Normal Distribution Calculator
The probability of a value falling below, above or between any points on a normal distribution — from the mean and standard deviation.
A normal distribution is fixed by two numbers: where it is centred (the mean) and how spread out it is (the standard deviation).
How the normal distribution calculator works
A normal distribution is fixed by two numbers: where it is centred (the mean) and how spread out it is (the standard deviation). Any value converts to a z-score, (x − μ) ÷ σ, which says how many standard deviations from the mean it sits, and the probability follows from the standard normal curve.
The familiar landmarks are the empirical rule: about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. That is why a result three standard deviations out is remarkable and one standard deviation out is not.
Formula: z = (x − μ) ÷ σ; P(X < x) = Φ(z)
Worked examples
| Inputs | P(X < x) | Note |
|---|---|---|
| IQ of 130 (mean 100, sd 15) | 97.725% | z = 2, so 97.7% score below |
| One standard deviation above | 84.134% | 84.13% |
| The mean itself | 50% | exactly half |
FAQFrequently asked questions
What is a z-score?
How many standard deviations a value sits from the mean: (x − μ) ÷ σ. A z of 2 means two standard deviations above.
What is the 68-95-99.7 rule?
About 68% of a normal distribution lies within one standard deviation of the mean, 95% within two and 99.7% within three.
What percentile is an IQ of 130?
The 98th — z is 2, so 97.7% of people score below it.
Is real data ever exactly normal?
Almost never, but many measurements are close enough for the approximation to be useful. Heavily skewed data is not, and needs a different distribution.
How accurate is the calculation?
The normal tail here is accurate to better than one part in ten million, far beyond what any real dataset justifies.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.