Normal Distribution Calculator
Calculate the cumulative probability Φ(z) for a standard normal distribution.
How to use this tool
- Enter z-score in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your p(z ≤ z) — cumulative probability and the full breakdown beneath it.
Calculate P(Z ≤ z) for the standard normal distribution using Abramowitz & Stegun approximation (max error <7.5×10⁻⁸).
Formula
Standard normal PDF: φ(z) = (1/√(2π)) × e−z²/2
Cumulative probability (Abramowitz & Stegun approximation):
Φ(z) ≈ 1 − φ(z) × poly(t) for z ≥ 0, where t = 1 / (1 + 0.2316419 × z) and poly(t) is a 5th-degree polynomial in t. For z < 0, Φ(z) = 1 − Φ(−z).
How it works
This calculator estimates the cumulative distribution function Φ(z) — the probability that a standard normal variable is less than or equal to a given z-score — using the Horner-evaluated rational approximation from Abramowitz and Stegun (formula 26.2.17).
The approximation has a maximum error of about 7.5 × 10−8, making it accurate to seven decimal places for practical purposes. The upper-tail probability P(Z > z) is returned as 1 − Φ(z).
Worked example
Worked example
- Input: z = 0.0 (the mean of the standard normal distribution).
- By symmetry, exactly half the distribution lies below zero.
- Φ(0) = 0.5; upper tail P(Z > 0) = 1 − 0.5 = 0.5.
P(Z ≤ 0) = 0.5, P(Z > 0) = 0.5
Key terms
- Standard normal distribution
- A normal distribution with mean 0 and standard deviation 1, often denoted N(0,1); the reference distribution for z-score calculations.
- Z-score
- The number of standard deviations a value lies above or below the mean; computed as z = (x − μ) / σ for a general normal variable.
- Cumulative distribution function (CDF)
- Φ(z) gives the probability that a standard normal variable takes a value less than or equal to z.
- Upper-tail probability
- P(Z > z) = 1 − Φ(z); the probability of observing a value more extreme than z in the positive direction.
- p-value
- In hypothesis testing, the probability of obtaining a result at least as extreme as the observed statistic under the null hypothesis; often read from normal or t-distribution tables.
Frequently asked questions
- What does Φ(z) represent?
- Φ(z) is the cumulative distribution function of the standard normal distribution — the probability that a standard normal random variable is less than or equal to z. For example, Φ(1.96) ≈ 0.975, corresponding to the 95% confidence interval.