Normal (Z) Distribution Calculator

What is the standard normal distribution?

The standard normal distribution is a bell-shaped curve with mean 0 and standard deviation 1. A z-score measures how many standard deviations an observation falls from the mean. It is used in three common situations: when the population standard deviation is known, when sample sizes are large enough for the Central Limit Theorem to apply, and when testing or estimating proportions (e.g. one-proportion and two-proportion z-tests), where the standard error is derived from the proportion itself rather than a measured σ.

How to use this calculator

• Z → P mode: Enter your z-statistic to get the exact corresponding p-value. The shaded area on the graph equals the p-value.
• P → Z mode: Enter a significance level (α, e.g. 0.05) to find the critical z-value.
• Two-tailed is used for Hₐ: μ ≠ μ₀, p ≠ p₀ or p₁ − p₂ ≠ 0. Use one-tailed when your hypothesis specifies a direction (e.g. μ > μ₀, μ < μ₀, p > p₀, p < p₀, p₁ − p₂ > 0, p₁ − p₂ < 0).