American Journal of Theoretical and Applied Statistics

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A Brief Review of Tests for Normality

Received: 24 December 2015    Accepted: 05 January 2016    Published: 27 January 2016
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Abstract

In statistics it is conventional to assume that the observations are normal. The entire statistical framework is grounded on this assumption and if this assumption is violated the inference breaks down. For this reason it is essential to check or test this assumption before any statistical analysis of data. In this paper we provide a brief review of commonly used tests for normality. We present both graphical and analytical tests here. Normality tests in regression and experimental design suffer from supernormality. We also address this issue in this paper and present some tests which can successfully handle this problem.

DOI 10.11648/j.ajtas.20160501.12
Published in American Journal of Theoretical and Applied Statistics (Volume 5, Issue 1, January 2016)
Page(s) 5-12
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Power, Empirical Cdf, Outlier, Moments, Skewness, Kurtosis, Supernormality

References
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[2] Bera, A. K., and Jarque, C. M. 1982. ‘‘Model specification tests: A simultaneous approach.’’ Journal of Econometrics 20: 59-82.
[3] Bowman, K. O., and Shenton, B. R. 1975. ‘‘Omnibus test contours for departures from normality based on √(b_(1) ) and b<𝑠𝑢𝑏>2.’’ Biometrika 64: 243-50.
[4] Chambers, J. M., Cleveland, W. S., Kleiner, B., and Tukey, P. A. 1983. Graphical Methods for Data Analysis. Boston. Duxbury Press.
[5] Cook, R. D., and Weisberg, S. 1982. Residuals and Influence in Regression. New York. Chapman and Hall.
[6] D'Agostino, R. B. 1971. ‘‘An omnibus test of normality for moderate and large sample sizes.’’ Biometrika 58(August): 341-348.
[7] D'Agostino, R. B. 1986. ‘‘Tests for normal distribution.’’ In Goodness-of-fit Techniques, edited by D'Agostino, R. B., and Stephens, M. A. 367-420. New York. Marcel Dekker.
[8] DʼAgostino R, and Pearson E. S. 1973. ‘‘Tests for departure from normality. Empirical results for the distributions of b2 and √(b_(1) ).’’ Biometrika. 60(3), 613-622.
[9] David, H. A., Hartley, H. O., and Pearson, E. S. 1954. ‘‘The distribution of the ratio, in a single normal sample of range to standard deviation.’’ Biometrika 41: 482-93.
[10] Fisher, R. A. 1930. ‘‘The moments of the distribution for normal samples of measures of departure from normality.’’ Proceedings of the Royal Society of London 130(December): 16-28.
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[12] Geary, R. C. 1947. ‘‘Testing for normality.’’ Biometrika 34: 209-242. http://webspace.ship.edu/pgmarr/Geo441/Readings/Geary%201947%20%20Testing%20for%20Normality.pdf.
[13] Gnanadesikan, R. 1977. Methods for Statistical Analysis of Multivariate Data. New York. Wiley.
[14] Huber, P. J. 1973. ‘‘Robust regression: Asymptotics, conjectures, and Monte Carlo.’’ The Annals of Statistics 1(5): 799-821. DOI: 10.1214/aos/1176342503.
[15] Imon, A. H. M. R. 2003. ‘‘Simulation of errors in linear regression: An approach based on fixed percentage area.’’ Computational Statistics 18(3): 521–531.
[16] Imon, A. H. M. R. 2003. ‘‘Regression residuals, moments, and their use in tests for normality.’’ Communications in Statistics—Theory and Methods, 32(5): 1021–1034.
[17] Imon, A. H. M. R. 2015. ‘‘An Introduction to Regression, Time Series, and Forecasting.’’ (To appear).
[18] Imon, A. H. M. R., and Das, K. 2015. ‘‘Analyzing length or size based data: A study on the lengths of peas plants.’’ Malaysian Journal of Mathematical Sciences 9(1): 1-20. http://einspem.upm.edu.my/journal/fullpaper/vol9/1.%20imon%20&%20keya.pdf.
[19] Judge, G. G., Griffith, W. E., Hill, R. C., Lutkepohl, H., and Lee, T. 1985. Theory and Practice of Econometrics. 2nd. Ed. New York. Wiley.
[20] Koenker, R. W. 1982. ‘‘Robust methods in econometrics.’’ Econometric Reviews 1: 213-290.
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[22] Mardia, K. V. 1980. ‘‘Tests of univariate and multivariate normality.’’ In Handbook of Statistics 1: Analysis of Variance, edited by Krishnaiah, P. R. 279-320. Amsterdam. North-Holland Publishing.
[23] Pearson, K. 1905. ‘‘On the general theory of skew correlation and non-linear regression.’’ Biometrika 4: 171-212.
[24] Pearson, E. S. 1930. ‘‘A further development of tests for normality.’’ Biometrika 22(1-2): 239-249. doi: 10.1093/biomet/22.1-2.239.
[25] Pearson, E. S., and Please, N. W. 1975. ‘‘Relation between the shape of population distribution and the robustness of four simple statistical tests.’’ Biometrika 62: 223-241.
[26] Rana, M. S., Habshah, M. and Imon, A. H. M. R. 2009. ‘‘A robust rescaled moments test for normality in regression.’’ Journal of Mathematics and Statistics 5 (1): 54–62.
[27] Royston, J. P. 1982. ‘‘An extension of Shapiro-Wilk's W test for non-normality to large samples.’’ Applied Statistics 31: 115-124.
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Author Information
  • Department of Statistics, Bangabandhu Sheikh Mujibur Rahman Agricultural University, Gazipur, Bangladesh

