Testing experimental data for univariate normality

BackgroundMany experimentally-derived data sets are generated in the practice of clinical chemistry. Graphical presentation is essential to assess the data distribution. The distribution must also be assessed quantitatively. These approaches will determine if the data is Normal or not. Finally the results of these tests of Normality must be shown to be free of sample size effects.MethodsFour experimentally-derived data sets were used. They represented normal, positive kurtotic, positive- and negatively-skewed distributions. These data sets were examined by graphical techniques, by moment tests, by tests of Normality, and monitored for sample size effects.ResultsThe preferred graphical techniques are the histogram and the box-and-whisker plots that may be supplemented, with advantage, by quantile–quantile or probability–probability plots. Classical tests of skewness and kurtosis can produce conflicting and often confusing results and, as a consequence, the alternative use of the newer L-moments is advocated. Normality tests included the Kolmogorov–Smirnov (Lilliefors modification), Cramér-von Mises and Anderson–Darling tests (empirical distribution function statistics) and the Gan–Koehler, Shapiro–Wilk, Shapiro–Francia, and Filliben tests (regression/correlation techniques). Of these only the Anderson–Darling, Shapiro–Wilk, and Shapiro–Francia tests correctly classified all four test samples. The effect of sample size on the resulting p-value was investigated using Royston's V′ / v′ graphical test.ConclusionsA systematic approach to Normality testing should follow the route of graphical presentation, the use of L-moments, the use of Anderson–Darling, Shapiro–Wilk, or Shapiro–Francia testing, and Royston's sample size monitoring.