Small subsets of data with disproportionate influence on model outcomes can have dramatic impacts on conclusions, with a few data points sometimes overturning key findings. While recent work has developed methods to identify these most influential sets, no formal theory exists to determine when their influence reflects genuine problems rather than natural sampling variation. We address this gap by developing a principled framework for assessing the statistical significance of most influential sets. Our theoretical results characterize the extreme value distributions of maximal influence and enable rigorous hypothesis tests for excessive influence, replacing current ad-hoc sensitivity checks. We demonstrate the practical value of our approach through applications across economics, biology, and machine learning benchmarks.
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