  • Department of Mathematical Sciences, Ball State University, Muncie, IN, USA

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    Keya Rani Das, A. H. M. Rahmatullah Imon. (2016). A Brief Review of Tests for Normality. American Journal of Theoretical and Applied Statistics, 5(1), 5-12. https://doi.org/10.11648/j.ajtas.20160501.12

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    ACS Style

    Keya Rani Das; A. H. M. Rahmatullah Imon. A Brief Review of Tests for Normality. Am. J. Theor. Appl. Stat. 2016, 5(1), 5-12. doi: 10.11648/j.ajtas.20160501.12

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    AMA Style

    Keya Rani Das, A. H. M. Rahmatullah Imon. A Brief Review of Tests for Normality. Am J Theor Appl Stat. 2016;5(1):5-12. doi: 10.11648/j.ajtas.20160501.12

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  • @article{10.11648/j.ajtas.20160501.12,
      author = {Keya Rani Das and A. H. M. Rahmatullah Imon},
      title = {A Brief Review of Tests for Normality},
      journal = {American Journal of Theoretical and Applied Statistics},
      volume = {5},
      number = {1},
      pages = {5-12},
      doi = {10.11648/j.ajtas.20160501.12},
      url = {https://doi.org/10.11648/j.ajtas.20160501.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajtas.20160501.12},
      abstract = {In statistics it is conventional to assume that the observations are normal. The entire statistical framework is grounded on this assumption and if this assumption is violated the inference breaks down. For this reason it is essential to check or test this assumption before any statistical analysis of data. In this paper we provide a brief review of commonly used tests for normality. We present both graphical and analytical tests here. Normality tests in regression and experimental design suffer from supernormality. We also address this issue in this paper and present some tests which can successfully handle this problem.},
     year = {2016}
    }
    

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    AB  - In statistics it is conventional to assume that the observations are normal. The entire statistical framework is grounded on this assumption and if this assumption is violated the inference breaks down. For this reason it is essential to check or test this assumption before any statistical analysis of data. In this paper we provide a brief review of commonly used tests for normality. We present both graphical and analytical tests here. Normality tests in regression and experimental design suffer from supernormality. We also address this issue in this paper and present some tests which can successfully handle this problem.
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