Empirical Rule Calculator (68-95-99.7)
The empirical rule describes the concentration of data in a normal distribution: approximately 68% of values fall within 1 standard deviation of the mean, about 95% fall within 2 standard deviations, and about 99.7% fall within 3 standard deviations. This rule is widely used in quality control (the 3-sigma rule), finance (value at risk), medicine (reference ranges), and education (standardized test score interpretation). Enter the mean and standard deviation of a normally distributed variable to see the corresponding value ranges for each level.
Empirical rule intervals
1 SD: [μ - σ, μ + σ] covers ~68.27%
2 SD: [μ - 2σ, μ + 2σ] covers ~95.45%
3 SD: [μ - 3σ, μ + 3σ] covers ~99.73%
These percentages come from the cumulative normal distribution: P(-k less than Z less than k) evaluated at k = 1, 2, 3. The rule is exact for a perfect normal distribution.
Applications of the empirical rule
- Quality control: the 3-sigma rule says process outputs beyond 3 standard deviations are considered defective (about 0.27% of output).
- Medicine: clinical lab reference ranges often span the middle 95% (mean plus or minus 2 standard deviations) of a healthy population.
- Finance: value at risk (VaR) often uses the 2-sigma interval (95% confidence) for one-day loss estimates.
- Education: IQ scores (mean = 100, SD = 15): 68% of people score between 85 and 115; 95% score between 70 and 130.
Frequently asked questions
What is the empirical rule?
The empirical rule (also called the 68-95-99.7 rule) states that for a normally distributed variable: approximately 68% of values fall within 1 standard deviation of the mean, about 95% fall within 2 standard deviations, and about 99.7% fall within 3 standard deviations.
Is the empirical rule exact?
The exact percentages are 68.27%, 95.45%, and 99.73% for 1, 2, and 3 standard deviations respectively. These are the areas of the standard normal distribution within those intervals. The rule is a useful approximation but applies only to data that is approximately normally distributed.
How do I use the empirical rule in practice?
For data with mean 100 and standard deviation 15 (such as IQ scores), the empirical rule tells you that 68% of scores fall between 85 and 115, 95% fall between 70 and 130, and 99.7% fall between 55 and 145. Values outside 3 standard deviations are extremely rare (about 3 in 1,000).
Can I use this rule for non-normal data?
No. The empirical rule applies specifically to normal (bell-curve) distributions. For non-normal distributions, use Chebyshev's inequality, which states that at least 75% of values fall within 2 standard deviations and at least 89% within 3, regardless of the distribution shape.
What is the connection between the empirical rule and z-scores?
The intervals in the empirical rule correspond to z-scores of plus or minus 1, 2, and 3. The probability P(-1 less than Z less than 1) equals 0.6827, P(-2 less than Z less than 2) equals 0.9545, and P(-3 less than Z less than 3) equals 0.9973.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Normal Distribution.
- NIST/SEMATECH e-Handbook: Engineering Statistics Handbook.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.