Abstract <p>For many statistical procedures, it is important to assume the source data have a normal distribution. However, if this assumption is insufficiently justified, the use of these procedures can lead to false conclusions. The problem of normality testing has therefore received much attention in the literature. This work considers normality testing in a case where the data consist of a number of small independent samples, in each of which the observations are independent and identically distributed but have different shift and scale parameters from sample to sample. In such cases, we must use statistics that are independent of the parameters. A natural way of eliminating the shift parameter is to substitute the observations in each small sample according to their differences. This work estimates the stability of such decompositions and compares the power of several normality tests based on transformed data.</p>