SavedAdd your journal

Journals / İstatistik ve Uygulamalı Bilimler Dergisi / 2026 / Sayı: 11

Refined Normality Test Based on the Parametric Seven-Number Summary

Parametrik Yedi Sayılı Özet Temelli Geliştirilmiş Bir Normallik Testi

Jose Moral De La Rubia(Universidad Autónoma de Nuevo León)0000-0003-1856-1458
Pages
1–39
Publication Date
2025-06-30
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Conclusions: QB is a suitable inferential extension of the seven-number summary for testing normality.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In 2022, a normality test based on the parametric seven-number summary was proposed.
📊
Key Findings & Data: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).
💡
Conclusion & Impact:Conclusions: QB is a suitable inferential extension of the seven-number summary for testing normality.
🔬Design:Araştırma Makalesi
📍Scope:2022 · 31 sample
⚙️Variables:normallik testi · normal dağılım · kare biçimler

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.

Özet

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.

🔖

Cite This Article

Rubia, J. M. D. L. (2025). Refined Normality Test Based on the Parametric Seven-Number Summary. İstatistik ve Uygulamalı Bilimler Dergisi, Sayı: 11, 1–39. https://doi.org/10.52693/jsas.1670945

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (72)

  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