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.

中文翻译：

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当异常值不同时

某个测量属性中异常值的存在何时表明该异常对象与其表观类别的其他成员在质量上而不是数量上不同？历史上的天文例子包括多种类型的超新星和短与长的伽马射线暴。质的差异意味着某些参数具有特征尺度，因此它的分布不能是幂律（不能有这样的尺度）。如果分布是幂律，则对象仅在数量上有所不同。幂律对经验分布的适用性可以通过将最极端的成员与其下一个最极端的成员进行比较来测试。计算它们比率的概率分布，并与恒星、无线电和 X 射线源的数据以及通量、通量、和快速射电暴 (FRB) 的旋转测量。发现具有高度统计意义的是，软伽马中继器 SGR 1806-20 的巨大爆发与其较小的爆发存在质的差异，而 FRB 200428 与其他 FRB 存在质的差异（按银河系中的位置），但在一些超新星残骸的旋转模型中测量 FRB 121102 在统计学上不是异常值。