Updated 12 September 2026

Cauchy distribution

The Cauchy distribution looks like a Normal from the centre but has radically heavier tails. Its mean is undefined and its second moment is infinite. The usual finite-mean Law of Large Numbers cannot be applied to its sample average.

What to notice

• PDFs look similar near the peak. Both distributions are symmetric and bell-shaped. The difference is in the tails: Cauchy’s tails decay as $1/x^2$, while Normal’s decay exponentially.
• Running means behave differently. The normal running mean approaches zero as sampling continues, without a fixed finishing point. The Cauchy mean wanders indefinitely, with occasional violent jumps driven by extreme outliers.
• Resample to see how different Cauchy runs can look. The wandering is not a quirk of one bad draw — it is intrinsic to the distribution.

Why no mean?

The Cauchy mean integral diverges:

The positive and negative tails are equally heavy, so the integral does not converge absolutely. The usual i.i.d. Law of Large Numbers assumes a finite absolute first moment. For the Cauchy specifically, the averaging identity below shows why its averages do not concentrate at a constant.

The ratio connection

If X ~ Normal(0,1) and Y ~ Normal(0,1) independently, then X/Y ~ Cauchy(0,1). Dividing by a near-zero normal draw can give an arbitrarily large result, which is why heavy tails appear in ratios.

Symmetry gives a median, not a mean

The standard Cauchy CDF is F(x)=1/2+arctan(x)/π. Therefore F(0)=1/2 and its quartiles are −1 and 1. Those statements remain meaningful even though the expected value does not exist. Integrating x/(π(1+x²)) from zero to R gives log(1+R²)/(2π), which grows without bound. The negative side diverges in the opposite direction; choosing equal cutoffs to make them cancel is a principal value, not an expectation.

The experiment computes an ordinary finite-sample mean and median. Both numbers exist for the displayed draws. That does not supply the missing population mean. The median estimates the distribution’s center using ranks, so an extremely large observation has much less influence on it than on the average.

What averaging preserves

For independent standard Cauchy variables, the characteristic function of their average is [exp(−|t|/n)]ⁿ=exp(−|t|). Thus the average has exactly the original Cauchy law for every positive integer n. In particular, P(−1≤average≤1)=1/2, even at n=2,000. A sequence converging in probability to zero would instead have that probability tend to one.

At the default n=500, repeat the experiment and compare the fraction of means and medians inside [−1,1]. The exact one-half reference applies to means; the median uses a different sampling distribution. Each point represents a separate sample, not another observation appended to the running trajectory above. A quiet stretch of one trajectory is not evidence for concentration.

Keep extreme values visible in the accounting

The density window [−8,8] omits about 7.917% of standard Cauchy probability. The repeated-estimate plot clips at ±5, but the table retains those estimates and their extreme observations. Removing large observations would change the estimator and its distribution, so the experiment does not trim them.

Reference

Random Services: Cauchy distribution derives the CDF, divergent moments, and stable sum identity. Compare the finite-variance normal averages and the Student t family, which includes Cauchy at one degree of freedom.

The symmetric stable family places the Cauchy average beside models whose averages narrow or widen.