Calculates the value ranges covering roughly 68%, 95%, and 99.7% of a normal distribution, based on its mean and standard deviation.
How it works
Each range extends a whole number of standard deviations above and below the mean — one, two, and three standard deviations respectively.
What this does not include
This does not include the exact probability for any specific value — for that, use this site’s normal distribution calculator instead, which computes precise probabilities rather than these three fixed reference bands.
How to use this calculator
- Enter the distribution’s mean and standard deviation.
A worked example
Mean 100, standard deviation 15: about 68% of values fall within 85–115 (±1 SD), about 95% within 70–130 (±2 SD), and about 99.7% within 55–145 (±3 SD).
What the ranges mean
| Range | Approximate coverage |
|---|---|
| Mean ± 1 SD | ~68% |
| Mean ± 2 SD | ~95% |
| Mean ± 3 SD | ~99.7% |
Edge cases worth knowing
This only applies to data that’s actually normally distributed. A skewed or unusual distribution won’t match these percentages, even approximately — the “empirical” in the name refers to normal distributions specifically.
Only about 0.3% of values fall outside the ±3 SD range — a value that far from the mean is genuinely unusual under a normal distribution, not just “somewhat rare.”
Frequently asked questions
Why is it called the “empirical” rule?
Because these percentages were observed to reliably describe normal distributions in practice, even before being derived mathematically from the distribution’s formula.
Does the empirical rule apply to any dataset?
No — it specifically applies to data that follows (or closely approximates) a normal distribution; skewed or unusual distributions won’t match these percentages.
What falls outside the ±3σ range?
Only about 0.3% of the data — values that far from the mean are considered quite unusual under a normal distribution.
