P-Value Calculator
Convert a z-score into a p-value from the normal distribution for a left, right or two-tailed test, with a significance verdict at 0.05.
How it works
Left: p = Φ(z) · Right: p = 1 − Φ(z) · Two-tailed: p = 2 × (1 − Φ(|z|)), where Φ is the standard normal CDF
The p-value is the probability of seeing a test statistic at least as extreme as yours if the null hypothesis were true. Starting from a z-score, the standard normal CDF Φ gives the area in the relevant tail: one tail for a directional test, both tails doubled for a two-sided test. A small p-value means the observed result would be surprising under the null, which is the evidence that leads to rejecting it.
Worked example
A z-score of 2.0 in a two-tailed test: Φ(2.0) ≈ 0.9772, so p = 2 × (1 − 0.9772) ≈ 0.0455. Since 0.0455 is below 0.05, the result is statistically significant and the null hypothesis is rejected at the 5% level.
Frequently asked questions
Does a small p-value prove my hypothesis is true?
No. It only measures how incompatible the data is with the null hypothesis. A low p-value is evidence against the null, not proof of your alternative, and it says nothing about the size or importance of an effect.
Why choose 0.05 as the cutoff?
It is a long-standing convention, not a law of nature. Depending on the stakes, fields use 0.01, 0.001 or other thresholds — and many statisticians now urge reporting the exact p-value and effect size rather than a pass/fail at 0.05.
Should I use a one-tailed or two-tailed test?
Use two-tailed unless you have a firm, pre-registered reason to test only one direction. A two-tailed test is more conservative because it splits the significance level across both tails; choosing one-tailed after seeing the data inflates false positives.
Related calculators
For education — check assumptions (sample size, distribution) before using statistics in real decisions.