Cauchy Sample-Mean Stability
See why averaging independent Cauchy observations does not narrow their sampling law.
Free theory articleSee why averaging independent Cauchy observations does not narrow their sampling law.
Concept & interpretation
The Central Limit Theorem asserts that sums (or averages) of independent random variables with a finite variance converge to a normal distribution. However, the Cauchy distribution has undefined expectation and undefined variance. Consequently, averaging multiple independent Cauchy variables does not narrow down their uncertainty. Instead, the sample mean has the exact same distribution as a single observation.
Experiment
Use the interactive module to compare the stability of the Cauchy mean against a hypothetical normal sample mean that matches the scale parameter.
Link to the interactive module: https://riano.app/module/cauchy-sample-mean-stability
- Change the Sample size from 1 to 4, then to 16. Notice that the Cauchy Mean PDF curve remains unchanged. The Cauchy scale stays exactly the same.
- Observe the Normal comparison density curve. Its peak gets narrower and taller as you increase the sample size because its standard error shrinks by 1/√n.
- Adjust the Window radius to see how the probability mass around the center stays constant for the Cauchy distribution.