Comparison of Normality Tests in Terms of Sample Sizes under Different Skewness and Kurtosis Coefficients

Öz

This study aims to compare normality tests in different sample sizes in data with normal distribution under different kurtosis and skewness coefficients obtained simulatively. To this end, firstly, simulative data were produced using the MATLAB program for different skewness/kurtosis coefficients and different sample sizes. The normality analysis of each data type was conducted using the MATLAB program and ten different normality tests; namely, (Kolmogorov Smirnov (KS) Test, KS Stephens Modification, KS Marsaglia, KS Lilliefors Modification, Anderson-Darling Test, Cramer- Von Mises Test, Shapiro-Wilk Test, Shapiro-Francia Test, Jarque-Bera Test, and D’Agostino & Pearson Test). As a result of the analyses conducted according to ten different normality tests, it was found that normality tests were not affected by the sample size when the skewness and kurtosis coefficients were equal to or close to zero; however, in cases where the skewness and kurtosis coefficients moved away from zero, it was found that normality tests are affected by the sample size, and such tests tend to give significant results. Therefore, in large samples, it may be suggested that critical values for skewness and kurtosis coefficients’ z-scores as proposed by Kim (2013) and Mayers (2013) or the histogram graphs be used.

Anahtar Kelimeler

• Normal distribution
• Skewness
• Kurtosis
• Normality tests

References

1. Abbott, M.L. (2011). Understanding educational statistics using Microsoft Excel and SPSS. Wiley & Sons, Inc.
2. Ahad, N.A., Yin, T.S., Othman, A.R., & Yaacob, C.R. (2011). Sensitivity of normality tests to non normal data. Sains Malaysiana, 40(6), 637 641. https://core.ac.uk/download/pdf/11491563.pdf
3. Anderson, T.W., & Darling, D.A. (1952). Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes. The Annals of Mathematical Statistics, 23(2), 193-212. https://doi.org/10.1214/aoms/1177729437
4. Anderson, T.W., & Darling, D.A. (1954). A test of goodness of fit. Journal of the American Statistical Association, 49(268), 765 769. https://doi.org/10.1080/01621459.1954.10501232
5. Baykul, Y., & Güzeller, C.O. (2013). Sosyal bilimler için istatistik: SPSS uygulamalı [Statistics for social sciences: SPSS applied]. Pegem Akademi.
6. Bulmer, M.G. (1979). Principles of Statistics. Dover.
7. Byrne, B.M. (2010). Structural Equation Modeling with AMOS: Basic Concepts, Applications, and Programming. Taylor and Francis Group Publication.
8. Csörgö, S., & Faraway, J.J. (1996). The exact and asymptotic distributions of Cramer-von Mises statistics. Journal of Royal Statistical Society. Series B (Methodological), 58(1), 221-234.

