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Empirical Rule Calculator (68-95-99.7)
The empirical rule says that for bell-shaped (normal) data about 68% of values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three. Enter the mean and standard deviation to get the three ranges, and optionally two values to find the percentage of data between them.
When the two values are a whole number of standard deviations from the mean, the calculator adds the standard segments of the rule (34%, 13.5%, 2.35% and 0.15% on each side) the way a textbook does. For IQ scores with mean 100 and SD 15, the share between 85 and 130 is 34 + 34 + 13.5 = 81.5%. It also gives the exact normal-curve area, 81.86%, so you can see how close the rule is. When a value is not a whole number of SDs away, only the exact area is shown.
Leave either value blank for an open-ended range (above or below a single value). For any probability under a normal curve, including values between the whole-SD marks, use the normal distribution calculator; for a single score’s position, the z-score calculator.
Result and step-by-step solution
By the empirical rule (exact normal area: 81.8595%).
| Interval | Range | Empirical rule | Exact (normal) |
|---|---|---|---|
| μ ± 1σ | 85 to 115 | 68% | 68.2689% |
| μ ± 2σ | 70 to 130 | 95% | 95.45% |
| μ ± 3σ | 55 to 145 | 99.7% | 99.73% |
- RuleAbout 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ (bell-shaped data).
- Intervalsμ ± 1σ = 100 ± 15; μ ± 2σ = 100 ± 30; μ ± 3σ = 100 ± 45
- Convert the limits to z-scoresz = (x − μ) ÷ σ: lower z = −1, upper z = 2
- AreaAdd the rule segments (34%, 13.5%, 2.35%, 0.15% on each side) between the two z values = 81.5%. Exact normal area Φ(2) − Φ(−1) = 81.8595%
Frequently asked questions
- What is the 68-95-99.7 rule?
- For normally distributed data, about 68% of values fall within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3.
- What percent of data is above 2 standard deviations?
- About 2.5% (half of the 5% outside ±2σ). By the exact normal curve it is 2.28%.
- Can I use the empirical rule for skewed data?
- No. It only holds for roughly bell-shaped data. For any distribution, Chebyshev’s inequality guarantees at least 75% within 2 SD.
- Why does the calculator show two percentages?
- The rule uses rounded values (68, 95, 99.7). The exact normal area is slightly different, for example 68.27% within 1 SD.
Last reviewed: October 7, 2026. Calculations run in your browser and were validated against SciPy.