Is formally testing for normality largely a waste of time?

A common assumption of many statistical techniques is that the data are conditionally normally distributed. For example, an assumption made when performing a t-test is that the variable being tested is normally distributed in each group. In the example of linear regression, one of the assumptions is that the residuals are normally distributed.

Clearly it is a good idea to examine the validity of any assumptions made when performing a statistical test. With the assumption of normality we can either examine this assumption graphically (e.g. by producing histograms or QQ plots) or we can perform formal statistical test of normality (e.g. with a a Kolmogorov–Smirnov test). I would argue that the former is a much better idea and the latter is largely a waste of time. The reasons behind this dismissal of formal testing falls broadly into three camps:

  1. Formal tests of normality are notoriously under powered. With a small sample size only major departures from normality are detected.
  2. Major departures from normality are blindingly obvious in graphical methods and so confirming what is before your eyes with a formal test is not very helpful.
  3. When the sample size is large enough to detect small deviations from normality it almost certainly doesn’t matter. This is due to the wonder of the central limit theorem (which I won’t go into here), which means that when sample sizes are large, inferences (by which I mean p-values and confidence intervals) are robust to modest deviations from normality. When sample sizes are very large, like the General Practice Patient Survey data we use with one to two million responses per year, our inferences are robust to quite large deviations from normality.

So In conclusion plotting a histogram is probably a wise move, but formal tests for normality are unlikely to give any further insight.

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