Refined Normality Test Based on the Parametric Seven-Number Summary

Year 2025, Issue: 11, 1 - 39, 30.06.2025

Jose Moral De La Rubia

https://doi.org/10.52693/jsas.1670945

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

testing normality, normal distribution, quadratic forms, bootstrap, generalized chi-square distribution, inference statistics

Ethical Statement

The author declares no conflict of interest. The data are generated, therefore no ethics committee approval is required.

Supporting Institution

This research received no external funding.

Thanks

The author expresses his gratitude to the reviewers and editors for the suggestions and corrections received for the improvement of the manuscript.

Parametrik Yedi Sayılı Özet Temelli Geliştirilmiş Bir Normallik Testi

Abstract

2022 yılında, parametrik yedi sayılı özet temelli bir normallik testi önerilmiştir. Test istatistiği, karelenmiş standartlaştırılmış çeyreklerin toplamından oluşmakta olup, başlangıçta çeyrekler arasındaki korelasyon göz önünde bulundurulmadan yedi serbestlik dereceli ki-kare dağılımı ile yaklaşık olarak değerlendirilmiştir. Amaç, bu korelasyonları dikkate alarak testi geliştirmektir. Bu doğrultuda iki alternatif önerilmiştir: (1) Q-istatistiğinin örnekleme dağılımının bootstrap yöntemiyle tahmin edilmesi (QB) ve (2) normallik altında çeyreklerin korelasyon matrisini içeren bir karelik formun kullanılması (QT). Üç varyant (Q, QB, QT), doğruluk oranı (başarı oranı) ve istatistiksel güç açısından Shapiro–Wilk W-testi ile karşılaştırılmıştır. On iki sürekli dağılımdan, otuz bir farklı örneklem büyüklüğünde toplam 372 rastgele örnek üretilmiştir. Doğru sınıflandırmalar Cochran’ın Q testiyle karşılaştırılmış, güç ise tekrarlı ölçümler ANOVA’sı ile değerlendirilmiştir. QB, anlamlı şekilde en yüksek doğruluğa sahip olmuş ve Q ile QT’ye kıyasla en yüksek ortalama gücü göstermiştir. Doğruluk açısından Shapiro–Wilk W-testi ile eşdeğer olsa da, bu test ortalama güç bakımından üç Q varyantının tamamından üstün performans sergilemiştir. Sonuç olarak, QB, normallik testi için yedi sayılı özetin uygun bir çıkarımsal genişletmesi olarak değerlendirilmektedir.

Keywords

normallik testi, normal dağılım, kare biçimler, bootstrap, genelleştirilmiş ki-kare dağılımı, çıkarımsal istatistikler

Ethical Statement

Yazar, herhangi bir çıkar çatışması olmadığını beyan etmektedir. Veriler üretildiğinden etik kurul onayı gerekmemektedir.

Supporting Institution

Bu araştırma herhangi bir dış fon tarafından desteklenmemiştir.

Thanks

Yazar, makalenin geliştirilmesine yönelik olarak sunulan öneriler ve düzeltmeler için hakemlere ve editörlere teşekkürlerini sunmaktadır.

