Empirical Rule Calculator
Enter a mean and standard deviation to calculate the ranges that contain approximately 68%, 95%, and 99.7% of values in a normally distributed data set — commonly called the empirical rule or 68-95-99.7 rule.
How the Empirical Rule formula works
Standard ranges based on multiples of standard deviation:
68% range: mean ± 1 standard deviation 95% range: mean ± 2 standard deviations 99.7% range: mean ± 3 standard deviations
Step-by-step calculation
- Multiply the standard deviation by 1, 2, and 3 respectively.
- Add and subtract each result from the mean to get the three ranges.
Worked example
Mean of 100, standard deviation of 15: the 68% range is 85–115, the 95% range is 70–130, and the 99.7% range is 55–145.
Frequently asked questions
Does the empirical rule apply to any data set?
It specifically applies to data that's approximately normally distributed (bell-curve shaped) — for heavily skewed data, these percentage ranges don't hold accurately.
What's this used for in practice?
It's commonly used to quickly assess how unusual a particular value is relative to a normal distribution — a value beyond 2 standard deviations from the mean, for instance, falls outside the 95% range and is comparatively rare.