Written by Dackohn; last reviewed November 2026.
In a normal distribution, about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three. On a test with mean 100 and SD 15, about 95% of people score between 70 and 130. The rule applies only to roughly bell-shaped data.
The exact numbers
| Range | Rule | Exact | Outside |
|---|---|---|---|
| Mean ± 1 SD | 68% | 68.27% | about 1 in 3 |
| Mean ± 2 SD | 95% | 95.45% | about 1 in 22 |
| Mean ± 3 SD | 99.7% | 99.73% | about 1 in 370 |
Worked example
Test scores have mean 100 and SD 15.
- 68% lie between 85 and 115.
- 95% lie between 70 and 130.
- 99.7% lie between 55 and 145.
What share scores above 130? About 95% lie between 70 and 130, leaving 5% outside, split equally between the two tails. So about 2.5% score above 130 (the exact figure is 2.28%). That places 130 at about the 97.7th percentile.
What share scores between 100 and 115? Half of the 68% band: about 34%.
Spotting unusual values
A value beyond 2 SDs is uncommon, and one beyond 3 SDs is rare in normal data. But don’t rely on this as an outlier test in small samples, where an outlier inflates the SD itself. See how to detect outliers.
When the rule fails
It applies only to roughly normal data. The 20 reaction times used in our normality test example are right-skewed, with mean 460.75 and SD 82.45. In reality, 90% of them lie within 1 SD, not 68%, because two slow values inflate the SD. A quick check like this can reveal non-normal data.
A rule for any distribution
Chebyshev’s theorem says that for any distribution, at least 1 − 1/k² of values lie within k SDs. So at least 75% lie within 2 SDs and at least 88.9% within 3. These guarantees are much weaker but always hold, whatever the shape.
Why it matters
Many procedures rest on these numbers. The familiar 1.96 in a 95% confidence interval is the exact version of “about 2 SDs”. Thanks to the Central Limit Theorem, sample means are roughly normal even when the data are not, so the rule applies to them.
Try it yourself
Empirical Rule Calculator →Get the ranges, or check your own data against the rule. Normal Distribution Calculator →Exact areas for any cut-off.Frequently asked questions
What is the 68-95-99.7 rule?
In a normal distribution, about 68%, 95% and 99.7% of values lie within one, two and three standard deviations of the mean.
Is the empirical rule exact?
No, it is rounded. The exact figures are 68.27%, 95.45% and 99.73%.
Why is 1.96 used instead of 2?
Exactly 95% of a normal distribution lies within 1.96 SDs. The rule rounds this to 2.
Related reading: Z-table · Skewness and kurtosis · Distribution explorer
Sources & further reading: NIST/SEMATECH e-Handbook of Statistical Methods · OpenStax Introductory Statistics. How we compute & validate: methodology. Written & reviewed by Dackohn; last reviewed August 2026.