The Empirical Rule — 68-95-99.7 for Normal Distributions
Foundations of Statistics
The 68-95-99.7 Rule Every Analyst Needs
The empirical rule provides instant intuition about data spread in normal distributions, enabling quick assessments without complex calculations. This simple framework is the foundation for outlier detection, quality control, and rapid data analysis.
- Quality Assurance — Six Sigma methodologies use the rule to identify process deviations
- Risk Management — Financial institutions apply it to estimate VaR and expected loss ranges
- Clinical Research — Researchers quickly assess whether patient measurements fall within expected ranges
Three numbers that capture the essence of normal variability.
Core Concepts
The empirical rule provides exact percentages for how probability mass concentrates around the mean in a normal distribution. It is a direct consequence of the Gaussian pdf's structure.
Rigorous Derivation
Chebyshev's Inequality (The Universal Bound)
Higher-Order Concentration: The 6σ Rule
Generalization: Higher-Order Concentration Inequalations
Worked Example
Relationship to the Normal Distribution Family
The empirical rule is a special property of the Gaussian family. Other distributions have their own concentration behavior:
- Laplace ():
- Uniform (): — exactly 100% for
- Cauchy: No finite variance exists, so the empirical rule doesn't apply at all
Specific Applications
- Six Sigma manufacturing — Defect rates are computed using from the empirical rule.
- Process capability indices — and are defined in terms of tolerance limits.
- Outlier detection — Values beyond are flagged as potential outliers (0.27% false positive rate for normal data).
- Standardized testing — IQ scores (): 68% score between 85–115, 95% between 70–130.