Finite-sample tail distortion under normal approximation

Normal approximations justified by the central limit theorem constitute a cornerstone of statistical inference, particularly in settings where tail probabilities govern decision thresholds and error calibration. Although asymptotically valid under minimal conditions, such approximations exhibit non-uniform convergence, with tail regions displaying markedly slower rates of convergence in the presence of mild distributional asymmetry. This short communication investigates the finite-sample behavior of standardized sample means and establishes that Gaussian tail approximations systematically underestimate upper-tail exceedance probabilities when skewness is present. The effect persists at sample sizes routinely encountered in applied work and admits a precise asymptotic characterization via Edgeworth expansions. A simple monotone corrective approximation, constructed to preserve closed-form evaluation, invertibility, and probabilistic admissibility, is proposed as a practical finite-sample refinement.