Empirical Rule Calculator
Find the 1σ, 2σ and 3σ ranges around a normal-distribution mean.
The 68-95-99.7 Rule
For an approximately normal distribution, about 68% of observations fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
How the Empirical Rule Is Used
The empirical rule summarizes how values are distributed around the mean in an approximately normal distribution: about 68% within one standard deviation, 95% within two and 99.7% within three.
Empirical Rule Example
If a variable has mean 100 and standard deviation 15, roughly 95% of values are expected between 70 and 130 when the distribution is close to normal.
Important Limitations
The rule is an approximation and should not be applied blindly to skewed, heavy-tailed or multimodal data.
Frequently Asked Questions
Does the rule require normal data?
It is intended for approximately bell-shaped normal distributions.
What range covers about 68%?
Mean plus or minus one standard deviation.
Is 99.7% exactly guaranteed?
No, it is an approximation for normal data.
Methodology, Assumptions and Interpretation
The 68–95–99.7 rule is an approximation for normally distributed data: about 68%, 95% and 99.7% lie within 1, 2 and 3 standard deviations of the mean. It should not be applied automatically to strongly non-normal distributions.