Estimating departure from normality

Indices of departure based on the Shapiro-Francia W' and the Shapiro -Wilk W statistics are derived, and shown to have a natural interpretation in relation to the normal probability plot.

Abstract

Departure of a sample from a normal distribution should be assessed by a quantity that is meaningful in terms of the data, rather than merely by the P-value from a test statistic. Indices of departure based on the Shapiro-Francia W' and the Shapiro-Wilk W statistics are derived, and shown to have a natural interpretation in relation to the normal probability plot. A new diagnostic plot is proposed. An example is given which shows the relationship between one of the new indices and errors in calculated reference ranges due to non-normality of the data.