Statistics
Empirical rule calculator (68–95–99.7)
Enter a mean and a standard deviation to get the one, two and three sigma ranges, with the exact percentage of a normal distribution each one covers.
The 68–95–99.7 rule is a rounding of the exact coverages shown above. It holds only for data that is roughly normal.
Show the working, step by step
The rule
| Range | Quoted | Exact | Values outside |
|---|---|---|---|
| μ ± 1σ | 68% | 68.269% | About 1 in 3 |
| μ ± 2σ | 95% | 95.450% | About 1 in 22 |
| μ ± 3σ | 99.7% | 99.730% | About 1 in 370 |
The right-hand column is the part people find most surprising. "68% within one standard deviation" sounds like most of the data, but it means roughly a third of all observations sit outside that band — a value one standard deviation from the mean is entirely ordinary.
Each range is built from bands of fixed size on either side of the mean. Halving 68.27% gives the 34.13% between the mean and one standard deviation; the next band out adds 13.59%, the one after that 2.14%, and 0.13% lies beyond three standard deviations on each side, which is why 100 − 99.73 = 0.27% falls outside ±3σ in total.
| Band, on each side of the mean | Share |
|---|---|
| Mean to 1σ | 34.13% |
| 1σ to 2σ | 13.59% |
| 2σ to 3σ | 2.14% |
| Beyond 3σ | 0.13% |
The three-sigma rule
"Three-sigma rule" is another name for the same fact, focused on its last band: for normal data, 99.73% of values lie within three standard deviations (3σ) of the mean, so only about 1 value in 370 falls outside. That is why a point beyond ±3σ is treated as unusual enough to investigate, and why quality-control charts draw their limits at ±3σ. The three-sigma control limits behind process capability indices rest on it. For data that is not normal, Chebyshev's inequality still guarantees at least 88.9% (1 − 1/3²) within three standard deviations.
A worked example
Adult IQ scores are modelled with μ = 100 and σ = 15:
- 85 to 115 holds about 68% of people.
- 70 to 130 holds about 95%.
- 55 to 145 holds about 99.7% — a score above 145 is roughly one in 740.
To see how closely real measurements follow these bands, check the rule against your own data.
Two and 1.96
Textbooks say "95% within two standard deviations" and then confidence intervals use 1.96. Both are right, and they answer slightly different questions: two standard deviations covers 95.45%, while covering exactly 95% takes 1.96. The rule rounds for memorability; inference uses the precise value.
When it does not apply
The rule is a property of the normal distribution, not of data in general. Three failure modes worth watching for:
- Skewed data. With a long right tail, far more than 68% may sit within 1σ while the upper reaches extend much further than 3σ.
- Bounded data. If values cannot go below zero, a mean of 3 with σ = 2 gives a nonsensical lower bound of −3.
- Heavy tails. Financial returns produce "six-sigma" events far more often than the rule allows — which is precisely why models assuming normality fail in market crises.
When you cannot assume normality, Chebyshev's inequality still holds for any distribution at all: at least 1 − 1/k² of values lie within k standard deviations. That gives 75% at 2σ and 88.9% at 3σ — much weaker guarantees, but ones that are never wrong. To check whether your data is close enough to normal for the 68–95–99.7 bands, see standard deviation and the normal distribution.
Related calculators
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Normal distribution
Exact probabilities for any range, not just whole σ steps.
-
Z-score calculator
How many standard deviations one value sits from the mean.
-
Bell curve generator
Draw the curve these bands sit on.
-
Standard deviation
Get the σ this rule needs from raw data.
Common questions
What is the 3 sigma rule?
For normally distributed data, 99.73% of values fall within three standard deviations (3σ) of the mean, so only about 0.27% — roughly 1 in 370 — lie beyond. Quality control uses ±3σ as its control limits for that reason.
What is the empirical rule?
For roughly normal data, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. It is also called the 68–95–99.7 rule or the three-sigma rule.
Are 68, 95 and 99.7 exact?
No — they are rounded. The exact coverages are 68.269%, 95.450% and 99.730%. The calculator shows both so you can see the size of the rounding.
Note that the often-quoted "95% within two standard deviations" is really 95.45%; the figure that gives exactly 95% is 1.96 standard deviations, which is where that number in confidence intervals comes from.
Does the rule work for any data?
No. It requires an approximately normal distribution. On strongly skewed data the percentages can be far off, and the lower bound may even be impossible — a mean of 3 with an SD of 2 puts −3 inside the three-sigma range, which is meaningless for a count.
Chebyshev's inequality is the distribution-free fallback: for any distribution, at least 75% of values lie within 2σ and 88.9% within 3σ. Much weaker, but always true.
What is the empirical rule used for?
Quick sanity checks, mostly. It tells you whether a value is ordinary or unusual without a table, and it underpins the ±3σ control limits used in manufacturing quality control.