Interactive module guide

Cauchy Distribution Calculator & Guide

This Cauchy distribution calculator lets you explore the classic heavy-tailed counterexample whose mean and variance do not exist, with exact quantiles and tails.

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The Cauchy distribution is probability theory’s standard counterexample: a symmetric, bell-shaped curve whose tails are so heavy that its mean, variance, and every higher moment fail to exist.

The average of n independent Cauchy observations follows the same Cauchy distribution as a single observation, so averaging never stabilizes anything—the Law of Large Numbers and the Central Limit Theorem both fail here.

This page provides exact densities, cumulative probabilities, and quantiles (which all exist and are perfectly well behaved) while the summary panel reports every moment as a dash, because printing a number there would be a lie.

Is this the right module for my question?

Use it when

Use it to teach the boundary of the classical limit theorems: every statement of the LLN and CLT carries a finite-mean or finite-variance condition, and the Cauchy family is the cleanest demonstration of why those conditions are not decorative.
Use it as a stress-test distribution for methods that claim robustness: an estimator or procedure that behaves sensibly under Cauchy noise has earned the claim, because sample means, sample variances, and moment-based machinery all fail here.
Use it when a model genuinely produces normal ratios: the ratio of two independent centered normal variables is exactly Cauchy, so instrument readings formed as ratios can inherit this behavior, complete with wild outliers that are features rather than errors.

Choose another method when

Avoid it as a routine model for data that merely have a few outliers; a t distribution with low degrees of freedom keeps heavy tails while retaining a finite mean, which most applications quietly require.
Avoid any workflow that summarizes Cauchy data with sample means or standard deviations—those statistics exist for any finite sample but estimate nothing, and they will swing wildly between samples.
Avoid confusing "no mean" with "no center": the distribution has a perfectly good median and mode at x₀; what fails is specifically the expectation integral, not the notion of a typical value.

Interactive tool

The calculator loads as you approach this section so the guide remains fast on mobile connections.

Example preview

A ratio with no stable average

Inputs: X ~ Cauchy(x₀=0, γ=1); evaluate the central interval −1 to 1.
Representative result: P(−1 < X ≤ 1) = 0.5 exactly, yet the mean does not exist and sample averages never settle.

Illustrative only. Load the interactive tool to enter your own values and review assumptions.

Watch the explanation

2:59 min

The player loads only after you press play. You can also watch on YouTube.

Read the complete transcript

The Cauchy Distribution. Why can averaging a bell-shaped distribution forever fail to produce a stable center? The Cauchy family uses location x zero, positive scale gamma, and the whole real line. Its density is one over pi gamma times one plus the squared standardized distance. Integrating gives one half plus arctangent of standardized distance divided exactly by pi. Inverting the CDF gives x zero plus gamma tangent of pi times p minus one half. Quartiles sit one gamma from the center, making the interquartile range exactly two gamma.

Symmetry places both median and mode at x zero, even though the mean does not exist. The first moment fails because absolute tail contribution behaves like one over distance and diverges. Without an existing mean, variance and every higher positive integer moment also fail. Its exponential characteristic function reveals exact stability under independent sums and rescaled averages. Consequently every independent Cauchy sample average has exactly the same distribution as one observation. This does not disprove limit theorems; Cauchy data violate their finite-moment hypotheses. Against a normal curve, Cauchy tails remain visibly and numerically enormous far from center.

Let X be the ratio of two independent standard normals, creating a standard Cauchy measurement. Between minus one and one, the exact probability is one half, bounded by the quartiles. The ninety-ninth percentile is thirty-one point eight two, far beyond the central scale. Beyond three scale units, Cauchy probability is point one zero two four, not rare. A matching normal tail is only point zero zero one three five, about seventy-six times smaller. Use median zero and interquartile range two; sample mean and standard deviation are misleading.

In physics the same curve is a Lorentzian resonance profile with half-width gamma. Robust methods often face Cauchy noise as a deliberate worst-case stress test. A few outliers alone do not justify Cauchy; a low-degree t model may fit better. Computed sample means are finite numbers, but they estimate no finite expectation here. No mean never means no center; median, mode, quartiles, and probabilities remain valid. Plotted windows truncate infinite support, so extreme quantiles may lie far beyond the chart. Reject nonpositive gamma, and keep location finite before evaluating any probability or quantile. Report x zero, gamma, endpoint inclusion, and exact probability or quantile for reproducibility. Explore heavy tails, exact probabilities, and Cauchy quantiles free at Distri Scope dot com.

How to read the result

Compare tails with the normal: a Cauchy variable exceeds x₀ + 3γ with probability about 0.102, while a normal variable exceeds its mean plus three standard deviations with probability 0.0013—two orders of magnitude apart.

