Cauchy Distribution Calculator

Averaging a thousand values is no better than taking one

This is the property that makes the Cauchy distribution worth a page of its own, and it is exact rather than approximate.

The sample mean of n Cauchy draws has exactly the same distribution as a single draw. Not approximately, and not only asymptotically — for every n, as an algebraic result about the characteristic function. Averaging is a no-op.

The tool measures it rather than stating it. 1,200 simulated samples at each of four sizes, with the spread of the sample mean reported as an interquartile range — because its variance does not exist either. The Cauchy column reads 1.85, 2.07, 1.95, 2.09 at n = 1, 10, 100 and 1,000.

The normal column beside it is what you expected to happen. 1.365, 0.4313, 0.1339, 0.0435 — each within about 1% of 1.349/√n. That law is what the Cauchy column is ignoring.

The law of large numbers needs a finite mean to converge TO. There is none here, so there is nothing for the average to approach. The central limit theorem needs a finite variance, which is also absent. Both theorems are silent rather than approximately true.

So every routine built on averaging fails. Confidence intervals from a standard error, t-tests, least squares, control limits: all of them assume a quantity that does not exist for this distribution.

What “has no mean” actually means

It is a stronger statement than “the mean is hard to estimate” or “the mean is infinite”.

The integral diverges at both ends, in opposite directions. ∫x·f(x)dx runs to +∞ on the right and −∞ on the left, and the two do not cancel in any well-defined way. The expected value is undefined, not infinite.

Which is why the sample mean wanders rather than growing. A distribution with an infinite mean would give a sample average that climbs steadily. This one gives an average that jumps around forever, dragged by whichever extreme value arrived most recently.

The median is perfectly well behaved. It exists, it equals the location parameter, and the sample median converges to it at the usual rate. Everything that works on Cauchy data is rank-based.

The interquartile range is exactly 2γ. The quartiles sit one scale either side of the location, which follows immediately from arctan(±1) = ±π/4. So a perfectly good measure of spread exists — it is just not the standard deviation.

And the parameters are called location and scale for that reason. Calling x₀ “the mean” and γ “the standard deviation” is not a shorthand; it names two things that do not exist.

How heavy the tails really are

“Heavy-tailed” is qualitative until you put two distributions on the same footing and compare.

Match a normal to the same interquartile range and look five sigma out. The normal puts about 1 in 3.5 million beyond that point. The Cauchy puts about 1 in 12 — the tool computes the exact ratio for your parameters.

The density falls as 1/x² rather than exponentially. That single difference in decay rate is the whole story: a normal tail dies faster than any polynomial, and this one does not die at all in the sense the moment integrals need.

So the 99th percentile is already far out. At γ = 1 it sits at 31.8, against 2.33 for a standard normal. A distribution whose middle half spans 2 units has a 1-in-100 event at 32.

This is why it is used as a stress model. Where a normal treats a ten-sigma move as impossible, a Cauchy treats it as a Tuesday. Neither is a description of any particular market; the point is that the choice of tail decides what your model calls impossible.

Where it turns up, on purpose and by accident

Three places it is the correct model, and one where it appears without being invited.

The ratio of two independent standard normals is exactly Cauchy. Not approximately. Any statistic that divides one roughly-normal quantity by another roughly-normal quantity centred near zero inherits this behaviour, which is a common and unrecognised source of wild estimates.

Resonance line shapes are Lorentzian, which is the same function. In spectroscopy and particle physics the distribution is written with a full width at half maximum of 2γ and called a Breit-Wigner or Lorentzian curve. The physics named it independently.

The angle of a randomly pointed beam gives one. Spin a line uniformly and record where it crosses a fixed axis: the intercepts are Cauchy distributed. That construction is the shortest proof that the tails must be this heavy — a nearly parallel line lands arbitrarily far away.

And it appears as the t-distribution with one degree of freedom. A t-distribution on 1 df IS a standard Cauchy, which is why a t-test on two observations has no useful power: the reference distribution has no variance.

Working with data that might be Cauchy

You rarely know in advance. What you can do is notice, and then stop using the tools that assume otherwise.

The diagnostic is the running mean. Plot the sample average as observations accumulate. Normal-ish data settles into a narrowing band; Cauchy-ish data keeps jumping to wherever the latest extreme value drags it, with no sign of convergence at any n.

A running standard deviation does the same thing, more visibly. Each new extreme value ratchets it upward and it never comes back down, which is the signature of an infinite or undefined second moment.

Switch to the median and the interquartile range. Both exist, both converge, and both are what the trimmed mean family generalises. A trimmed mean at any positive fraction has a finite variance even here, which is the practical reason robust estimators exist.

And be careful with ratios. Dividing by a quantity that can be near zero is the most common way ordinary data acquires Cauchy tails, and it is invisible until the estimates start misbehaving.

Reporting heavy-tailed data

Four habits, and the first one removes most of the damage.

Do not report a mean and a standard deviation. They will be computed without error and mean nothing. The median and the interquartile range are the correct pair, and they are no harder to read.

Say which parameters you fitted. Location and scale are not mean and standard deviation, and a reader who assumes they are will get the tail behaviour badly wrong.

Report a quantile rather than a multiple of sigma. “Three sigma” has no meaning without a variance. “The 99th percentile is 31.8” is exact and needs no moments.

And show the running-mean plot if you are claiming convergence. It is the cheapest evidence that an average is settling, and its absence is the cheapest sign that it is not.

Sources and methodology

References for the Cauchy distribution and heavy tails.

• Tukey, “A Survey of Sampling from Contaminated Distributions”, 1960 — where the cost of heavy tails to classical methods is set out
• NIST/SEMATECH e-Handbook, 1.3.6.6.3 — the Cauchy distribution, with its undefined moments stated explicitly
• Breit & Wigner, “Capture of Slow Neutrons”, Physical Review 49, 1936 — the same function under its physics name
• Central Limit Theorem Calculator — the theorem whose conditions this distribution violates
• T-Distribution Calculator — on 1 degree of freedom, it is exactly this distribution
• How we calculate — methodology

Method. The PDF, CDF and quantile function are the exact closed forms, so no numerical integration or inversion is involved and the quartile identity IQR = 2γ holds to machine precision. The averaging experiment uses a seeded generator, so the table is identical on every render and inside the verification suite — which checks both that a repeated run reproduces it and that the Cauchy interquartile range fails to shrink while the normal one tracks 1.349/√n to within 12% at every size tested. The density is confirmed to integrate to its own CDF range by Simpson’s rule over a wide window, and the CDF is checked monotone across 400 points. That engine is verified on every change against 75 assertions. The count and the per-case breakdown are published on the formula verification page.

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