P-Value Calculator
Find the p-value from a z-score for left-, right-, or two-tailed tests.
Statistically significant at α = 0.05.
The p-value is the tail area beyond your test statistic under the null. From a z-score, a two-tailed p is 2 × P(Z ≥ |z|). For z = 1.96, P(Z ≥ 1.96) = 0.0250, so p = 2 × 0.0250 = 0.0500 — exactly the 5% threshold, making the result borderline significant.
What a p-value means
The p-value is the probability of observing a result at least as extreme as yours, assuming the null hypothesis is true. It’s the area in the tail (or tails) of the distribution beyond your test statistic. A small p-value means your data would be surprising if nothing were going on — so it counts as evidence against the null. It is not the probability that the null is true.
right-tailed = P(Z ≥ z); left-tailed = P(Z ≤ z)
Worked example
A z-statistic of 1.96 on a two-tailed test.
- 1 Find the cumulative area to the left. From the normal table, P(Z ≤ 1.96) = 0.9750.
- 2 Take the upper-tail area. P(Z ≥ 1.96) = 1 − 0.9750 = 0.0250.
- 3 Double it for two tails. 2 × 0.0250 = 0.0500 — exactly the 5% threshold, so this is borderline significant.
Common two-tailed critical z-values
The z beyond which the two-tailed p falls below each α.
| Significance α | Two-tailed |z| | p at that z |
|---|---|---|
| 0.10 | 1.645 | 0.10 |
| 0.05 | 1.960 | 0.05 |
| 0.01 | 2.576 | 0.01 |
| 0.001 | 3.291 | 0.001 |
Reading and reporting the result
Choosing a tail. Use a two-tailed test when a difference in either direction matters, and a one-tailed test only when you have a directional prediction made before seeing the data. A one-tailed p is half the two-tailed p in the predicted direction.
p < 0.05 is a convention, not a verdict. The 0.05 cutoff is arbitrary. Report the exact p-value and your chosen α so readers can judge for themselves; a p of 0.049 and 0.051 are practically identical.
A large p-value isn’t proof of no effect. Failing to reject the null means you lack evidence for an effect — often because the sample is too small — not that the effect is zero.
Significance is not size. With a big enough sample, trivial differences become statistically significant. Always pair the p-value with an effect size or a confidence interval. If your statistic comes from raw data rather than a known z-score, compute it first.