Standard Normal Distribution Calculator
Probabilities under the standard normal curve — below, above, between or outside any pair of z-values, with the density at each point.
The standard normal has mean 0 and standard deviation 1, and every normal distribution reduces to it by the substitution z = (x − μ)/σ.
How the standard normal distribution calculator works
The standard normal has mean 0 and standard deviation 1, and every normal distribution reduces to it by the substitution z = (x − μ)/σ. That is the entire reason one table could serve all of statistics.
Areas under the curve are probabilities. There is no closed form for them, so the CDF is evaluated numerically and the between-area is simply Φ(z₂) − Φ(z₁).
Formula: φ(z) = e^(−z²/2)/√(2π); P(z₁ < Z < z₂) = Φ(z₂) − Φ(z₁)
Worked examples
| Inputs | P(z₁ < Z < z₂) | Note |
|---|---|---|
| −1.96 to 1.96 | 95.000435% | exactly 95% |
| −1 to 1 | 68.268948% | 68.3% |
| −3 to 3 | 99.730007% | 99.7% |
FAQFrequently asked questions
What is the standard normal distribution?
The normal distribution with mean 0 and standard deviation 1. Every other normal reduces to it by standardising.
What is the 68-95-99.7 rule?
The shares within one, two and three standard deviations. The exact figures are 68.27%, 95.45% and 99.73%.
Why is 95% at 1.96 rather than 2?
Because ±2σ actually covers 95.45%. Exactly 95% needs ±1.96, which is where the familiar constant comes from.
What is the peak height?
1/√(2π) ≈ 0.3989. The density is not a probability — the area under it is.
Why does everything use the normal distribution?
The central limit theorem: sums and averages of many independent contributions tend to normal, whatever they started as.
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.