Refined Normality Test Based on the Parametric Seven-Number Summary

Authors : Jose Moral De La Rubia

Pages : 1-39

Doi:10.52693/jsas.1670945

View : 58 | Download : 30

Publication Date : 2025-06-30

Article Type : Research Paper

Abstract :In 2022, a normality test based on the parametric seven-number summary was proposed. Its test statistic is the sum of squared standardized quantiles and was initially approximated by a chi-square distribution with seven degrees of freedom, without accounting for correlation among quantiles. Objective: To improve the test by incorporating these correlations. Two alternatives were proposed: (1) estimating the sampling distribution of the Q-statistic via bootstrap (QB), and (2) using a quadratic form with a correlation matrix of quantiles under normality (QT). Methods: The three variants (Q, QB, QT) were compared with the Shapiro-Wilk W-test in terms of accuracy (hit ratio) and statistical power. A total of 372 random samples were generated across 31 sample sizes from twelve continuous distributions. Correct classifications were compared using Cochran’s Q test, and power was assessed via repeated-measures ANOVA. Results: QB was significantly the most accurate and showed the highest average power compared to Q and QT. Its accuracy was equivalent to that of the Shapiro–Wilk W-test, although the latter outperformed all three Q variants in average power. Conclusions: QB is a suitable inferential extension of the seven-number summary for testing normality.
Keywords : normallik testi, normal dağılım, kare biçimler, bootstrap, genelleştirilmiş ki-kare dağılımı, çıkarımsal istatistikler

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