Thought Toys · Chance & inference · Exhibit 31
Everyone knows the law of averages: keep averaging more numbers and the answer settles down. Here is a curve for which it never does. Its bell shape looks perfectly tame, yet a rare monster draw keeps flinging the running average clean off the chart — no matter how many numbers you pile on. Drag the tail heavier and watch the law of averages quietly break.
The average never settles
running average now — · biggest single draw so far —
Draw a random number, then another, and keep a running average of all of them. The amber line is that average as the pile of draws grows to the right (the axis counts draws on a log scale, so the whole life of the run fits on screen). The grey funnel is where the law of large numbers promises the average will be: a band that pinches toward the true centre like one over the square root of how many numbers you've drawn. Draw enough and you should be trapped in the throat of the funnel.
With a light tail — slide the dial right, toward the bell curve — that's exactly what happens. The trace wobbles early, then homes in and stays. Now drag the dial back to the left. The tail gets heavier: extreme draws that a bell curve would essentially never produce start showing up. Past ν = 2 the funnel vanishes entirely, because the spread — the variance, the average squared distance from the middle — is now infinite; the promise has no width to make. And at the far left, ν = 1, you meet the Cauchy: a curve that looks like a slightly sharper bell, but whose mean does not exist.
Watch what the Cauchy does. The average creeps toward zero, looks like it's settling — and then one draw lands at +400, or −1,900, and the average jumps, erasing everything the previous thousand draws had earned. It never settles because it can't: the average of a million Cauchy numbers is, provably, no more reliable than a single one. This is why "just take more data" is not always an answer. When the tails are heavy enough — as they can be for market crashes, city sizes, or wealth — the one freak event isn't noise around the average. It is the average.
improve/verify/31-cauchy.js): for ν = 8 the run-to-run scatter of the sample mean
shrinks about 4× when the sample grows 16× (one over root n, the law of large numbers), while for
the Cauchy it does not shrink at all — its sample mean still has the Cauchy's own interquartile range
of 2 at every sample size. Counter-example: clip the very same Cauchy tail (throw out draws
beyond ±50) and the average settles like one over root n again — proving it is precisely the heavy
tail, not the peaked shape, that breaks the law of averages. (That clipping quietly
makes a different distribution — a truncated Cauchy has finite variance by construction. It's a diagnosis
of the cause, not a licence to delete the outliers you meet in the wild, where you can't know in advance which
extremes are real.) (Each draw here is
t = Z·√(ν/χ²ν), the same generator the proof uses.)
Also in Chance & inference: The birthday paradox →
← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 31.