International Encyclopedia of Statistical Science

2011 Edition
| Editors: Miodrag Lovric

Normality Tests

Reference work entry
DOI: https://doi.org/10.1007/978-3-642-04898-2_423

The Importance of Testing for Normality

Many statistical procedures such as estimation and hypothesis testing have the underlying assumption that the sampled data come from a normal distribution. This requires either an effective test of whether the assumption of normality holds or a valid argument showing that non-normality does not invalidate the procedure. Tests of normality are used to formally assess the assumption of the underlying distribution.

Much statistical research has been concerned with evaluating the magnitude of the effect of violations of the normality assumption on the true significance level of a test or the efficiency of a parameter estimate. Geary (1947) showed that for comparing two variances, having a symmetric non-normal underlying distribution can seriously affect the true significance level of the test. For a value of 1.5 for the kurtosis of the alternative distribution, the actual significance level of the test is 0.000089, as compared to the nominal level...

This is a preview of subscription content, log in to check access.

References and Further Reading

  1. Anderson TW, Darling DA (1954) A test of goodness of fit. J Am Stat Assoc 49:765–769zbMATHMathSciNetGoogle Scholar
  2. Box GEP (1953) Non-normality and tests on variances. Biometrika 40:318–335zbMATHMathSciNetGoogle Scholar
  3. D’Agostino RB, Lee AFS (1977) Robustness of location estimators under changes of population kurtosis. J Am Stat Assoc 72:393–396Google Scholar
  4. D’Agostino RB, Stephens MA (eds) (1986) Goodness-of-fit techniques. Marcel Dekker, New YorkzbMATHGoogle Scholar
  5. David HA, Hartley HO, Pearson ES (1954) The distribution of the ratio, in a single normal sample, of the range to the standard deviation. Biometrika 41:482–493zbMATHMathSciNetGoogle Scholar
  6. Franck WE (1981) The most powerful invariant test of normal versus Cauchy with applications to stable alternatives. J Am Stat Assoc 76:1002–1005zbMATHGoogle Scholar
  7. Geary RC (1936) Moments of the ratio of the mean deviation to the standard deviation for normal samples. Biometrika 28:295–305zbMATHGoogle Scholar
  8. Geary RC (1947) Testing for normality. Biometrika, 34:209–242zbMATHMathSciNetGoogle Scholar
  9. Jarque C, Bera A (1980) Efficient tests for normality, homoskedasticity and serial independence of regression residuals. Econ Lett 6:255–259MathSciNetGoogle Scholar
  10. Jarque C, Bera A (1987) A test for normality of observations and regression residuals. Int Stat Rev 55:163–172zbMATHMathSciNetGoogle Scholar
  11. Johnson ME, Tietjen GL, Beckman RJ (1980) A new family of probability distributions with application to Monte Carlo studies. J Am Stat Assoc 75:276–279zbMATHMathSciNetGoogle Scholar
  12. Pearson ES, Please NW (1975) Relation between the shape of population distribution and the robustness of four simple test statistics. Biometrika 62:223–241zbMATHMathSciNetGoogle Scholar
  13. Shapiro SS, Wilk MB (1965) An analysis of variance test for normality (complete samples). Biometrika 52:591–611zbMATHMathSciNetGoogle Scholar
  14. Subrahmaniam K, Subrahmaniam K, Messeri JY (1975) On the robustness of some tests of significance in sampling from a compound normal distribution. J Am Stat Assoc 70:435–438zbMATHGoogle Scholar
  15. Thode HC Jr (2002) Testing for normality. Marcel-Dekker, New YorkzbMATHGoogle Scholar
  16. Tukey JW (1960) A survey of sampling from contaminated distributions. In: Olkin I, Ghurye SG, Hoeffding W, Madow WG, Mann HB (eds) Contributions to probability and statistics. Stanford University Press, CA, pp 448–485Google Scholar
  17. Uthoff VA (1970) An optimum test property of two well-known statistics. J Am Stat Assoc 65:1597–1600zbMATHMathSciNetGoogle Scholar
  18. Uthoff VA (1973) The most powerful scale and location invariant test of the normal versus the double exponential. Ann Stat 1:170–174zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Stony Brook UniversityStony BrookUSA