A Brief Review of Tests for Normality
American Journal of Theoretical and Applied Statistics
Volume 5, Issue 1, January 2016, Pages: 5-12
Received: Dec. 24, 2015; Accepted: Jan. 5, 2016; Published: Jan. 27, 2016
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Authors
Keya Rani Das, Department of Statistics, Bangabandhu Sheikh Mujibur Rahman Agricultural University, Gazipur, Bangladesh
A. H. M. Rahmatullah Imon, Department of Mathematical Sciences, Ball State University, Muncie, IN, USA
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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.
Keywords
Power, Empirical Cdf, Outlier, Moments, Skewness, Kurtosis, Supernormality
To cite this article
Keya Rani Das, A. H. M. Rahmatullah Imon, A Brief Review of Tests for Normality, American Journal of Theoretical and Applied Statistics. Vol. 5, No. 1, 2016, pp. 5-12. doi: 10.11648/j.ajtas.20160501.12
Copyright
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
Anderson, T. W., and 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. http://www.cithep.caltech.edu/~fcp/statistics/hypothesisTest/PoissonConsistency/AndersonDarling1952.pdf.
[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.
[11]
Geary, R. C. 1935. ‘‘The ratio of mean deviation to the standard deviation as a test of normality.’’ Biometrika 27: 310-332.
[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.
[21]
Kolmogorov, A. 1933. ‘‘Sulla determinazione empirica di una legge di distribuzione.’’ G. Ist. Ital. Attuari 4, 83–91.
[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.
[28]
Shapiro, S. S., and Francia, R. S. 1972. ‘‘An approximate analysis of variance test for normality.’’ Journal of the American Statistical Association 67(337): 215-216. DOI: 10.1080/01621459.1972.10481232.
[29]
Shapiro, S. S., and Wilk, M. B. 1965. ‘‘An analysis of variance test for normality (complete samples).’’ Biometrika 52(3/4): 591-611. http://sci2s.ugr.es/keel/pdf/algorithm/articulo/shapiro1965.pdf.
[30]
Smirnov, N. 1948. ‘‘Table for estimating the goodness of fit of empirical distributions.’’ Annals of Mathematical Statistics 19(2): 279–281. doi: 10.1214/aoms/1177730256.
[31]
Stephens, M. A. 1974. ‘‘EDF statistics for goodness of fit and some comparisons.’’ Journal of the American Statistical Association 69(347): 730-737.
[32]
Weisberg, S., and Bingham, C. 1975. ‘‘An approximate analysis of variance test for non-normality suitable for machine calculation.’’ Technometrics 17(1): 133-134.
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