Tests for Normal distribution

Tests available in MedCalc

MedCalc offers 4 tests for Normal distribution:

  • The Shapiro-Wilk test (Shapiro & Wilk, 1965; Royston, 1995) and the Shapiro-Francia test (Shapiro & Francia, 1972; Roysten, 1993a) calculate a W and W' statistic, respectively, that tests whether a random sample comes from a Normal distribution. Small values of W or W' are evidence of departure from normality. The Shapiro-Wilk W statistic can only be computed when sample size is between 3 and 5000 (inclusive) (Royston, 1995), the Shapiro-Francia W' statistic can be computed when sample size ranges from 5 to 5000 (Roysten, 1993a & 1993b).
  • The D'Agostino-Pearson test (Sheskin, 2011) computes a single P-value for the combination of the coefficients of Skewness and Kurtosis.
  • The Kolmogorov-Smirnov test (Neter et al., 1988) with Lilliefors significance correction (Dallal & Wilkinson, 1986) is based on the greatest discrepancy between the sample cumulative distribution and the Normal cumulative distribution.
  • The Chi-squared goodness-of-fit test is applied to binned data (the data are put into classes) (Snedecor & Cochran, 1989) and requires a larger sample size than the other two tests.


The result of this test is expressed as 'accept Normality' or 'reject Normality', with P value.

  • If P is higher than 0.05, it may be assumed that the data have a Normal distribution and the conclusion ‘accept Normality’ is displayed.
  • If P is less than 0.05, then the hypothesis that the distribution of the observations in the sample is Normal, should be rejected, and the conclusion ‘reject Normality’ is displayed. In the latter case, the sample cannot accurately be described by arithmetic mean and standard deviation, and such samples should not be submitted to any parametrical statistical test or procedure, such as e.g. a t-test. To test the possible difference between not Normally distributed samples, the Wilcoxon test should be used, and correlation can be estimated by means of rank correlation.
  • When the sample size is small, it may not be possible to perform the selected test and an appropriate message will appear. In this case you can visually evaluate the symmetry and peakedness of the distribution using the Histogram, Cumulative frequency distribution, or Box-and-whisker plot.


  • Dallal GE, Wilkinson L (1986) An analytic approximation to the distribution of Lilliefors' test for normality. The American Statistician 40: 294–296.
  • Neter J, Wasserman W, Whitmore GA (1988) Applied statistics. 3rd ed. Boston: Allyn and Bacon, Inc.
  • Royston P (1993a) A pocket-calculator algorithm for the Shapiro-Francia test for non-normality: an application to medicine. Statistics in Medicine 12: 181–184.
  • Royston P (1993b) A Toolkit for Testing for Non-Normality in Complete and Censored Samples. Journal of the Royal Statistical Society. Series D (The Statistician) 42: 37-43.
  • Royston P (1995) A Remark on Algorithm AS 181: The W-test for Normality. Journal of the Royal Statistical Society. Series C (Applied Statistics) 44: 547-551.
  • Shapiro SS, Francia RS (1972) An approximate analysis of variance test for normality. Journal of the American Statistical Association 67: 215-216.
  • Shapiro SS, Wilk MB (1965) An analysis of variance test for normality (complete samples). Biometrika 52: 3-4.
  • Sheskin DJ (2011) Handbook of parametric and nonparametric statistical procedures. 5th ed. Boca Raton: Chapman & Hall /CRC.
  • Snedecor GW, Cochran WG (1989) Statistical methods, 8th edition. Ames, Iowa: Iowa State University Press.

See also

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