Cauchy Distribution Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The Cauchy distribution has density f(x) = 1 ÷ (πγ(1+((x−x0)/γ)²)), where x0 is the location parameter and γ the scale parameter. Remarkably: unlike almost every other statistical distribution, neither its expected value nor its variance is mathematically defined, although it has a perfectly defined median, equal to x0.
Explanation
The Cauchy distribution visually resembles a bell curve, like the normal distribution, but with "tails" — the extremities of the distribution, far from the center — much thicker: extreme values there stay significantly more probable than for a normal distribution of the same apparent width. This difference has a spectacular mathematical consequence often cited as a textbook case: the integral that would define the expected value of this distribution does NOT converge (it diverges to infinity in both directions), which means the Cauchy distribution simply has NO mathematically defined mean — and by extension, no variance either. It is a result that often surprises at first: you might naively expect the mean of a large number of observations drawn from this distribution to gradually stabilize around a central value, as the law of large numbers predicts for most usual distributions — but for the Cauchy distribution, this empirical mean never converges and keeps fluctuating erratically, however large the number of observations. The Cauchy distribution nonetheless has a perfectly well-defined MEDIAN, exactly equal to the location parameter x0: half the probability mass lies on each side of that point, a property that illustrates well that median and expected value, often intuitively confused, are two distinct statistical notions that do not always coincide. This distribution appears naturally in physics (it describes, for example, the shape of spectral lines broadened by collision, where it is called the Lorentz distribution) and serves in statistics as a reference teaching counterexample to illustrate the limits of the central limit theorem and the law of large numbers. It thus joins, on this site, other distributions noted for their thick tails or atypical behavior, such as the log-normal distribution calculator or the Pareto distribution calculator, which also describes phenomena where extreme values weigh far more heavily than a normal distribution would suggest.
Example: density and CDF at the location point
Inputs
Value x = 0. Location parameter x₀ = 0. Scale parameter γ = 1.
Calculation
f(0) = 1 ÷ (π×1×(1+0²)) = 1/π ≈ 0.3183. F(0) = (1/π)×arctan(0) + 0.5 = 0 + 0.5 = 0.5.
Result
At the location point, the density reaches its maximum (≈0.3183) and the CDF is exactly 0.5: x₀ is indeed the median of the distribution.
Frequently asked questions
Why is the Cauchy distribution's expected value not simply equal to x₀ by symmetry?
The distribution is indeed perfectly symmetric about x0, which might suggest its expected value should be x0 by a simple symmetry argument. But the mathematical definition of the expected value requires the integral ∫x×f(x)dx to converge to a FINITE value, which is not the case here: the two halves of this integral (from x0 to +∞ and from −∞ to x0) each diverge to infinity, and their formal difference (+∞ minus ∞) is mathematically undefined, even though it "should" intuitively cancel by symmetry. It is a classic pitfall that illustrates why the convergence of an integral must be rigorously verified, not just assumed from symmetry arguments.
Does the parameter γ correspond to the distribution's standard deviation?
No, and it is a frequent confusion: since the variance of the Cauchy distribution is not defined, γ CANNOT be interpreted as a standard deviation in the usual sense. It is rather a geometric scale parameter, corresponding precisely to the distance between x0 and the point where the density drops to exactly half its maximum value (the half-width at half-maximum) — a different dispersion measure, which stays well defined even when the classic standard deviation is not.
In what physical context does this distribution appear naturally?
Under the name Lorentz distribution, it describes the profile of emission or absorption spectral lines broadened by collisions between atoms or molecules (collisional broadening, a mechanism distinct from Doppler broadening, which instead produces a Gaussian profile). It also appears as the distribution of the ratio of two random variables each following a centered normal distribution, a mathematical result that partly explains why its tails are so heavy compared with those of a simple normal distribution.