References

• [1] A. L. Bowley, Elements of Statistics. P. S. King, London, 1901.
• [2] A. L. Bowley, Elementary Manual of Statistics. P. S. King, London, 1910.
• [3] P. Muthudoss, S. Kumar, E. Y. C. Ann, K. J. Young, R. L. R. Chi, R. Allada, B. Jayagopal, A. Dubala, I. B. Babla, S. Das, S. Mhetre, I. Saraf, and A. Paudel, "Topologically directed confocal Raman imaging (TD-CRI): Advanced Raman imaging towards compositional and micromeritic profiling of a commercial tablet components," J. Pharm. Biomed. Anal., vol 210, article 114581, February 2022. https://doi.org/10.1016/j.jpba.2022.114581
• [4] J. W. Tukey, Exploratory Data Analysis. Addison and Wesley, Reading, MA, 1977.
• [5] J. Moral, "Testing for normality from the parametric seven number summary," Open J. Stat., vol. 12, no. 1, pp. 118‑154, February 2022. https://doi.org/10.4236/ojs.2022.121009
• [6] A. Avdović, and V. Jevremović, "Quantile-zone based approach to normality testing," Mathematics, vol. 10, no. 11, article 1828, May 2022. https://doi.org/10.3390/math10111828
• [7] G. Dikta, and M. Scheer, Bootstrap Methods: with Applications in R. Springer Nature, Cham, Switzerland, 2021. https://doi.org/10.1007/978-3-030-73480-0
• [8] B. Efron, and B. Narasimhan, "The automatic construction of bootstrap confidence intervals," J Comput Graph Stat., vol. 29, no. 3, pp. 608-619, Mach 2020. https://doi.org/10.1080/10618600.2020.1714633
• [9] G. Rousselet, C. R. Pernet, and R. R. Wilcox, "An introduction to the bootstrap: a versatile method to make inferences by using data-driven simulations", Meta-Psychology, vol. 7, article 2058. https://doi.org/10.15626/MP.2019.2058
• [10] A. Stuart, and J. K. Ord, Kendall’s Advanced Theory of Statistics. Volume 1. Distribution Theory, 6th ed., Edward Arnold, London, 1994.
• [11] P. Lafaye de Micheaux, Package CompQuadForm: Distribution function of quadratic forms in normal variables, 2025. https://cran.r-project.org/web/packages/CompQuadForm/CompQuadForm.pdf
• [12] S. S. Shapiro, and M. B. Wilk, "An analysis of variance test for normality (complete samples)," Biometrika, vol. 52, no. 3‑4, pp. 591‑611, December 1965. https://doi.org/10.1093/biomet/52.3-4.591
• [13] J. P. Royston, "Approximating the Shapiro-Wilk W-test for non-normality," Statistics and Computing, vol. 2, no. 3, pp. 117‑119, September 1992. https://doi.org/10.1007/BF01891203
• [14] J. Arnastauskaitė, T. Ruzgas, and M. Bražėnas, "An exhaustive power comparison of normality tests," Mathematics, vol. 9, no. 7, article 788, April 2021. https://doi.org/10.3390/math9070788
• [15] N. Khatun, "Applications of normality test in statistical analysis," Open J. Stat., vol. 11, no. 1, pp. 113-122, January 2021. https://doi.org/10.4236/ojs.2021.111006
• [16] S. S. Shapiro, and R. S. Francia, "An approximate analysis of variance test for normality," J. Am. Stat. Assoc., vol. 67, no 337, pp. 215-216, June 1972. https://doi.org/10.1080/01621459.1972.10481232
• [17] J. P. Royston, "A toolkit of testing for non-normality in complete and censored samples," J. Roy. Stat. Soc. Ser. D-The Statistician, vol. 42, no. 1, pp. 37‑43, March 1993. https://doi.org/10.2307/2348109
• [18] R. M. Bagby, J. D. Parker, and G. J. Taylor, "The twenty-item Toronto Alexithymia Scale--I. Item selection and cross-validation of the factor structure," J. Psychosom. Res., vol. 38, no. 1, pp. 23-32, Janaury 1994. https://doi.org/10.1016/0022-3999(94)90005-1
• [19] J. Moral, "Propiedades psicométricas de la Escala de Alexitimia de Toronto de 20 reactivos en México," Revista Electrónica de Psicología Clínica de Iztacala, vol. 11, no. 2, pp. 97-114, July 2008. Available at: https://www.iztacala.unam.mx/carreras/psicologia/psiclin/
