Empirical rule

The empirical rule states that roughly 68 %, 95 % and 99.7 % of observations in a normal distribution fall within one, two and three standard deviations of the mean. It provides a quick gauge of how unusual any value is.

ByHoang TruongUpdated July 2, 2026

FrameworkNormal distribution

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For a normal distribution with mean €500 and standard deviation €50, about 68% of values fall between €450 and €550, roughly 95% fall between €400 and €600, and about 99.7% fall between €350 and €650. Values further than three standard deviations from the mean are rare under normality.

Where it fits

TopicProbability & DistributionsCoreTopicDescriptive StatisticsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

\mu \pm 1\sigma \approx 68\% \quad \mu \pm 2\sigma \approx 95\% \quad \mu \pm 3\sigma \approx 99.7\%\text{ of observations}

Variables

$μ$: Population mean
$σ$: Population standard deviation

Valid only for approximately normal distributions. Financial data often show higher kurtosis, so tail events occur more frequently than the 0.3% implied by the third interval.

Check yourself

PracticeCORE

A normally distributed dataset has mean μ = 200 and standard deviation σ = 25. Applying the empirical rule, approximately what percentage of observations fall between 150 and 250?

Select an answer to check your understanding.