Sains Malaysiana 40(6)(2011): 637–641

Sensitivity of Normality Tests to Non-normal Data

(Kepekaan Ujian Kenormalan Terhadap Data Tidak Normal)

Nor Aishah Ahad, Teh Sin Yin*, Abdul Rahman Othman & Che Rohani Yaacob

Robust Statisties Computational Laboratory, School of Distance Education, Universiti Sains Malaysia

11800 Minden, Penang, Malaysia

UUM College of Arts and Sciences, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia

Teh Sin Yin* & Che Rohani Yaacob

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Minden, Penang, Malaysia

Abdul Rahman Othman

Institute of Postgraducte Studies, Universiti Sains Malaysia, 11800 Minden, Penang, Malaysia

Received: 9 June 2010 / Accepted: 29 September 2010

ABSTRACT

In many statistical analyses, data need to be approximately normal or normally distributed. The Kolmogorov-Smirnov test, Anderson-Darling test, Cramer-von Mises test, and Shapiro-Wilk test are four statistical tests that are widely used for checking normality. One of the factors that influence these tests is the sample size. Given any test of normality mentioned, this study determined the sample sizes at which the tests would indicate that the data is not normal. The performance of the tests was evaluated under various spectrums of non-normal distributions and different sample sizes. The results showed that the Shapiro-Wilk test is the best normality test because this test rejects the null hypothesis of normality test at the smallest sample size compared to the other tests, for all levels of skewness and kurtosis of these distributions.

Keywords: Monte Carlo simulation; sample size; sensitivity; tests of normality

ABSTRAK

Dalam kebanyakan analisis statistik, data perlu tertabur secara normal atau menghampiri taburan normal. Empat ujian statistik yang digunakan secara meluas untuk memeriksa kenormalan data adalah Kolmogorov-Smirnov, Anderson-Darling, Cramer-von Mises dan Shapiro-Wilk. Salah satu daripada faktor yang mempengaruhi ujian-ujian ini ialah saiz sampel. Untuk sebarang ujian kenormalan seperti yang dinyatakan, kajian ini akan menentukan saiz sampel dan ujian-ujian akan menunjukkan bahawa data tersebut adalah tidak normal. Prestasi ujian-ujian ini dinilai pada pelbagai spektrum data yang tidak normal dan saiz sampel yang berbeza. Keputusan kajian menunjukkan bahawa ujian Shapiro-Wilk adalah ujian kenormalan terbaik kerana ujian ini menolak hipotesis nol bagi ujian kenormalan pada saiz sampel terkecil berbanding dengan ujian-ujian yang lain, untuk semua peringkat kepencongan dan kurtosis setiap taburan.

Kata kunci: Kepekaan; saiz sampel; simulasi Monte Carlo; ujian kenormalan

REFERENCES

Anderson, T.W. & Darling, D.A. 1952. Asymptotic theory of certain “goodness-of-fit” criteria based on stochastic processes. Ann. Math. Stat. 23: 193-212.

Anderson, T.W. 1962. On the distribution of the two-sample Cramer-von Mises criterion. Ann. Math. Stat. 33(3): 1148-1169.

Chambers, J.M., Cleveland, W.S., Kleiner, B. & Tukey, P.A. 1983. Graphical Methods for Data Analysis. Belmont, CA: Wadsworth International Group.

D’Agostino, R.B. & Stephens, M A. 1986. Goodness-of-Fit Techniques. New York: Marcel Dekker, Inc.

D’Agostino, R.B. 1971. An omnibus test of normality for moderate and large size samples. Biometrika 58(2): 341-348.

Fleishman, A.I. 1978. A method for simulating non-normal distributions. Psychometrika. 43: 521-532.

Gravetter, F.J. & Wallnau, L.B. 2000. Statistics for the Behavioral Sciences, 5th ed. Belmont, CA: Wadsworth.

Hoaglin, D.C. 1985. Summarizing Shape Numerically: The g-and-h Distributions, 461-513. In Exploring Data Tables, Trends, and Shapes. D. Hoaglin, F. Mosteller and J. Tukey (Eds), New York: Wiley.

Kenney, J.F. & Keeping, E.S. 1962. Skewness. §7.10 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand.

Kolmogorov, A. 1956. Foundations of the Theory of Probability, 2nd ed. Chelsea: New York.

Miles, J. & Shevlin, M. 2001. Applying Regression and Correlation. London: Sage.

SAS Institute Inc. 2004. SAS OnlineDoc® 9.1.2., SAS Institute Inc., Cary, NC.

Shapiro, S.S. & Wilk, M.B. 1965. An analysis of variance test for normality. Biometrika 52: 591-611.

Smirnov, S. 1936. Beschreibung einer neuen Acartia-Art aus dem Japanischen Meer nebst einiger Bemerkungen über die Untergattung Euacartia Steuer. Zool Anz. 114: 87-92.

*Correspondence author; email: syin.teh@gmail.com

 content