TITLE:
Improving Accuracy of Normal Approximation
AUTHORS:
Jan Vrbik
KEYWORDS:
Central Limit Theorem, Distribution’s Cumulants, Edgeworth Series, Sample-Mean Transformation, Skewness Removal
JOURNAL NAME:
Open Journal of Statistics,
Vol.15 No.6,
December
16,
2025
ABSTRACT: The Central Limit Theorem occasionally results in extremely slow convergence to the Normal distribution, making the corresponding approximation virtually useless (a typical example being the sample correlation coefficient). In most of these cases, there is a relatively easy way of substantially improving the approximation by including a few extra terms of the corresponding Edgeworth expansion to match not only the distribution’s mean and variance, but also its skewness and kurtosis. A further improvement is possible by transforming the studied sample statistic to fully remove the skewness; this simplifies the resulting approximation, making it also more accurate. All this is amply demonstrated by many practical examples.