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Shapiro-Wilk test

Like the Kolmogorov-Smirnov test, the Shapiro-Wilk test is a commonly used test to check whether a variable is normally distributed. A statistically significant result indicates that the variable under investigation is not normally distributed.

Both tests are only useful for research with a small number of cases. From 25 or more cases, the Shapiro-Wilk test almost always produces statistically significant results. It is therefore actually more surprising if there is no statistically significant result.

For small numbers (n < 30), non-parametric tests are preferred. For larger numbers, the other tests are preferred. This preference is independent of the test result of the Shapiro-Wilk test.

Better indicators for the normal distribution of a variable are skewness and kurtosis.

Moreover, the question is why a variable should be tested for normal distribution. The population in a country will never be normally distributed at the variable age. That would amount to many people about 40 years old and hardly any babies or old people. The same applies to the income of people or the turnover of companies. It is very unlikely that these variables will be normally distributed. Therefore, testing for normal distribution of variables that are not normally distributed by nature, is very strange. For a research, it is better to test for representativity than for normal distribution.

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