Why does the Cauchy distribution not have a mean or variance?
Swiss commemorative stamp of mathematician Jakob Bernoulli, issued 1994, displaying the formula and the graph for the law of large numbers, first proved by Bernoulli in 1713.
The Cauchy distribution lacks a defined mean and variance because the mathematical integrals used to calculate these values do not converge.1
In probability theory, the expectation (or mean) of a random variable is defined by an integral of the variable multiplied by its probability density function. For the Cauchy distribution, the "tails" of the distribution do not taper off quickly enough for this integral to result in a finite value. Because the mean is undefined, the variance—which measures the spread of data around the mean—also fails to exist.21
This characteristic leads to unusual statistical behaviors. For instance, unlike many other distributions where the average of multiple observations becomes more stable as the sample size increases, the average of independent Cauchy-distributed variables retains the same variability as a single observation.1