Empirical Rule (68-95-99.7) Calculator
The empirical rule tells you what share of a normal distribution lies within one, two and three standard deviations of the mean.
- 1
x − μ
130 − 100 = 30 - 2
z = (x − μ) ÷ σ
30 ÷ 15 = 2Number of standard deviations x lies from the mean.
How does this calculator work?
For a normal distribution, about 68.27% of values fall within one standard deviation of the mean, 95.45% within two, and 99.73% within three. Enter the mean and standard deviation to get the exact μ±σ, μ±2σ and μ±3σ intervals, and optionally a value x to see its z-score and which band it lands in.
Formula
How this is calculated
Enter the mean (μ) and standard deviation (σ) of a roughly normal (bell-shaped) dataset. The calculator builds three symmetric intervals centred on the mean: μ±σ covers about 68.27% of the data, μ±2σ about 95.45%, and μ±3σ about 99.73%. These fixed percentages come from the area under the standard normal curve and apply to any normal distribution regardless of its scale.
The optional value x is converted to a standard score z = (x − μ) / σ, which measures how many standard deviations x sits from the mean. Its absolute value decides the band: |z| ≤ 1 is inside the 68% range, |z| ≤ 2 inside 95%, |z| ≤ 3 inside 99.7%, and anything larger is treated as an outlier beyond three sigma.
The rule assumes the data are approximately normal and symmetric; σ must be positive. For skewed or heavy-tailed data the percentages are only rough guides. Units of the intervals match the units of μ and σ (for example points, cm, or dollars), while z and the percentages are unitless.
Frequently asked questions
No. It is exact only for a normal (bell-shaped) distribution and a close approximation for nearly normal data. For strongly skewed data use Chebyshev’s inequality instead.
These are the areas under the standard normal curve within ±1, ±2 and ±3 standard deviations of the mean. The fuller values are 68.27%, 95.45% and 99.73%.
z = (x − μ) / σ is the number of standard deviations x lies from the mean. A z of 2 means x is two standard deviations above the mean, placing it at the edge of the 95% band.
Also known as
TG we-Calculate Editorial Team. (2026). Empirical Rule (68-95-99.7) Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/empirical-rule-calculator
TG we-Calculate Editorial Team. "Empirical Rule (68-95-99.7) Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/empirical-rule-calculator.
TG we-Calculate Editorial Team, "Empirical Rule (68-95-99.7) Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/empirical-rule-calculator
@misc{wecalculate_empirical_rule_calculator, title = {Empirical Rule (68-95-99.7) Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/empirical-rule-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?