Normal Distribution Probability Calculator
Convert a z-score into a probability under the standard normal curve — left tail, right tail, or the area between ±z — using an erf-based CDF.
How it works
Φ(z) = P(Z < z) = ½·[1 + erf(z ÷ √2)] · Right tail = 1 − Φ(z) · Between = 2·Φ(|z|) − 1
The standard normal distribution has mean 0 and standard deviation 1, and its cumulative function Φ(z) gives the area to the left of a z-score. This calculator evaluates Φ with an error-function approximation accurate to about seven decimals, then derives the right tail, the central area between ±z, and the combined two-tail area from it. Any raw value converts to a z-score first with z = (x − μ) ÷ σ.
Worked example
For z = 1.96, Φ(1.96) ≈ 0.975, so 97.5% of values fall below it and the right tail is 0.025. The two-tail area P(|Z| > 1.96) = 2 · 0.025 = 0.05 — which is exactly why ±1.96 marks the 95% confidence bounds.
Sources: Normal distribution — Wikipedia
Frequently asked questions
How do I turn a real measurement into a z-score?
Subtract the mean and divide by the standard deviation: z = (x − μ) ÷ σ. A value one standard deviation above the mean has z = 1; then read its probability here.
Why does z = 1.96 give 0.975?
It is the point below which 97.5% of the standard normal curve lies, leaving 2.5% in the upper tail. Because the curve is symmetric, ±1.96 encloses the middle 95%, which is the basis of the familiar 95% confidence interval.
Is this exact?
It uses the Abramowitz & Stegun erf approximation with a maximum error near 1.5 × 10⁻⁷, so for practical purposes the probabilities match standard z-tables to the digits shown.
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For education — check the assumptions (sample size, independence, distribution) before using these results in real decisions.