Identifying data that are not normally distributed

Many of the statistical techniques that we are going to look at in later chapters only work properly if the data being analysed follow a normal distribution. The requirement is more than just a statistical nicety. In some cases, we can get quite ludicrous results by applying methods that assume normality to severely non-normal data. An unfortunate side-effect of the very term normal distribution is a subliminal suggestion that such distributions are what we would normally expect to obtain and anything else is, in some way, abnormal. That sadly is not the case. 

There are a number of statistical tests that will allegedly help in deciding whether data deviate from a normal distribution. However, in the author s experience the results of such tests tend to be difficult to interpret in real-life situations. What I am going to suggest is a simple visual inspection of histograms of data, and I have highlighted specific features that indicate non-normality. 

There are three visual characteristics that any true normal distribution will possess. Strictly speaking, possession of all three of these does not guarantee normality, but it is rare to find data sets that fit all three criteria and still manage to be sufficiently nonnormal to cause serious practical difficulties. The three characteristics are  

Byway of illustration, the three cases below show infractions of each of the criteria in turn. You do need to be aware that, unless samples are very large, histograms will not exactly follow the classical, pleasing-to-the-eye bell curve. What you need to check for are obvious and gross deviations from a normal distribution. 

Any attempt to apply statistical methods that rely upon a normal distribution to data like this is likely to end in nonsensical conclusions. 

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