Quantiles explode near the extremes: Q(0.99) sits about 31.8γ above the center and Q(0.999) about 318γ, a direct display of the power-law-like tail.

Half of all probability lies within x₀ ± γ; the family is not diffuse in the center—its pathology lives entirely in the tails.

Inputs and parameter meaning

Two parameters: the location x₀ (the center, median, and mode) and the scale γ, which is the half-width at half maximum of the density peak.

The plotted window covers x₀ ± 8γ; the tails continue far beyond it with non-negligible mass, which is precisely the point of this family.

The interval and quantile calculators work exactly—probabilities and percentiles exist even though moments do not.

Formula and available parameters

Density or probability mass

f(x) = \frac{1}{\pi\gamma\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}

Cumulative distribution

F(x) = \frac{1}{2} + \frac{1}{\pi}\arctan\!\left(\frac{x-x_0}{\gamma}\right)

Mean

\text{does not exist}

Variance

\text{does not exist}

Location (x₀): interactive range -10 to 10; default 0.
Scale (γ): interactive range 0.1 to 5; default 1.

What the method is doing

The density, CDF, and quantile function are closed-form: F(x) = 1/2 + arctan((x−x₀)/γ)/π and Q(p) = x₀ + γ·tan(π(p−1/2)); no numerical approximation is involved.

The summary panel deliberately reports mean, variance, standard deviation, skewness, and kurtosis as dashes: the defining integrals diverge, and the honest display is refusal, not a large number.

The interquartile range is exactly 2γ (quartiles sit at x₀ ± γ), which is the scale reading that replaces the nonexistent standard deviation.

Worked example: the interval a ratio cannot escape

A measurement is formed as the ratio of two independent, centered normal readings, so it follows a standard Cauchy distribution with x₀ = 0 and γ = 1. Find the probability of landing in the central interval from −1 to 1, the 99th percentile, and explain what happens to the running average of repeated measurements.

1Central interval: P(−1 < X ≤ 1) = F(1) − F(−1) = (1/2 + 1/4) − (1/2 − 1/4) = 0.5 exactly—the quartiles of the standard Cauchy are ±1.
2Upper percentile: Q(0.99) = tan(0.49π) ≈ 31.82, so one measurement in a hundred lands beyond roughly 32 scale units despite half the mass sitting inside ±1.
3Tail comparison: P(X > 3) ≈ 0.102, against 0.0013 for a standard normal—Cauchy “outliers” are routine events.
4Averaging: the mean of n independent standard Cauchy measurements is again standard Cauchy for every n, so the running average wanders forever and the LLN never engages.

Interpretation

The example shows both faces of the family: probabilities and percentiles are exact and easy, while every moment-based summary is undefined. A practitioner facing genuine ratio noise should summarize with the median and quartiles, never the mean—and should expect occasional readings dozens of scale units out that are not data errors but the model working as specified.

Common mistakes

Do not compute and report a sample mean of Cauchy-like data as if it estimated something; it has the same distribution as one raw observation, no matter how large the sample.
Do not trim a few extreme values and then treat the rest as normal; with genuine Cauchy tails the “outliers” carry structural information, and trimming changes the question.
Do not fit it casually because data look heavy-tailed; establishing that no mean exists is a strong scientific claim, and a low-df t distribution is usually the more defensible model.

Limits and independent validation

The summary panel cannot display moment-based quantities because they do not exist; comparisons with other families must go through quantiles and interval probabilities.

The plotted window necessarily truncates tails that matter; numerical readings of extreme quantiles are exact even where the chart cannot show them.

This page covers the location-scale Cauchy only; skewed and wrapped variants used in specialized fields are out of scope.

Before using the result

Check the quartile identity: F(x₀ ± γ) must equal 0.75 and 0.25 exactly; the point calculator verifies it in two clicks.
Recompute one quantile by hand with Q(p) = x₀ + γ·tan(π(p−1/2)) and compare with the quantile lookup.
If simulating, plot the running mean of your draws: its refusal to settle is the family’s signature, and a stabilizing mean indicates the noise was never Cauchy.

t distribution →Cauchy is Student’s t with one degree of freedom; higher df restores finite moments.LLN / CLT simulator →See the finite-variance sample-mean behavior that Cauchy data refuse to follow.normal distribution →Compare with the light-tailed reference whose ratio construction produces Cauchy.

← Hypergeometric Distribution Pareto Distribution →

See every option in the Probability Distribution Explorer or review the DistriScope methodology. Educational information; last reviewed 2026-08-09. Verify consequential calculations independently.