The normality test is a test, which once computed tells you an assessment of whether or not the range of your data is normally distributed, which is a prerequisite for many statistical tests because normal data is an underlying assumption in parametric testing. There are two main methods of assessing normality: graphically and numerically. What more could we draw from normality test?
A normality test can usually NOT be used to demonstrate that the data is normally distributed. It is typically ONLY able to indicate that the observed data is unlikely when the variability is explained by the normal distribution model. Therfore, such tests are typically NOT suited to check if assumptions are met.(1) (2)
The normal distribution is a "least information distribution". This means, if the data looks different from a normal distribution, some information is available from the data. A normality test can threfore be used to screen data sets for those which may indicate some useable information. This is the only practical use I can imagine (although I have no concrete example). Maybe others have more ideas.
(1) A (statistically) non-significant result does not meen that the assumptions are met, and a significant result does not say that they are not met. One may use significant results to reject the null that the assumption is met, but one still has no idea if the violation of the assumption is relevant.
(2) The regular use of "tests of assumption" which results dictate what final tests to use render the tests useless because they change the error-rates. To give a (rather stupid) example: consider the variable is normal, then it will happen that the normality test is significant. These cases are likely cases with relatively high variance where the "normal-based test" will likey fail to be significant. These cases are tested with a different test, maybe a rank-based test, that is more likely to give significant results in just such cases. Sounds nice for the researcher, but it is an unsound effective increase of the type-I error rate that is clearly not controlled at the nominal level anymore. But is the error rate is not controlled, the whole objective of performing a test is lost.
Great points. Some references:
http://rt.uits.iu.edu/visualization/analytics/docs/normality-docs/normality.pdf
http://www.unm.edu/~marcusj/testingassumptions.pdf
A few years ago, I gave a conference presentation on why I think it is rather pointless to use a statistical test of normality as a precursor to a t-test (or other parametric test). You can view a PDF of the slides at the link below. HTH.
https://www.nosm.ca/uploadedFiles/Research/Northern_Health_Research_Conference/Weaver,%20Bruce_Silly%20or%20Pointless%20Things.pdf
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