The Verification Venue · pointed at the average you were taught to trust
The Average of a Million Is Worth One
You were taught that averaging tames noise: more data, tighter mean. For Cauchy data that lesson is not merely strained, it is exactly backwards: the mean of n standard Cauchys has precisely the standard Cauchy distribution, for every n. This page proves that against the closed-form CDF over billions of draws, then measures what the percentile bootstrap does about it.
Every heavy-tail demo shows a running average jumping around and lets you believe it would settle with more data. At a few hundred draws, a jumpy average and a slowly settling one look identical, so the demonstration cannot show the difference. Drag the tail slider first: the thing below is true within seconds, and the full run behind the button is what turns the suspicion into a measurement.
Current running mean
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0 draws
Small-run bootstrap coverage (illustrative)
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a few hundred experiments: useless precision
Drag toward 1.0: the mean stops settling and leaps to a new plateau each time one enormous draw lands. At 2.0 the law is Gaussian and the average behaves.
Draws per frame for the running trace. The coverage meter on the right runs its own small study alongside.
The mechanism is one line of probability. Averages concentrate when the variance is finite: the mean of n draws wanders like 1/√n. The Cauchy has no variance to speak of, and its stability law says the sum of n of them is Cauchy with scale n, so dividing by n hands you back scale 1. The mean of a million is distributed exactly like the first draw. Do not confuse it with the sum, which is Cauchy with scale n: get that backwards and the whole finding inverts.
What breaks downstream is anything that resamples the data. The percentile bootstrap builds its intervals from the sample's own extremes, and under an infinite-variance law one observation owns every resample. Below, the full run measures its coverage of the truth at n = 100, and the subsampling repair at block size b = 10.
The full run
Billions of draws across every core: the CDF arm at n = 1, 10, 1000 and 1000000, the bootstrap arm at n = 100 with B = 2000 resamples and R = 2000 subsamples of b = 10, and the scaling ladder at five values of alpha. Default sizes: 200000, 50000 and 2500 experiments on the CDF arm, 16000 on the bootstrap arm.
idle
The check · run in front of you, or nothing is shown
Before any headline number, both implementations (the compiled C engine and a slow obvious JavaScript twin written from the same contract) run a battery of small instances and must agree fingerprint for fingerprint. The reproducible-build check is offline: it lives in research/the-average-of-a-million-is-one/verify-the-average-of-a-million-is-one.mjs.
| battery row | engine | reference | match |
|---|---|---|---|
| loading engine… | |||
The control: press to break the reference's inverse-CDF centre constant and watch the agreement check go red.
not yet run
Full-size checks, run as part of the launch above:
The binary this page fetched:
What none of this rules out: the two implementations share their transcendental formulas by declaration, so a bug in a shared series would pass the battery; the closed-form CDF check is what guards the law itself, and two agreeing decompositions rule out boundary errors and very little else. Run it yourself: node research/the-average-of-a-million-is-one/verify-the-average-of-a-million-is-one.mjs.
The cost, counted not estimated
What's exact here, and what's a measurement
Exactly true. The mean of n standard Cauchys is standard Cauchy, a stability theorem, checked here against F(x) = 1/2 + arctan(x)/pi. The scaling exponent of a symmetric alpha-stable mean is 1/alpha − 1, also a theorem. The CMS generator reduces to the Cauchy at alpha = 1 and to a mean-zero Gaussian of variance 2 at alpha = 2, which is how the cross-method check works.
Measured on this page, attributable to no one. Every coverage figure, every KS distance, every fitted slope. Seeds are fixed and counter-based, so these are deterministic: rerun and you get these digits.
Idealised. The bootstrap failure claimed here is specific to infinite-variance data; under finite variance the bootstrap is consistent and this page does not dispute that. The subsampling repair carries conditions of its own: block size b = 10 at n = 100 is an illustration, not a prescription, and consistency needs b growing with n. The binned KS statistic is an upper bound on the true sup-distance, and the critical value is the asymptotic Kolmogorov limit inverted at significance 0.001.