Extending the Three-Sigma Limit for Non-Gaussian Data Using Extreme Value Theory

The three-sigma limit is a fundamental concept in quality control, ensuring that 99.865% (one-sided) or 99.73% (two-sided) of data fall within defined thresholds when the data follow a Gaussian distribution. However, the conventional interpretation of three-sigma limits may no longer be valid for non-Gaussian data. This discrepancy can result in inaccurate assessments of process capability and anomaly detection, ultimately undermining the reliability of quality control. In this paper, we propose a new method for setting three-sigma limits based on extreme value theory (EVT) that ensures the validity of the conventional quality control interpretation even for non-Gaussian data. Our proposed method, based on return levels within EVT, extends the conventional three-sigma limit. We demonstrate the effectiveness of our method through numerical experiments on both synthetic datasets and anomaly scores from AI-based visual inspection.