This p-value calculator approximates a normal p-value from a z-score. Choose two-tailed, left-tailed, or right-tailed, enter z, and read the approximate p-value from the standard normal model.
Students and analysts use it as a quick check beside homework or software output when the test statistic is already a z. It is not a full hypothesis testing suite and does not compute t or other distribution p-values.
How the p-value is computed
Let Φ be the standard normal CDF. Two-tailed uses 2 * (1 – Φ(|z|)). Left-tailed uses Φ(z). Right-tailed uses 1 – Φ(z). Values are clamped between 0 and 1.
Worked example
Z-score 1.96, two-tailed test. Approximate p-value is about 0.05.
| Input | Value |
|---|---|
| Z-score | 1.96 |
| Test | Two-tailed |
| P-value | ~0.05 |
Choosing a tail
- Two-tailed when the alternative allows deviations in either direction.
- Right-tailed when only large positive z values support the alternative.
- Left-tailed when only large negative z values support the alternative.
Reading the result
Compare p to a prechosen alpha. If p is smaller than alpha, the result is statistically significant at that level under the normal model. Significance is not the same as practical importance. Always report effect size and study design context beside p.
Common mistakes
- Using a z table p-value for a t statistic with small samples
- Switching tails after seeing the data
- Treating p = 0.049 and p = 0.051 as deeply different stories
- Ignoring multiple testing when many z scores are scanned
Assumptions
The normal model assumes the z statistic is standard normal under the null. That can fail with tiny samples, heavy tails, or dependent observations. If your software reports a t, F, or chi-square p-value, use that distribution instead of this page.
Classroom workflow
Compute the test statistic from your data first. Enter only the finished z here. Keep a written null and alternative so the tail choice stays honest. Cross check one classic value (z = 1.96 ≈ 0.05 two-tailed) to confirm you selected the right tail mode.
Reporting with context
When you write results, state the null hypothesis, the tail choice, the z value, the approximate p-value, and the sample design in plain language. Readers should not have to guess whether a test was one sided. If you ran many related tests, say so and describe any adjustment plan.
Effect size and confidence intervals often communicate practical meaning better than a bare p-value. Use this calculator for the normal p step, then return to your full analysis for intervals and visuals.
Software cross checks
- Compare a few known z values with your stats package once per project.
- Watch for sign errors when software returns a one sided p and you needed two sided.
- Do not convert a t statistic with a z table when degrees of freedom are small.
- Keep a screenshot or log of inputs when p-values drive a grade or a paper claim.
Those habits keep the approximate normal p-value honest inside a larger reproducible workflow.
Teaching sequence
First state the null. Second compute z from your data using the correct standard error. Third choose the tail that matches the alternative you wrote before seeing data. Fourth enter z here. Fifth compare p with the alpha you chose in advance.
Skipping the written alternative invites one sided tests chosen after the fact. Skipping alpha invites moving goalposts around 0.05. The calculator cannot fix those process errors; it only maps z to an approximate normal p-value.
When results are borderline, report the p-value with enough digits to show it clearly, and discuss practical significance honestly instead of hunting for a star.
What this page is not
It is not a sample size planner, not a confidence interval builder, and not a t distribution tool. If your statistic is t with small degrees of freedom, use software built for that distribution. If you still need a quick normal approximation for teaching, say so explicitly in your writeup.
Keep the scope narrow and the calculator stays trustworthy as a z to p helper.
Limitations
Results are approximate normal p-values only. They do not adjust for multiple comparisons, sequential testing, or Bayesian alternatives. Use them as a transparent normal CDF helper beside your full analysis.
For related unit and percent checks while you plan materials, try the unit converter or the percentage calculator.