9. D’Agostino, R.B., & Pearson, E.S. (1973). Tests for departures from normality. Empirical results for the distribution of b2 and √b1. Biometrika, 60(3), 613-622. https://doi.org/10.1093/biomet/60.3.613
10. Dellal, G.E., & Wilkinson, L. (1986). An analytic approximation to the distribution of Lilliefors’s test statistic for normality. The American Statistician, 40(4), 294-296. https://doi.org/10.1080/00031305.1986.10475419
11. Demir, E., Saatcioğlu, Ö., & İmrol, F. (2016). Uluslararası dergilerde yayımlanan eğitim araştırmalarının normallik varsayımları açısından incelenmesi [Examination of educational researches published in international journals in terms of normality assumptions]. Current Research in Education, 2(3), 130-148.
12. Douglas G.B., & Edith, S. (2002). A test of normality with high uniform power. Journal of Computational Statistics and Data Analysis 40(3), 435 445. https://doi.org/10.1016/S0167-9473(02)00074-9
13. Facchinetti, S. (2009). A procedure to find exact critical values of Kolmogorov-Smirnov test. Statistica Applicata – Italian Journal of Applied Statistics, 21(3-4), 337-359.
14. Field, A. (2013). Discovering statistics using SPSS. Sage Publications.
15. Frain, J.C. (2007). Small sample power of tests of normality when the alternative is an α-stable distribution. Trinity Economics Papers TEP-0207, Trinity College Dublin, Department of Economics. http://www.tcd.ie/Economics/TEP/2007/TEP0207.pdf
16. George, D., & Mallery, M. (2010). SPSS for Windows Step by Step: A Simple Guide and Reference, 17.0. Pearson.
17. Gravetter, F., & Wallnau, L. (2014). Essentials of statistics for the behavioral sciences. Wadsworth.
18. Hair, J.F., Black, W.C., Babin, B.J., & Anderson, R.E. (2010). Multivariate data analysis: A global perspective. Prentice Hall.
19. Harter, H.L. (1961). Expected values of normal order statistics, Biometrika, 48, 151-65.
20. Howell, D.C. (2013). Statistical methods for psychology. Belmont, Wadsworth/Cengage Learning.
21. Jarque, C.M., & Bera, A.K. (1987). A test for normality of observations and regression residuals. International Statistical Review, 55(2), 163 172. https://doi.org/10.2307/1403192
22. Keskin, S. (2006). Comparison of several univariate normality tests regarding type I error rate and power of the test in simulation based small samples. Journal of Applied Science Research 2(5), 296-300.
23. Kim, H.Y. (2013). Statistical notes for clinical researchers: assessing normal distribution (2) using skewness and kurtosis. Restor Dent Endod, 38(1), 52 4. https://doi.org/10.5395/rde.2013.38.1.52
24. Kline, R.B. (2011). Methodology in the Social Sciences: Principles and practice of structural equation modeling. Guilford Press.
25. Kundu, M.G., Mishra, S., & Khare, D. (2011). Specificity and Sensitivity of Normality Tests. In Proceedings of VI International Symposium on Optimisation and Statistics. Anamaya Publisher.
26. Lee, C., Park, S., & Jeong, J. (2016). Comprehensive Comparison of Normality Tests: Empirical Study Using Many Different Types of Data. Journal of the Korean Data and Information Science Society, 27(5), 1399 1412. https://doi.org/10.7465/jkdi.2016.27.5.1399
27. Lilliefors, H.W. (1967). On the Kolmogorov-Smirnov test for normality with mean and variance unknown. Journal of the American Statistical Association, 62(318), 399-402. https://doi.org/10.1080/01621459.1967.10482916
28. Lumley, T., Diehr, P., Emerson, S., & Chen, L. (2002). The importance of the normality assumption in large public health data sets. Annual Review of Public Health, 23, 151–169. https://doi.org/10.1146/annurev.publhealth.23.100901.140546
29. Marsaglia, G., Tsang, W.W., & Wang, J. (2003). Evaluating Kolmogorov’s distribution. Journal of Statistical Software, 8(18). https://doi.org/10.18637/jss.v008.i18
30. Martin, W., & Bridgmon, K. (2012). Quantitative and statistical research methods: from hypothesis to results. Jossey-Bass.
31. Mayers, A. (2013). Introduction to statistics and SPSS in psychology. Pearson Education Limited.
32. Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 105(1), 156–166. https://doi.org/10.1037/0033-2909.105.1.156
33. Nor-Aishah H., & Shamsul R. A (2007, 12-14 December). Robust Jacque-Bera Test of Normality. The 9th Islamic Countries Conference on Statistical Sciences, University Malaya, Malaysia.
34. Nornadiah, M.R., & Yap, B.W. (2011). Power comparison of Shapiro-Wilk, Kolmogorov-Smirnov, Lillieforsand Anderson-Darling tests. Journal of Statistical Modeling and Analytics, 2(1), 21-33.
35. Orcan, F. (2020). Parametric or non-parametric: skewness to test normality for mean comparison. International Journal of Assessment Tools in Education, 7(2), 255-265. https://doi.org/10.21449/ijate.656077
36. Öztuna, D., Elhan, A.H., & Tuccar, E. (2006). Investigation of four different normality tests in terms of Type I error rate and power under different distributions, Turk. J. Med. Sci. 36(3), 171–176.
37. Ö̈ner, M., & Kocakoç, İ.D. (2017). JMASM 49: A compilation of some popular goodness of fit tests for normal distribution: their algorithms and MATLAB codes (MATLAB). Journal of Modern Applied Statistical Methods, 16(2), 547 575. https://doi.org/10.22237/jmasm/1509496200
38. Rinnakorn, C., & Kamon, B. (2007). A power comparison of goodness-of-fit tests for normality based on the likelihood ratio and the non-likelihood ratio. Thailand Statistician, 5, 57-68.
39. Shapiro, S.S., & Francia, R.S. (1972). An approximate analysis of variance test for normality. Journal of the American Statistical Association, 67(337), 215 216. https://doi.org/10.1080/01621459.1972.10481232
40. Stephens, M.A. (1986). Tests based on EDF statistics. In R.B.D’Agostino & M. A. Stephens (Eds.), Goodness-of-fit techniques (pp. 97-194). Marcel Dekker.
41. Stephens, M.A. (1974), EDF Statistics for Goodness of Fit and Some Comparisons. Journal of the American Statistical Association, 69, 730-737. https://doi.org/10.2307/2286009
42. Shapiro, S.S., & Wilk, M.B. (1965). An analysis of variance test for normality. Biometrika, 52, 591-611. https://doi.org/10.2307/2333709
43. Smirnov, N.V. (1948). Table for estimating the goodness of t of empirical distributions. The Annals of Mathematical Statistics, 19, 279-281.
44. Tabachnick, B.G., & Fidell, L.S. (2013). Using Multivariate Statistics. Pearson.
45. Trochim, W.M., & Donnelly, J.P. (2006). The research methods knowledge base. Atomic Dog.
46. Ukponmwan, H.N., & Ajibade, F.B. (2017). Evaluation of techniques for univariate normality test using monte carlo simulation. American Journal of Theoretical and Applied Statistics, 6(5), 51-61. https://doi.org/10.11648/j.ajtas.s.2017060501.18
47. Wilcox, R.R. (2010). Fundamentals of modern statistical methods: Substantially improving power and accuracy. Springer-Verlag.
48. Yap, B.W., & Sim, C.H. (2011). Comparisons of various types of normality tests. Journal of Statistical Computation and Simulation, 81(12), 2141 2155. https://doi.org/10.1080/00949655.2010.520163