Normality tests, rounded data and random sets: Imprecision meets uncertainty

The measurement and representation of a continuous variable necessarily involves some degree of rounding and inevitably results in a loss of information that is potentially dangerous for some statistical inference tasks. In the present paper, we study the particular case of normality tests and show that these tests are typically doomed when applied to rounded data if the sample size is large or the scale is too small in comparison to the digit to which the data is rounded. For such purpose, we formalize the problem by using random sets to model both the uncertainty associated with the realization of the random variable and the imprecision caused by rounding. We end by providing some guidelines of use for practitioners that need to apply normality tests with rounded data.