Turn a test statistic into a decision.
Enter a z-score, pick your tail, and set your significance level. See the exact p-value, whether you reject the null, and the rejection region drawn on the normal curve.
Your test
The p-value uses the standard normal distribution: Φ(z) = 0.5(1 + erf(z / √2)). Left p = Φ(z), right p = 1 - Φ(z), two-tailed p = 2(1 - Φ(|z|)).
What this result means
The normal curve and your rejection region
Where your statistic falls
Decision scorecard
P-values around your statistic
How the p-value shifts as the test statistic moves, using your selected tail.
| Test statistic z | Φ(z) | P-value | vs α |
|---|
Critical z at common significance levels
Reject the null when your statistic is more extreme than the critical value for your tail.
| Significance α | One-tailed critical z | Two-tailed critical z |
|---|
P-values, explained
What Is a P-value?
A p-value is the probability of obtaining results at least as extreme as the observed results, assuming the null hypothesis is true. It measures the strength of evidence against the null hypothesis. A smaller p-value indicates stronger evidence that the observed effect is real, not due to random chance.
How to Interpret P-values
Common significance thresholds: p < 0.05 (statistically significant), p < 0.01 (highly significant), p < 0.001 (very highly significant). If p < α (your chosen significance level), reject the null hypothesis. However, statistical significance does not necessarily mean practical significance, always consider effect size.
One-Tailed vs Two-Tailed Tests
A one-tailed test checks for an effect in a specific direction (e.g., "is A greater than B?"). A two-tailed test checks for any difference in either direction (e.g., "is A different from B?"). Two-tailed tests are more conservative and are the default in most research. One-tailed p-values are half of two-tailed p-values.
Common Misinterpretations
A p-value is NOT the probability that the null hypothesis is true. It is NOT the probability that your results occurred by chance. It IS the probability of seeing data this extreme if the null hypothesis were true. Also, "not significant" does not mean "no effect", it means insufficient evidence to conclude an effect.
Common questions
What p-value is considered significant?
The most common threshold is p < 0.05 (5%), but this is a convention, not a universal rule. Some fields use stricter thresholds: particle physics uses p < 0.0000003 (5σ), while exploratory studies may use p < 0.10.
When should I use a z-test vs t-test?
Use a z-test when the population standard deviation is known or the sample size is large (n > 30). Use a t-test when the population SD is unknown and the sample is small. For large samples, both give nearly identical results.
What are degrees of freedom?
Degrees of freedom (df) represent the number of independent values that can vary. For a one-sample t-test, df = n - 1. For a two-sample t-test, the calculation depends on the method used. More degrees of freedom make the t-distribution closer to the normal distribution.
Can a p-value be exactly zero?
Theoretically no, but computationally it can round to zero for extremely large test statistics. In practice, very small p-values are reported as p < 0.001 or p < 0.0001 rather than as zero.
For learning and planning only. This calculator uses the standard normal (z) distribution. For small samples with an unknown population standard deviation, a t-distribution p-value will differ slightly. Always confirm the test assumptions before drawing conclusions.