Shapiro-Wilk Test for Non-Normality

A topic in the lecture "Assumption of Normality"

SPSS provides two statistical tests for deviation from normality, the 'Kolomogorov-Smirnov' test and the 'Shapiro-Wilk' test. Of the two, the Shapiro-Wilk (1965) test seems to be preferable (see the comparative study by Shapiro, Wilk, & Chen, 1968). The null hypothesis of the tests is that the population is normally distributed, and the alternative hypothesis is that it is not normally distributed. There are some important limitations to the usefulness of these tests.

  1. Because it involves null hypothesis significance testing, if you reject H0 you can conclude that the population is not normally distributed, but if you don't reject H0 then you only conclude that you failed to show the population is not normally distributed. In other words, you can prove the population is not normally distributed but you can't prove it is normally distributed.
  2. Rejecting H0 means that the population is not normally distributed, but it doesn't tell you whether it is because it is a fat-tailed distribution, a thin-tailed distribution, a skewed distribution, or something else. As we have seen, some of these deviations from normality are much more a problem than others.
  3. The tests are influenced by power. If you have a small sample the test may not have enough power to detect non-normality in the population (and it is when N is small that we usually have to worry the most because of the Central Limit Theorem). If you have a very large sample the test will detect even trivial deviations from normality, those we don't really have to worry about.

The tests for non-normality are available in SPSS under the Analyze>> DescriptiveStatistics>>Explore menu as ‘Normality plots with tests’ available when you click on the ‘Plots’ button.

Examples


Normal Distribution


  • NormalHist.jpg Shapiro-Wilk: p=0.521

Negatively Skewed Distribution


  • NegativeHist.jpg Shapiro-Wilk: p<.001

Fat-Tailed Distribution


  • FatHist.jpg Shapiro-Wilk: p<.001

Skinny-Tailed Distribution


  • SkinnyHist.jpg Shapiro-Wilk: p<.001


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