Z-Score Probability Calculator
Given a z-score (a standardized value from a normal distribution), this calculator computes the corresponding probabilities: the cumulative probability to the left of the z-score P(Z less than z), the right-tail probability P(Z greater than z), and the two-tailed probability 2 times min(P(Z less than z), P(Z greater than z)). These are the areas under the standard normal curve. Z-score probabilities underpin hypothesis testing, confidence intervals, and quality control (process capability). The calculator uses a high-accuracy rational approximation of the normal CDF.
Normal CDF approximation
P(Z < z) = Φ(z) = 1 - φ(z)(b1t + b2t² + b3t³)
where t = 1/(1 + 0.33267|z|), φ(z) = e^(-z²/2)/√(2π)
This calculator uses the Abramowitz and Stegun (1964) polynomial approximation, which provides approximately 7 significant figures of accuracy for all z values.
Common z-score reference values
- z = 1.645: P(Z < z) = 0.9500 (90% one-tailed or critical value for 90% CI).
- z = 1.960: P(Z < z) = 0.9750 (95% CI critical value, two-tailed alpha = 0.05).
- z = 2.326: P(Z < z) = 0.9900 (99% one-tailed).
- z = 2.576: P(Z < z) = 0.9950 (99% CI critical value, two-tailed alpha = 0.01).
- The empirical rule: P(-1 < Z < 1) = 68.3%, P(-2 < Z < 2) = 95.4%, P(-3 < Z < 3) = 99.7%.
Frequently asked questions
What is a z-score?
A z-score is the number of standard deviations a value is above or below the mean of a normal distribution. A z-score of 0 is the mean, a z-score of 1.0 is one standard deviation above the mean, and a z-score of -1.5 is 1.5 standard deviations below the mean.
What does P(Z < z) mean?
P(Z less than z) is the cumulative probability: the proportion of a standard normal distribution that falls below the given z-score. For example, P(Z less than 1.96) equals approximately 0.975, meaning 97.5% of the distribution falls below z = 1.96.
How is the normal CDF computed?
The normal cumulative distribution function does not have a closed form. This calculator uses the Abramowitz and Stegun rational approximation (formula 26.2.17), which gives results accurate to about 7 significant figures.
What are the common z-score probability pairs?
z = 1.645 gives P = 0.95 (90% CI critical value). z = 1.960 gives P = 0.975 (95% CI critical value, two-tailed). z = 2.326 gives P = 0.99. z = 2.576 gives P = 0.9950 (99% CI critical value). These correspond to common significance levels.
Can I use this for one-tailed tests?
Yes. For a one-tailed test, the p-value is 1 minus P(Z less than z) for an upper-tailed test, or P(Z less than z) for a lower-tailed test. For a two-tailed test, the p-value is 2 times (1 minus P(Z less than |z|)).
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Normal Distribution.
- NIST Digital Library of Mathematical Functions: Chapter 7: Error Functions, Dawson's and Fresnel Integrals.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.