Quick guide
How to use this calculator
- Enter the observations, probabilities, model parameters, or summary statistics requested by the visible labels.
- Keep every value on the same scale and confirm that the selected sampling relationship, distribution, and tail convention match the question you are investigating.
- Read the result together with its assumptions and interpretation. Statistical output summarizes uncertainty under a model; it does not repair biased data or establish causation.
Calculation method
How the cauchy distribution calculator works
F(x)=1/2+arctan((x−x₀)/γ)/π, with location x₀ and scale γ>0.
Cauchy tails are so heavy that the population mean and variance do not exist, even though the median and location are defined.
Worked example
Cauchy Distribution example
The standard Cauchy model has cumulative probability 0.75 and density 1/(2π) at x=1.
F(x)=1/2+arctan((x−x₀)/γ)/π, with location x₀ and scale γ>0.
Supported inputs
Precision and limits
Model and design
Do not summarize a Cauchy model with ordinary mean or standard deviation; the entered location and scale follow the stated convention.
Numerical scope
Inputs use double-precision numerical methods with guarded domains. Datasets accept up to 10,000 finite plain-decimal values. Extremely large parameters or probabilities deep in a numerical tail may require specialist statistical software.
Interpretation
Cauchy tails are so heavy that the population mean and variance do not exist, even though the median and location are defined.
Decision boundary
The calculator does not validate how data were collected, diagnose dependence or bias, choose a scientifically meaningful effect, or replace review by a qualified statistician for consequential research, medical, regulatory, safety, or policy decisions.
Privacy
Entered values and calculated results stay in this browser and are not sent to an analytics service.
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