How to use this empirical rule calculator
Enter a mean and standard deviation above. The calculator returns the three ranges of the empirical rule: where roughly 68%, 95% and 99.7% of a normal distribution's values live.
The 68–95–99.7 rule
For bell-shaped data, nature keeps a tidy pattern: about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. It is the fastest sanity check in statistics. If someone reports a normal-ish dataset where half the values sit beyond 2σ, something is wrong with the data or the claim.
Exam marks with mean 70 and SD 10: about 68% of students scored between 60 and 80, about 95% between 50 and 90, and virtually everyone between 40 and 100. A mark of 25 would be beyond 3σ, expected in fewer than 3 papers per thousand, and a reasonable trigger for double-checking the marking.
When the rule applies: and when it lies
The rule is a property of the normal distribution specifically. Roughly bell-shaped data (heights, measurement errors, standardised test scores) obeys it well. Skewed data (incomes) breaks it cheerfully: Chebyshev's inequality is the honest fallback for arbitrary distributions, guaranteeing only 75% within 2σ and 89% within 3σ, weaker, but true for everything.
Frequently asked questions
What is the empirical rule in statistics?
For a normal distribution, about 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
Does the empirical rule work for any dataset?
Only for approximately normal (bell-shaped) data. For other shapes, Chebyshev's inequality gives weaker but universal guarantees.
What percentage lies beyond 3 standard deviations?
About 0.3% in total, roughly 0.15% in each tail, for a normal distribution.