No mean? No problem.

This is Cauchy data: its tails are so heavy that the distribution has no mean and no variance, so averaging never converges. Watch the red sample mean get thrown by single wild points while the closed-form quantile MAD estimator locks onto the true curve every time.

Method: E. Pinsky* and Q. Wen*, “Estimation of Distribution Parameters by Mean Absolute Deviations of a Truncated Distribution Using Quantile Functions”, Statistical Papers 67:31 (2026). For Cauchy: location = sample median; scale = π/(2 ln 2) times the mean absolute deviation over the interquartile-truncated sample. Closed form, no moments required.