Empirical Rule Calculator (68-95-99.7)
Apply the 68-95-99.7 rule to a normal distribution — get the 1σ, 2σ and 3σ intervals around the mean and where a value falls.
How it works
μ ± 1σ ≈ 68% · μ ± 2σ ≈ 95% · μ ± 3σ ≈ 99.7%
The empirical rule describes how data clusters in a normal (bell-shaped) distribution. About 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. It gives a fast mental model for spotting typical versus unusual values without computing exact probabilities.
Worked example
For test scores with mean 70 and SD 5, the empirical rule says about 68% of students score 65–75, about 95% score 60–80, and about 99.7% score 55–85. A score of 78 is 1.6σ above the mean, comfortably inside the 95% band.
Sources: 68–95–99.7 rule — Wikipedia
Frequently asked questions
Does the empirical rule work for any data?
Only for distributions that are approximately normal and symmetric. For skewed or unknown distributions, use Chebyshev’s theorem instead, which holds for any shape but gives looser bounds.
Where do 68, 95 and 99.7 come from?
They are the areas under the standard normal curve within ±1, ±2 and ±3 standard deviations. More precisely they are about 68.27%, 95.45% and 99.73%; the rule rounds them for easy recall.
What counts as an outlier under this rule?
Values beyond 3 standard deviations are very rare in a normal distribution — under 0.3% combined in both tails — so they are often flagged as potential outliers worth investigating.
Related calculators
For education — check the assumptions (sample size, distribution shape) before using statistics in real decisions.