3sigma

Three-sigma, or the three-sigma rule, is a statistical concept tied to the normal distribution. It is part of what is often called the empirical rule or the 68-95-99.7 rule, describing how data values spread around the mean when the distribution is approximately normal. The symbol σ denotes the standard deviation of the distribution.

For a normal distribution, about 68.3% of observations fall within one standard deviation of the mean (±1σ), about 95.4% fall within two standard deviations (±2σ), and about 99.7% fall within three standard deviations (±3σ). Consequently, roughly 0.3% of observations lie beyond ±3σ on either side. These percentages are exact for an ideal normal distribution and are approximate for real-world data that deviate from normality.

Applications of the three-sigma rule include quality control and statistical process monitoring. In control charts and process capability analysis, ±3σ limits are commonly used as thresholds for signaling potential special causes or out-of-control conditions. The concept also informs data analysis, anomaly detection, and risk assessment, where deviations beyond ±3σ may warrant further investigation.

Limitations should be noted: the rule relies on normality. Many real-world datasets exhibit skewness, heavy tails, or other deviations, which affect the exact percentages. Nevertheless, the three-sigma concept remains a widely used heuristic for assessing variation and identifying unusually distant observations in Gaussian-like processes.