Using Power Transformations to Enhance Descriptive Statistical Analyses
Professor Emeritus Víctor M. Guerrero Department of Statistics Instituto Tecnológico Autónomo de México (ITAM)
Abstract
This work introduces a statistical framework using power transformations to normalize skewed data, enabling more accurate application of classical descriptive methods. The Box-Cox power transformation is emphasized for its continuity and effectiveness in reducing skewness. New results include improved histogram construction through variable-width bins, with simulations showing enhanced clarity for Lognormal distributions. Other results are parametric averages and dispersion measures derived from transformed data, offering robust alternatives to traditional measures.
Parametric averages generalize classical sample means and are shown to be least squares estimators, with classical averages (arithmetic, geometric, harmonic) emerging as special cases. Parametric dispersion, both absolute and relative, captures variability as effectively as the standard deviation and coefficient of variation.
Using Mexican income data, the framework demonstrates how optimal the power transformation parameter yields interpretable, data-driven insights into central tendency, variability, and inequality. The approach aligns with established income inequality indices like Gini and Theil, while offering greater flexibility.
