Last updated: 2026-06-23

P-Value from Z-Score

A p-value turns a test statistic into a single number you can compare against a decision threshold. This calculator takes a z-statistic — a value already expressed in standard-normal units — and reports the corresponding p-value for a two-tailed or one-tailed test.

What the z-statistic represents

A z-statistic measures how far an observed result sits from what the null hypothesis predicts, counted in standard errors. A z of 0 means the observation lands exactly on the null expectation; a z of 2 means it is two standard errors above it. Because the statistic is already standardized, its sampling distribution under the null hypothesis is the standard normal distribution, with mean 0 and standard deviation 1. If you are starting from a raw observation rather than a test statistic, the Z-Score Calculator calculator shows how to standardize it first.

Tail areas and the normal CDF

The p-value is an area under the standard normal curve. Let $\Phi$ denote the cumulative distribution function of the standard normal — the area to the left of a given point:

The Normal Distribution Calculator calculator visualizes this area directly. How much area counts as "at least as extreme" depends on the alternative hypothesis:

A two-tailed test counts extreme outcomes in both directions, so it doubles the single upper-tail area beyond $|z|$. A one-tailed test commits to one direction in advance and counts only that tail.

Reading the p-value against α

A p-value is compared with a significance level $\alpha$ chosen before the analysis — most often the value $\alpha = 0.05$. If $p \lt \alpha$, the result is conventionally called statistically significant and the null hypothesis is rejected at that level. If $p \ge \alpha$, there is insufficient evidence to reject it. The threshold is a convention, not a law of nature: the levels $\alpha = 0.05$ and $\alpha = 0.01$ are both common, and the right choice depends on the cost of a false positive.

Worked example

Suppose a two-tailed z-test produces a z of 2.5. The upper-tail area beyond that point is $1 - \Phi(2.5) \approx 0.00621$, so:

Because $0.0124 \lt 0.05$, the result is significant at the five-percent level, but it is not significant at the one-percent level, since $0.0124 \gt 0.01$. The same statistic interpreted as a right-tailed test would give $p = 1 - \Phi(2.5) \approx 0.0062$.

Caveats

A p-value is widely misread, so a few cautions matter:

• It is not the probability that the null hypothesis is true. It is computed assuming the null is true, and it describes the data, not the hypothesis.
• It depends on the test's assumptions. Using the normal distribution requires either a known population standard deviation or a sample large enough for the normal approximation to hold. With a small sample and an estimated standard deviation, a t-distribution is more appropriate.
• Large samples make tiny effects significant. As the sample grows, the standard error shrinks, so even a negligible difference can yield a small p-value. Always pair the p-value with an effect size and a confidence interval to judge practical importance.

Statistical significance is a starting point for interpretation, not a verdict on its own.