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
Nor Aishah Ahad
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
Postgraduate Studies, Universiti Sains Malaysia, 11800 Minden, Penang, Malaysia
Diserahkan: 9 Jun 2010 / Diterima:
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
RUJUKAN
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.
*Pengarang untuk surat-menyurat; email:
syin.teh@gmail.com
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