Normal Distribution Calculator
Find probabilities under a normal (bell) curve for any mean and standard deviation, with the area shaded.
About the Normal Distribution Calculator
Heights, test scores, measurement errors and many natural quantities follow a normal distribution. The probability that a value falls in a range is the area under the curve over that range.
The calculator converts to a z-score and evaluates the standard normal distribution to about 15 significant digits, more precise than printed z-tables.
The formula
z = (x − μ) ÷ σ; P(X < x) = Φ(z)
- Φ
- standard normal cumulative distribution
- 68–95–99.7 rule
- share within 1, 2 and 3 σ of the mean
Worked example
IQ: μ = 100, σ = 15
- P(IQ > 130) = 1 − Φ(2)
- = 0.02275 (2.3%)
Questions people ask
What's the 68–95–99.7 rule?
About 68.27% of values lie within 1 σ of the mean, 95.45% within 2 σ and 99.73% within 3 σ.
Can probabilities be exactly 0?
Never for a normal distribution, but they can be astronomically small far in the tails.