• An, H. Z., , and B. Cheng, 1996: A Kolmogorov–Smirnov type statistic with application to test for normality of time series. Int. Stat. Rev., 59 , 4561.

    • Search Google Scholar
    • Export Citation
  • Berner, J., 2005: Linking nonlinearity and non-Gaussianity of planetary wave behavior by the Fokker–Planck equation. J. Atmos. Sci., 62 , 20982117.

    • Search Google Scholar
    • Export Citation
  • Branstator, G., , and J. Berner, 2005: Linear and nonlinear signatures in the planetary wave dynamics of an AGCM: Phase space tendencies. J. Atmos. Sci., 62 , 17921811.

    • Search Google Scholar
    • Export Citation
  • Burgers, G., , and D. B. Stephenson, 1999: The “normality” of El Niño. Geophys. Res. Lett., 26 , 10271030.

  • Colucci, S. J., 1976: Winter cyclone frequencies over the eastern United States and adjacent western Atlantic, 1964–1973. Bull. Amer. Meteor. Soc., 57 , 548553.

    • Search Google Scholar
    • Export Citation
  • D’Agostino, R. B., , and D. A. Stephens, 1986: Goodness-of-Fit Techniques. Marcel Dekker, 576 pp.

  • Dallal, G. E., , and L. Wilkinson, 1986: An analytic approximation to the distribution of Lilliefors’s test statistic for normality. Amer. Stat., 40 , 294296.

    • Search Google Scholar
    • Export Citation
  • Gershunov, A., , N. Schneider, , and T. Barnett, 2001: Low-frequency modulation of the ENSO–Indian monsoon rainfall relationship: Signal or noise? J. Climate, 14 , 24862492.

    • Search Google Scholar
    • Export Citation
  • Harman, J. R., 1991: Synoptic Climatology of the Westerlies: Process and Pattern. Association of American Geographers, 80 pp.

  • Hunt, B. R., , R. L. Lipsman, , and J. M. Rosenberg, 2001: A Guide to MATLAB: For Beginners and Experienced Users. Cambridge University Press, 416 pp.

    • Search Google Scholar
    • Export Citation
  • Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Reanalysis Project. Bull. Amer. Meteor. Soc., 77 , 437471.

  • Krause, A., , and M. Olson, 2002: The Basics of S-PLUS. 3d ed. Springer, 448 pp.

  • Lau, N-C., 1988: Variability of the observed midlatitude storm tracks in relation to low-frequency changes in the circulation pattern. J. Atmos. Sci., 45 , 27182743.

    • Search Google Scholar
    • Export Citation
  • Lilliefors, H. W., 1967: On the Kolmogorov–Smirnov test for normality with mean and variance unknown. J. Amer. Stat. Assoc., 62 , 399402.

    • Search Google Scholar
    • Export Citation
  • Massey Jr., F. J., 1967: The Kolmogorov–Smirnov test for goodness of fit. J. Amer. Stat. Assoc., 46 , 6878.

  • Mohymont, B., , G. R. Demarée, , and D. N. Faka, 2004: Establishment of IDF-curves for precipitation in the tropical area of Central Africa—Comparison of technique and results. Nat. Hazards Earth Syst. Sci., 4 , 375387.

    • Search Google Scholar
    • Export Citation
  • R Development Core Team, 2005: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2535 pp. [Available online at http://www.R-project.org.].

  • Sanders, M. A., , and A. S. Lea, 2005: Seasonal prediction of hurricane activity reaching the coast of the United States. Nature, 434 , 10051008.

    • Search Google Scholar
    • Export Citation
  • Stephens, M. A., 1974: EDF statistics for goodness of fit and some comparisons. J. Amer. Stat. Assoc., 69 , 730737.

  • Stephenson, D. B., , A. Hannachi, , and A. O’Neill, 2004: On the existence of multiple climate regimes. Quart. J. Roy. Meteor. Soc., 130 , 583605.

    • Search Google Scholar
    • Export Citation
  • Thode, H. C., 2002: Testing for Normality. Marcel Dekker, 368 pp.

  • Zishka, K. M., , and P. J. Smith, 1980: The climatology of cyclones and anticyclones over North America and surrounding ocean environs for January and July, 1950–77. Mon. Wea. Rev., 108 , 387401.

    • Search Google Scholar
    • Export Citation
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A Cautionary Note on the Use of the Kolmogorov–Smirnov Test for Normality

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  • 1 Nansen Environmental and Remote Sensing Center, and Bjerknes Centre for Climate Research, and Geophysical Institute, University of Bergen, Bergen, Norway
  • | 2 Department of Mathematics, University of Bergen, Bergen, Norway
  • | 3 Geophysical Institute, University of Bergen, and Bjerknes Centre for Climate Research, Bergen, Norway
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Abstract

The Kolmogorov–Smirnov goodness-of-fit test is used in many applications for testing normality in climate research. This note shows that the test usually leads to systematic and drastic errors. When the mean and the standard deviation are estimated, it is much too conservative in the sense that its p values are strongly biased upward. One may think that this is a small sample problem, but it is not. There is a correction of the Kolmogorov–Smirnov test by Lilliefors, which is in fact sometimes confused with the original Kolmogorov–Smirnov test. Both the Jarque–Bera and the Shapiro–Wilk tests for normality are good alternatives to the Kolmogorov–Smirnov test. A power comparison of eight different tests has been undertaken, favoring the Jarque–Bera and the Shapiro–Wilk tests. The Jarque–Bera and the Kolmogorov–Smirnov tests are also applied to a monthly mean dataset of geopotential height at 500 hPa. The two tests give very different results and illustrate the danger of using the Kolmogorov–Smirnov test.

Corresponding author address: Dr. Dag Johan Steinskog, Nansen Environmental and Remote Sensing Center, Thormøhlensgt. 47, N-5006 Bergen, Norway. Email: dag.johan.steinskog@nersc.no

Abstract

The Kolmogorov–Smirnov goodness-of-fit test is used in many applications for testing normality in climate research. This note shows that the test usually leads to systematic and drastic errors. When the mean and the standard deviation are estimated, it is much too conservative in the sense that its p values are strongly biased upward. One may think that this is a small sample problem, but it is not. There is a correction of the Kolmogorov–Smirnov test by Lilliefors, which is in fact sometimes confused with the original Kolmogorov–Smirnov test. Both the Jarque–Bera and the Shapiro–Wilk tests for normality are good alternatives to the Kolmogorov–Smirnov test. A power comparison of eight different tests has been undertaken, favoring the Jarque–Bera and the Shapiro–Wilk tests. The Jarque–Bera and the Kolmogorov–Smirnov tests are also applied to a monthly mean dataset of geopotential height at 500 hPa. The two tests give very different results and illustrate the danger of using the Kolmogorov–Smirnov test.

Corresponding author address: Dr. Dag Johan Steinskog, Nansen Environmental and Remote Sensing Center, Thormøhlensgt. 47, N-5006 Bergen, Norway. Email: dag.johan.steinskog@nersc.no

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