• [20] G. Pedersen, E. Normann‐Eide, I. U. M. Eikenæs, E. H. Kvarstein, and T. Wilberg, "Psychometric evaluation of the Norwegian Toronto Alexithymia Scale (TAS‐20) in a multisite clinical sample of patients with personality disorders and personality problems,” J. Clin. Psychol., June 2022, vol. 78, no. 6, pp. 1118-1136, 2022. https://doi.org/10.1002/jclp.23270
• [21] H. Zhang, J. Shen, and Z. Wu, "A fast and accurate approximation to the distributions of quadratic forms of Gaussian variables," J Comput. Graph. Stat., vol. 31, no. 1, pp. 304-311, January 2022. https://doi.org/10.1080/10618600.2021.2000423
• [22] M. Estaki, L. Jiang, N. A. Bokulich, D. McDonald, A. González, T. Kosciolek, C. Martino, Q. Zhu, A. Birmingham, Y. Vázquez-Baeza, M. R. Dillon, E. Bolyen, J. Caporaso, G., and R. Knight, "QIIME 2 enables comprehensive end-to-end analysis of diverse microbiome data and comparative studies with publicly available data," Curr. Protoc. Bioinformatics, vol. 70, article e100, April 2020. https://doi.org/10.1002/cpbi.100
• [23] J. M. O. Machado, Outlier detection in accounting, master thesis, University of Porto. Institutional repository of the University of Porto, 2018. https://sigarra.up.pt/fep/en/pub_geral.show_file?pi_doc_id=423163
• [24] S. Cai, J. Zhou, and J. Pan, "Estimating the sample mean and standard deviation from order statistics and sample size in meta-analysis," Stat. Methods Med. Res., 30, vol. 12, pp. 2701-2719, October 2021. https://doi.org/10.1177/09622802211047348
• [25] R. J. Hyndman, and Y. Fan, "Sample quantiles in statistical packages," Am. Stat., vol. 50, no. 4, pp. 361‑365, November 1996. https://doi.org/10.1080/00031305.1996.10473566
• [26] L. Makkonen, and M. Pajari, "Research article defining sample quantiles by the true rank probability," J. Probab. Stat., vol. 2014, article 326579, December 2014. http://dx.doi.org/10.1155/2014/326579
• [27] D. M. Hawkins, "Quantile-quantile methodology-detailed results," arXiv, article 2303.03215, October 2023. https://doi.org/10.48550/arXiv.2303.03215
• [28] W. G. Cochran, "The distribution of quadratic forms in a normal system, with applications to the analysis of covariance," Math. Proc. Camb. Philos. Soc., vol. 30, no. 2, pp. 178-191, April 1934. https://doi.org/10.1017/S0305004100016595
• [29] P. McCullagh, "Gaussian distributions," in Ten Projects in Applied Statistics, Springer Series in Statistics. Springer, Cham, pp. 251-277. https://doi.org/10.1007/978-3-031-14275-8_15
• [30] G. Casella, and R. Berger, Statistical Inference. CRC Press, Hoboken, NJ, 2024. https://doi.org/10.1201/9781003456285
• [31] W. Y. Chen, G. W. Peters, R. H. Gerlach, and S. A. Sisson, "Dynamic quantile function models," Quantitative Finance, vol. 22, no. 9, pp. 1665-1691, April 2022. https://doi.org/10.1080/14697688.2022.2053193
• [32] A. Canty, B. Ripley, and A. R. Brazzale, Package ‘boot’. Bootstrap functions, 2024 https://cran.r-project.org/web/packages/boot/boot.pdf
• [33] J. Cohen, "A power primer," in A. E. Kazdin, Ed., Methodological Issues and Strategies in Clinical Research, 4th ed. American Psychological Association, Washington, DC, 2016, pp. 279‑284. https://doi.org/10.1037/14805-018
• [34] R. B. Davies, "Algorithm AS 155: The distribution of a linear combination of chi-2 random variables." J. R. Stat. Soc. Ser. C Appl., vol. 29, no. 3, pp. 323‑333, December 1980. https://doi.org/10.2307/2346911
• [35] J. Cohen, Statistical Power Analysis for Behavioral Sciences, 2nd ed. Lawrence Erlbaum Associates, Hillsdale, NJ, 1988.
