When outliers are different

When does the presence of an outlier in some measured property indicate that the outlying object differs qualitatively, rather than quantitatively, from other members of its apparent class? Historical astronomical examples include the many types of supernovae and short versus long gamma-ray bursts. A qualitative difference implies that some parameter has a characteristic scale, and hence its distribution cannot be a power law (that can have no such scale). If the distribution is a power law, the objects differ only quantitatively. The applicability of a power law to an empirical distribution may be tested by comparing the most extreme member to its next-most extreme. The probability distribution of their ratio is calculated, and compared to data for stars, radio and X-ray sources, and the fluxes, fluences, and rotation measures of fast radio bursts (FRBs). It is found with high statistical significance that the giant outburst of soft gamma repeater SGR 1806-20 differed qualitatively from its lesser outbursts and FRB 200428 differed qualitatively from other FRBs (by location in the Galaxy), but that in some supernova remnant models of rotation measure FRB 121102 is not, statistically significantly, an outlier.

Automated summary

The text discusses the significance of outliers in measured properties and how to distinguish between qualitative and quantitative differences. By comparing the ratio of the most extreme member to the next-most extreme, the applicability of power law to empirical distributions can be tested.