Statistics
Empirical Rule Calculator
Type a mean and standard deviation and the calculator lays out the μ ± 1σ, μ ± 2σ, and μ ± 3σ ranges that hold roughly 68%, 95%, and 99.7% of a normal distribution.
Add an optional value x to see which band it lands in, its z-score, and the percent of data below it, all drawn on a bell curve.
Try an example
Result
68% range (μ ± 1σ)
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- 95% range (μ ± 2σ)
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- 99.7% range (μ ± 3σ)
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- z-score of x
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- Band containing x
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Study path
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What the empirical rule calculator solves
Type a mean and A number saying how far, on average, the data values sit from the mean. and the calculator lays out the μ ± 1σ, μ ± 2σ, and μ ± 3σ ranges that hold roughly 68%, 95%, and 99.7% of a normal distribution. Add an optional value x to see which band it lands in, its How many standard deviations a value sits above or below the mean., and the percent of data below it, all drawn on a bell curve.
Formula
How to use it
- Enter the mean and standard deviation of a roughly bell-shaped data set.
- Read the three ranges and the rounded percentages the rule assigns to each.
- Optionally enter a value x to see its z-score, its band, and the percent of data below it.
- For a cutoff that is not a whole number of standard deviations, use the normal distribution calculator.
How to read the answer
The empirical rule is a quick mental model: most data sits within one standard deviation, almost all within two, and values beyond three are rare (about 3 in 1,000). It is exact only for a normal distribution; skewed or heavy-tailed data can break it badly. The tails split evenly, so about 16% is below μ - σ and 16% above μ + σ.
Common mistakes and edge cases
- Applying the rule to skewed data such as incomes or wait times, where far more than 0.3% can sit beyond 3σ.
- Using 95% for μ ± 1.96σ and μ ± 2σ interchangeably; the rule rounds 95.45% to 95%.
- Forgetting that each tail holds half of the leftover: 32% outside ±1σ means 16% per tail.
- Entering the variance instead of the standard deviation.
Worked examples
IQ scores
μ = 100, σ = 15
68% range (μ ± 1σ)
85 to 115
Locate a value
Where does 130 fall?
68% range (μ ± 1σ)
85 to 115
Negative mean
Temperatures with μ = -5, σ = 4
68% range (μ ± 1σ)
-9 to -1
Decimal parameters
μ = 2.5, σ = 0.25
68% range (μ ± 1σ)
2.25 to 2.75
Standard normal
68% range (μ ± 1σ)
-1 to 1
Impossible: σ = 0
68% range (μ ± 1σ)
Error
Frequently asked questions
What is the empirical rule?+
For a normal (bell-shaped) distribution, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. It is also called the 68-95-99.7 rule or the three-sigma rule.
What percent of data is within 2 standard deviations?+
About 95% by the rule; the exact normal value is 95.45%. Exactly 95% corresponds to ±1.96 standard deviations.
What percent is above μ + 1σ?+
About 16%. The rule leaves 32% outside ±1σ, split evenly into 16% below and 16% above.
Does the empirical rule work for any data?+
Only for roughly normal data. For any distribution, Chebyshev's inequality guarantees at least 75% within 2σ and 89% within 3σ, which is much weaker.
How is the empirical rule used with z-scores?+
A z-score is the number of standard deviations from the mean, so z between -1 and 1 is the 68% band, -2 to 2 the 95% band, and -3 to 3 the 99.7% band.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026