Empirical Rule Calculator
This empirical rule calculator shows what share of a normal distribution lies within one, two and three standard deviations of the mean. Enter the mean and standard deviation: for 100 and 15 the ranges run from 85 to 115, 70 to 130 and 55 to 145.
Example: Mean μ 100 · Standard deviation σ 15 · Single value x (optional) 130 → Result: 68 %: 85–115 · 95 %: 70–130 · 99.7 %: 55–145. Source: NIST/SEMATECH e-Handbook – Normal distribution. Updated: .
How it is calculated
The empirical rule (68-95-99.7 rule) describes how far from the mean the values of a normal distribution fall. About 68 % lie within one standard deviation of the mean, about 95 % within two and about 99.7 % within three.
Formula and steps
The limits are μ − k·σ and μ + k·σ for k = 1, 2, 3. For a single value x you also form the z-score z = (x − μ) / σ; its absolute value shows which of the three ranges the value falls into. More precise than the rounded percentages are 68.27 %, 95.45 % and 99.73 %.
Worked example
An IQ test is scaled to a mean of 100 and a standard deviation of 15. Then about 68 % of people score between 85 and 115, about 95 % between 70 and 130 and about 99.7 % between 55 and 145. A score of 130 has z = (130 − 100) / 15 = 2, so it just sits inside the 95 % range.
- Only about 5 % of values lie beyond 2σ and only about 0.3 % beyond 3σ, which is why such values are often treated as outliers.
- The rule suits bell-shaped, roughly symmetric data. For skewed distributions the shares differ noticeably.
Frequently asked questions
What does the 68-95-99.7 rule say?
In a normal distribution about 68 % of values lie within one standard deviation of the mean, 95 % within two and 99.7 % within three.
How do I apply the rule to one value?
Compute z = (x − μ) / σ. If the absolute value of z is at most 2, the value is inside the 95 % range; with mean 100, σ = 15 and x = 130 you get z = 2.
Does the rule work for data that is not normal?
Only approximately, and only for bell-shaped data. For any distribution Chebyshev’s inequality gives much more cautious minimum shares, namely 75 % within 2σ.
Sources and legal basis
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