What is the Empirical Rule?
The empirical rule (also called the 68-95-99.7 rule or three-sigma rule) is a shorthand for remembering the percentage of values in a normal distribution that fall within 1, 2, and 3 standard deviations of the mean.
68%
within ±1σ
95%
within ±2σ
99.7%
within ±3σ
Visual Breakdown
The Classic Bell Curve
| Range | Percentage |
|---|---|
| μ ± 1σ | 68.27% |
| μ ± 2σ | 95.45% |
| μ ± 3σ | 99.73% |
Practical Applications
- Quick Probability Estimates:Without complex calculations, you can estimate that about 95% of data falls within 2 standard deviations of the mean.
- Outlier Detection:Data points beyond 3σ occur less than 0.3% of the time, making them statistical outliers worth investigating.
- Quality Control:Six Sigma methodology uses the rule to set quality thresholds and identify process variations.
Worked Examples
Example: SAT Scores
SAT scores are normally distributed with μ = 1050 and σ = 200.
- 68% of scores fall between 850 and 1250 (±1σ)
- 95% of scores fall between 650 and 1450 (±2σ)
- 99.7% of scores fall between 450 and 1650 (±3σ)
A score of 1450+ puts a student in the top ~2.5% of test-takers.
Limitations
Only Works for Normal Distributions
The empirical rule ONLY applies to data that follows a normal (Gaussian) distribution. For skewed or non-normal data, these percentages don't apply. Always check if your data is normally distributed before using this rule.
Further Reading
Sources
References and further authoritative reading used in preparing this article.