Apply the 68–95–99.7 empirical rule to find the exact probability and range for 1, 2, and 3 standard deviations from the mean.
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68.27%
Within 1σ
95.45%
Within 2σ
99.73%
Within 3σ
| Band | Range | Probability |
|---|---|---|
| μ ± 1σ | 85 – 115 | 68.27% |
| μ ± 2σ | 70 – 130 | 95.45% |
| μ ± 3σ | 55 – 145 | 99.73% |
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Probability & Statistics
The empirical rule states that for a normal distribution, approximately 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. This calculator computes the exact probabilities and the corresponding value ranges for any mean and standard deviation.
The rounded figures 68%, 95%, and 99.7% are approximations. The exact values from the normal CDF are: P(μ ± 1σ) ≈ 68.27%, P(μ ± 2σ) ≈ 95.45%, P(μ ± 3σ) ≈ 99.73%. This calculator returns the exact probabilities.
The empirical rule is useful for quick mental estimates. If a data distribution is approximately normal with μ = 100, σ = 15 (e.g. an IQ scale), then about 68% of people score between 85 and 115, 95% between 70 and 130, and 99.7% between 55 and 145.
The empirical rule only applies when the data is approximately normally distributed. It should not be used for heavily skewed data, multimodal distributions, or data with many outliers. Always check normality with a histogram or normality test before applying these rules.
For non-normal distributions, Chebyshev's inequality provides a weaker but universally valid bound: at least (1 − 1/k²) of values lie within k standard deviations of the mean, regardless of distribution shape.
Values beyond 3σ are very rare in a normal distribution (about 0.27%, or roughly 1 in 370). Values beyond 5σ are extraordinarily rare (about 1 in 3.5 million). In financial risk models and physics, sigma levels are used to quantify how unusual an event is — "five-sigma evidence" in particle physics indicates an extremely unlikely false positive.
Frequently asked questions
No. The "68–95–99.7" figures are rounded approximations. The exact values are 68.27%, 95.45%, and 99.73%, computed from the normal CDF. This calculator returns the exact values.
No — only to data that is approximately normally distributed. For skewed, bimodal, or heavy-tailed distributions, the rule can be significantly wrong. Always verify approximate normality before applying the empirical rule.
Chebyshev's inequality provides a distribution-free bound: for any distribution, at least (1 − 1/k²) of values lie within k standard deviations of the mean. For k=2, this gives at least 75% (much weaker than the normal distribution's 95.45%). It's a lower bound, not an approximation.
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