Testing for Normality When the Data Are Grouped Due to Rounding

A recent ANSI standard addressed the problem of assessing the assumption of normality using the W-test for normality (1). This same statistical test was presented in a TID publication devoted to statistical methodology in nuclear material control applications (2). Although the W-test has been demonstrated to be a superior test for normality against unspecified alternatives (3), this does not mean that the user is free to apply the test indiscriminately without considering the structure of a given data set. This statement, of course, is not restricted to application of this particular test; an all too frequent occurrence in data analysis and interpretation is the misapplication of statistical techniques to sets of data unsuited for such applications. This is because the user sometimes tends to overlook the assumptions underlying the use of each technique for one or more of a number of reasons, but often because of ignorance as to the crucial nature of some of the assumptions. The important assumption underlying the W-test for normality is that, under the null hypothesis, the data represent observations drawn from a normal distribution. In the strict sense, this implies that the measurements are made along a continuous scale, although in practice, all measurements are actually discrete in nature because of limitations of the measurement system. In many applications, this discrete nature of the data poses no problems when applying statistical procedures that assume continuity. However, in some applications, the limitations of the measurement system are such that the discreteness is quite severe, and failure to take this into account can give grossly misleading results. The purpose of this paper is to caution the user to use care in the application of the W-test to data that exhibit a discrete nature due to rounding imposed by the limitations of the measurement process. At the same time, it is hoped that the reader will be motivated to become more aware of the assumptions underlying statistical tests in general as he applies them, both from point of view of what these assumptions are and, perhaps more importantly, with respect to their effect on the conclusions suggested by the test