slow convergence you watch is the theorem, not an artefact. the fence →

tail α speed

draws n 0

running mean (bold path) —

true mean μ —

largest draw —

its share of the sum —

seed —

axes x 1→10⁶ (log; ticks 1, 10², 10⁴, 10⁶) · y 0→—

press Run

Exact, not staged: every purple path is real draws from a Pareto(α, x_m = 1), sampled by inverse transform — X = U^−1/α — from a seeded generator, so the same seed replays the same run (add ?seed=N to the URL to pin one). The blue path is a Gaussian handed the same true mean (μ = α/(α−1), σ = 1): same true mean, different tail — that pairing is a deliberate design choice, so the only thing that differs is the tail. Each upward stair in a purple path is, almost always, one single draw large enough to shove the whole average. (At α = 1 there is no mean, so the gold line is gone and the Gaussian twin sits out — it has no shared mean to copy.)

The rules are exact: the Pareto mean is finite exactly when α > 1, the variance exactly when α > 2. For 1 < α < 2 the law of large numbers still holds — but the error dies like n^1/α−1 instead of the Gaussian’s σ/√n. At α = 1.1 that exponent is −1/11: a million draws shrink the error scale by a factor of only ≈ 0.285, while the Gaussian’s standard error is down to 0.001·σ. At or below α = 1 there is no mean at all. Sums in this regime obey stable-law limits — the generalized central limit theorem — not the familiar bell-curve CLT. And the 80/20 rule lives on this same dial: the top fraction p of the population holds p^1−1/α of the total, which gives 80% from the top 20% exactly when α = ln 5/ln 4 ≈ 1.16096 — between the 1.1 and the 1.5 in the menu above.

The power law is Vilfredo Pareto’s, from the Cours d’économie politique (Lausanne: F. Rouge, 1896–97); the “80/20” name came later, from Joseph Juran — who retracted the attribution in a 1975 mea culpa in Quality Progress. The modern account of why averages misbehave under fat tails is Taleb’s Statistical Consequences of Fat Tails (Technical Incerto vol. 1, STEM Academic Press, 2020; free full text on arXiv). The limit theory itself is Gnedenko & Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, 1954).

the convexity lens →