Empirical Rule Calculator (68–95–99.7)
By Santaji GadepatilinUpdated October 27, 2026
For data with a normal distribution, about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three. This free empirical Rule calculator solves for whichever value you leave blank and shows the formula and steps.
How to use the empirical Rule calculator
Enter the mean and standard deviation. The 68%, 95% and 99.7% ranges are calculated.
The rule
μ ± 1σ ≈ 68% μ ± 2σ ≈ 95% μ ± 3σ ≈ 99.7%
Worked example — adult heights
Mean 170 cm, SD 8 cm:
- 68% are between 162 and 178 cm
- 95% are between 154 and 186 cm
- 99.7% are between 146 and 194 cm
Beyond three standard deviations
| Range | Share inside | Share outside | About 1 in … outside |
|---|---|---|---|
| μ ± 1σ | 68.269% | 31.7% | 3 |
| μ ± 2σ | 95.45% | 4.55% | 22 |
| μ ± 3σ | 99.73% | 0.27% | 370 |
| μ ± 4σ | 99.994% | 0.00633% | 15,787 |
| μ ± 5σ | 100% | 5.73e-05% | 1,744,278 |
| μ ± 6σ | 100% | 1.97e-07% | 506,797,317 |
“Six sigma” quality aims for so few defects that they lie beyond 6σ from the target; in practice the methodology allows a 1.5σ drift and quotes 3.4 defects per million.
Chebyshev’s inequality
For any distribution, not just normal ones, at least 1 − 1/k² of the data lies within k standard deviations:
| k | Normal distribution | Any distribution (at least) |
|---|---|---|
| 2 | 95.4% | 75% |
| 3 | 99.7% | 88.9% |
| 4 | 99.994% | 93.75% |
Use the empirical rule only when the data are roughly bell-shaped.
Spotting outliers
Values more than 3σ from the mean are rare in normal data (about 0.3%) and worth checking for errors. For skewed data, the quartile and IQR method is more robust.
Checking for normality
Before applying the rule, look at a histogram. Data that are skewed, have long tails or have two peaks will not follow 68–95–99.7.
Exact values
The precise percentages are 68.27%, 95.45% and 99.73%. For other ranges, use the normal distribution calculator.
Frequently asked questions
What is the empirical rule?
For a normal distribution, about 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
Does it work for any data?
Only for roughly bell-shaped (normal) data. For skewed data, the percentages can be very different.
What is "six sigma"?
A quality approach aiming for processes so consistent that defects are extremely rare — about 3.4 per million opportunities under its conventions.
What is the 68-95-99.7 rule?
In a normal distribution, about 68% of values are within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3.
Does the empirical rule work for all data?
No, only for data that are approximately normally distributed. For other data, Chebyshev’s inequality gives guaranteed minimums.