What Is the Empirical Rule?
The empirical rule (also called the 68-95-99.7 rule or the three-sigma rule) describes how data is distributed in a normal (bell-shaped) distribution:
• 68% of data falls within 1 standard deviation (μ ± σ)
• 95% of data falls within 2 standard deviations (μ ± 2σ)
• 99.7% of data falls within 3 standard deviations (μ ± 3σ)
How It Works
For any normally distributed dataset, you only need to know the mean (μ) and standard deviation (σ) to determine what percentage of data falls within any range. This makes the empirical rule incredibly useful for quick statistical analysis.
Applications
- Quality control: Manufacturing tolerances and defect rates
- Finance: Expected return ranges and risk assessment
- Education: Understanding test score distributions
- Science: Experimental data analysis and outlier detection
Identifying Outliers
Since 99.7% of data falls within 3 standard deviations, any value beyond 3σ from the mean is considered a statistical outlier, occurring less than 0.3% of the time.
How to Use
Enter the mean and standard deviation of your dataset. The calculator shows the ranges and data percentages for 1, 2, and 3 standard deviations.
Limitations
The empirical rule only applies to data that follows a normal distribution. Skewed or multimodal datasets do not follow this pattern. Always verify normality before applying the rule.