• [36] G. E. P. Box, "Some theorems on quadratic forms applied in the study of analysis of variance problems. II. Effects of inequality of variance and of correlation between errors in the two-way classification," Ann. Math. Stat., vol. 25, no. 3, pp. 484‑498, September 1954. https://doi.org/10.1214/aoms/1177728717
• [37] A. Das, "New methods to compute the generalized chi-square distribution," arXiv, article 2404.05062, February 2024. https://doi.org/10.48550/arXiv.2404.05062
• [38] H. Cramer, Mathematical Methods of Statistics. Princeton University Press, Princeton, NJ, 1946.
• [39] S. J. Peterson, and S. Foley, "Clinician's guide to understanding effect size, alpha level, power, and sample size," Nutr. Clin. Pract., vol. 36, no. 3, pp. 598‑605, June 2021. https://doi.org/10.1002/ncp.10674
• [40] C. Baumgarten, and T. Patel, "Automatic random variate generation in Python," Proceedings of the 21st Python in Science Conference, July 11, 2022., pp. 46‑51. https://doi.org/10.25080/majora-212e5952-007
• [41] R Core Team (2025). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org
• [42] N. Smeeton, N. Spencer, and P. Sprent, Applied Nonparametric Statistical Methods. CRC Press, Boca Raton, FL, 2025. https://doi.org/10.1201/9780429326172
• [43] C. F. Fey, T. Hu, and A. Delios, "The measurement and communication of effect sizes in management research," Manag. Organ. Rev., vol. 19, no. 1, pp. 176-197, February 2023. https://doi.org/10.1017/mor.2022.2
• [44] L. A. O. Orawo, "Confidence intervals for the binomial proportion: a comparison of four methods," Open J. Stat., vol. 11, no. 5, pp. 806-816, October 2021. https://doi.org/10.4236/ojs.2021.115047
• [45] H. Pham, Springer Handbook of Engineering Statistics, 2023. https://doi.org/10.1007/978-1-4471-7503-2
• [46] M. J. Blanca-Mena, R. Alarcón, J. Arnau, J. García-Castro and R. Bono, "How to proceed when normality and sphericity are violated in the repeated measures ANOVA," Ann. Psychol., vol. 40, no. 3, pp. 466-480, July 2024. https://doi.org/10.6018/analesps.594291
• [47] Microsoft Corporation. Microsoft Excel (version 2021) [Software], Microsoft Corp., 2021.
• [48] IBM Corporation, IBM SPSS Statistics (version 27) [Software], IBM Corp., 2020.
• [49] D. Freedman, and P. Diaconis, "On the histogram as a density estimator: L2 theory," Probab. Theory Relat. Fields, vol. 57, no. 4, pp. 453‑476, December 1981. https://doi.org/10.1007/BF01025868
• [50] V. A. Epanechnikov, "Non-parametric estimation of a multivariate probability density," Theory Probab. Appl., vol. 14, no. 1, pp. 153‑158, January 1969. https://doi.org/10.1137/1114019
• [51] S. J. Sheather, and Jones, M. C. "A reliable data-based bandwidth selection method for kernel density estimation," J. R. Stat. Soc. Ser. B Stat. Method., vol. 53, no. 3, pp. 683‑690, December 1991. https://doi.org/10.1111/j.2517-6161.1991.tb01857.x
• [52] H. Lilliefors, "On the Kolmogorov‑Smirnov test for normality with mean and variance unknown," J. Am. Stat. Assoc., vol. 62, no. 318, pp. 399‑402, June 1967. https://doi.org/10.1080/01621459.1967.10482916
• [53] T. W. Anderson, and D. A. Darling, "Asymptotic theory of certain “goodness-of-fit” criteria based on stochastic processes," Ann. Math. Stat., vol. 23, no. 2, pp. 193‑212, June, 1952. https://doi.org/10.1214/aoms/1177729437
• [54] F. J. Anscombe and W. J. Glynn, "Distribution of the kurtosis statistic b2 for normal samples," Biometrika, vol. 70, no. 1, pp. 227‑234, April 1983 https://doi.org/10.1093/biomet/70.1.227R.
• [55] R. B. D’Agostino, "Transformation to normality of the null distribution of g1," Biometrika, vol. 57, no. 3, pp. 679‑681, December 1970. https://doi.org/10.1093/biomet/57.3.679
• [56] F. E. Grubbs, "Sample criteria for testing outlying observations," Ann. Math. Statist. Vol. 21, no. 1, pp. 27‑58, March, 1950. https://doi.org/10.1214/aoms/1177729885
