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.