Given a z-statistic from a hypothesis test, this finds the associated p-value using the standard normal distribution.
How it works
The area under the standard normal curve beyond the z-statistic is computed using a published rational approximation to the normal distribution’s cumulative function; a two-tailed test doubles that area to account for both directions.
What this does not include
This does not include converting a raw value to a z-score first — for that starting step, use this site’s z-score calculator instead.
How to use this calculator
- Enter the z-statistic and choose one-tailed or two-tailed.
A worked example
A z-score of 1.96, two-tailed test: p-value = 0.05 — the classic threshold for statistical significance at the 95% confidence level.
A z-score of 1.645, one-tailed test: p-value = 0.05 as well — a smaller z-score reaches the same p-value threshold, since all the probability concentrates on one side.
What the variables mean
| Variable | Meaning |
|---|---|
| z-score | Standardized test statistic |
| Tails | Whether the test checks for a difference in one direction only, or either direction |
Edge cases worth knowing
A one-tailed test reaches a given p-value at a smaller z-score than a two-tailed test — the same underlying probability concentrates in one tail instead of splitting between both, which is why 1.645 (one-tailed) and 1.96 (two-tailed) both land on p=0.05.
A smaller p-value means stronger evidence against the null hypothesis — but the exact significance threshold used to call a result “significant” is a research-design choice, not a universal rule this calculator enforces.
Frequently asked questions
When should I use a one-tailed vs. two-tailed test?
A one-tailed test is used when only testing for an effect in one specific direction; a two-tailed test (more common) tests for an effect in either direction.
What does a small p-value mean?
Conventionally, a p-value below 0.05 is considered statistically significant, meaning the observed result would be unlikely under the null hypothesis — though the right threshold depends on the specific field and study.
Why does a two-tailed p-value double the one-tailed figure?
Because a two-tailed test accounts for an extreme result in either direction, not just the one the z-statistic happens to point toward.