• [57] DʼAgostino, and E. S. "Pearson, Tests for departure from normality: empirical results for the distributions of b2 and √b1," Biometrika, vol. 60, no. 3, pp. 613‑622, 01 December 1973. https://doi.org/10.1093/biomet/60.3.613
• [58] R. B. DʼAgostino, A. Belanger, and R. B. Jr. DʼAgostino, "A suggestion for using powerful and informative tests of normality," Am. Stat., vol. 44, no. 4, pp. 316‑321, December 1990. https://doi.org/10.1080/00031305.1990.10475751
• [59] C. M. Jarque, and A. Bera, "A test for normality of observations and regression residuals," Int. Stat. Rev., vol. 55, no. 2, pp. 163‑172, August 1987. https://doi.org/10.2307/1403192
• [60] C. Urzua, "On the correct use of omnibus tests for normality," Econ. Lett., vol. 53, no. 3, pp. 247‑251, December 1996. https://doi.org/10.1016/S0165-1765(96)00923-8
• [61] R. R. De Souza, M. Toebe, A. C. Mello, and K. C. Bittencourt, "Sample size and Shapiro-Wilk test: An analysis for soybean grain yield," Eur. J. Agron., vol. 142, no. 1, article 126666, January 2023. https://doi.org/10.1016/j.eja.2022.126666
• [62] Y. Şener, and Y. Günaydın, "Attitudes Toward Dating Violence, Social Impact, and Alexithymia in University Students: A Structural Equation Modeling," Psychol Reports, 0(0), July, 2024. https://doi.org/10.1177/00332941241265618
• [63] M. A. Terzioğlu, and A. Büber, "Alexithymia, internet addiction, and cyber-victimisation among high school students in Turkey: an exploratory study," Behav. Inform. Technol., vol. 44, no. 7, pp. 1350-1361, May 2024. https://doi.org/10.1080/0144929X.2024.2353273
• [64] M. Miniati, C. Poidomani, C. Conversano, G. Orrù, R. Ciacchini, D. Marazziti, G. Perugi, A. Gemignani, and L. Palagini, "Alexithymia and interoceptive confusion in Covid-19 pandemic distress," Clinical Neuropsychiatry: Journal of Treatment Evaluation, vol. 20, no. 4, pp. 264‑270, January 2023. https://doi.org/10.36131/cnfioritieditore20230405
• [65] J. L. Devore, K. N. Berk, and M. A. Carlton, Modern Mathematical Statistics with Applications. Springer Nature, Berlin, 2021. https://doi.org/10.1007/978-3-030-55156-8
• [66] S. Demir, "Comparison of normality tests in terms of sample sizes under different skewness and Kurtosis coefficients," IJATE, vol. 9, no. 2, pp. 397-409, June 2022. https://doi.org/10.21449/ijate.1101295
• [67] J. Vrbik, "Deriving CDF of Kolmogorov-Smirnov test statistic," Appl. Math., vol. 11, pp. 227-246, March 2020. https://doi.org/10.4236/am.2020.113018
• [68] S. S. Mangiafico, "G-test of goodness-of-fit," in An R Companion for the Handbook of Biological Statistics, Version 1.3.9, 2023. https://rcompanion.org/rcompanion/b_04.html
• [69] H. Wickham, W. Chang, L. Henry, K. Takahashi, C. Wilke, K. Woo, H. Yutani, D. Dunnington, T. van den Brand, and P. L. Pedersen, ggplot2: Create elegant data visualisations using the grammar of graphics, 2024. https://doi.org/10.32614/CRAN.package.ggplot2
• [70] S. Phuyal, M. Rashid, and J. Sarkar, GIplot: An R package for visualizing the summary statistics of a quantitative variable, 2021. https://cran.r-project.org/web/packages/GIplot/index.html
• [71] L. Gonzales-Fuentes, L. Barbé, K., Barford, L. Lauwers, and L. Philips, "A qualitative study of probability density visualization techniques in measurements," Measurement, vol. 65, no. 1, pp. 94‑111, April 2015. https://doi.org/10.1016/j.measurement.2014.12.022
• [72] A. J. Qasim, and F. Q. A. Alyousuf, "History of image digital formats using in information technology," Qalaai Zanist Scientific Journal, vol. 6, no. 2, pp. 1098-1112, June 2021. https://doi.org/10.25212/lfu.qzj.6.2.41