Normal Distribution Tests

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Normal Distribution Tests

The Kolmogorov-Smirnov test, the Shapiro-Wilk test (for sample sizes up to 2000), Stephens' test (for sample sizes greater than 2000), D'Agostino's test for skewness, the Anscombe-Glynn test for kurtosis, and the D'Agostino-Pearson omnibus test can be used to test the null hypothesis that the population distribution from which the data sample is drawn is a Gaussian (normal) distribution.


  • Within the sample, the values are independent, and identically distributed.
  • For the Kolmogorov-Smirnov test, the mean and variance of the ypothesized normal distribution should be specified in advance. If the mean and/or the variance must be estimated from the data, the Kolmogorov-Smirnov test becomes conservative, and thus less likely to reject the null hypothesis. The other normality tests listed above do not assume a specified mean or variance for the hypothesized normal distribution.


  • Ways to detect before performing the normality test whether your data violate any assumptions.
  • Ways to examine normality test results to detect assumption violations.
  • Possible alternatives if your data or normality test results indicate assumption violations.

To properly analyze and interpret results of normal distribution tests, you should be familiar with the following terms and concepts:

If you are not familiar with these terms and concepts, you are advised to consult with a statistician. Failure to understand and properly apply normal distribution tests may result in drawing erroneous conclusions from your data. Additionally, you may want to consult the following references:

  • Conover, W. J. 1980. Practical Nonparametric Statistics. 2nd ed. New York: John Wiley & Sons.
  • D'Agostino, R. B. and Stephens, M. A., eds. 1986. Goodness-of-fit Techniques. New York: Dekker.
  • Daniel, Wayne W. 1978. Applied Nonparametric Statistics. Boston: Houghton Mifflin.
  • Daniel, Wayne W. 1995. Biostatistics. 6th ed. New York: John Wiley & Sons.
  • Rosner, Bernard. 1995. Fundamentals of Biostatistics. 4th ed. Belmont, California: Duxbury Press.
  • Sokal, Robert R. and Rohlf, F. James. 1995. Biometry. 3rd. ed. New York: W. H. Freeman and Co.
  • Zar, Jerrold H. 1996. Biostatistical Analysis. 3rd ed. Upper Saddle River, NJ: Prentice-Hall.

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