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The Empirical Rule
The empirical rule (also called the 68-95-99.7 rule or three-sigma rule) states that for a normal distribution, approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. This powerful rule provides quick estimates without calculation.
The rule is fundamental in quality control (Six Sigma), IQ scoring, standardized testing, manufacturing tolerances, and any field that assumes normally distributed data. It helps identify outliers and set practical boundaries.
The 68-95-99.7 Rule
Breakdown
| Range | Contains | Outside |
|---|---|---|
| μ ± 1σ | 68.27% | 31.73% |
| μ ± 2σ | 95.45% | 4.55% |
| μ ± 3σ | 99.73% | 0.27% |
Applications
- IQ Scores: Mean 100, SD 15. About 68% score 85-115, 95% score 70-130.
- Quality Control: Manufacturing within 3σ means 99.7% of products meet spec (3.4 defects per million for 6σ).
- Finance: Stock returns within 2σ cover 95% of normal trading days.
Frequently Asked Questions
Does the rule work for non-normal data?
No. The empirical rule only applies to approximately normal (bell-shaped) distributions. For skewed or multimodal data, use Chebyshev's inequality instead, which guarantees at least 75% within 2σ for any distribution.
What is Six Sigma?
Six Sigma aims for processes within ±6σ, allowing only 3.4 defects per million. This exceeds the empirical rule's 3σ by targeting even tighter quality control.
How do I check if data is normal?
Use a histogram, Q-Q plot, or formal tests like Shapiro-Wilk or Anderson-Darling. If the data looks roughly bell-shaped with most values near the center and tails tapering symmetrically, the empirical rule is a reasonable approximation.