Foundations · The normal curve
Empirical Rule Calculator
The 68-95-99.7 rule made concrete. Give a mean and a standard deviation and see the three intervals — within 1, 2 and 3 standard deviations — and the share of a bell-shaped distribution each one holds. And because not every dataset is bell-shaped, it shows Chebyshev's bound too: what you can still guarantee when it isn't.
Result
In plain English
For data shaped like the familiar bell curve, almost everything sits within three standard deviations of the average — and the rule pins down how much: about 68% within one, 95% within two, and 99.7% within three. It turns a mean and a standard deviation into a quick mental map of where values live and how unusual any one of them is.
- standard deviation (σ)
- The typical distance of a value from the mean. The empirical rule measures everything in these units.
- 68-95-99.7
- The share of a normal distribution within 1, 2 and 3 standard deviations of the mean. The exact figures are 68.27%, 95.45% and 99.73%.
- the catch: it needs a bell curve
- The rule only holds for roughly normal data. For skewed or heavy-tailed data the percentages can be badly wrong.
- Chebyshev's inequality
- The honest fallback for any distribution: at least 1 − 1/k² of the data lie within k standard deviations — so at least 75% within 2σ and 89% within 3σ, no shape assumed. Weaker, but always true.
- where a value falls
- How many standard deviations a value sits from the mean (its z-score), and therefore which band it lands in and how rare it is.
Frequently asked
What is the 68-95-99.7 rule?
For a normal (bell-shaped) distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. It's a fast way to judge how typical or extreme a value is once you know the mean and standard deviation.
Does the empirical rule work for any data?
No — only for data that are roughly bell-shaped. For skewed or heavy-tailed distributions the 68-95-99.7 figures can be well off. When you can't assume normality, use Chebyshev's inequality instead: it guarantees at least 1 − 1/k² of the data within k standard deviations for any distribution (at least 75% within 2σ, 89% within 3σ). It's weaker, but it never lies.
How unusual is a value 2 standard deviations from the mean?
On a normal curve, only about 5% of values lie beyond ±2σ — so roughly 1 in 20 is that far out, and about 1 in 40 is that far above (or below). At ±3σ it's about 1 in 370. “Two sigma” is the common rough threshold for “unusual,” but it isn't automatically meaningful.
How is the empirical rule related to z-scores?
They are two views of the same thing. The empirical rule says about 68%, 95% and 99.7% of data fall within 1, 2 and 3 standard deviations of the mean; a z-score just counts exactly how many standard deviations a particular value sits from the mean. So “within 2 standard deviations” is the same as “|z| ≤ 2”, and a z-score lets you read the precise percentage for any value — not only the round 1-2-3 marks the rule quotes.