P-Value Calculator
Calculate p-values from z-scores or t-scores for hypothesis testing.
Result
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 helps determine statistical significance.
- • Small p-value (≤ 0.05): Strong evidence against null hypothesis
- • Large p-value (> 0.05): Weak evidence against null hypothesis
Common Significance Levels
| Alpha (α) | Confidence | Use Case |
|---|---|---|
| 0.10 | 90% | Exploratory research |
| 0.05 | 95% | Standard threshold |
| 0.01 | 99% | Stricter testing |
| 0.001 | 99.9% | Medical/safety research |
One-Tailed vs Two-Tailed Tests
Two-Tailed Test
Tests for any difference (greater or less). Use when you don't predict the direction.
One-Tailed Test
Tests for difference in one direction only. Use when you predict the direction.
Common Z-Scores and Their P-Values
The table below shows critical z-scores used in hypothesis testing. A z-score beyond the critical value means the result is statistically significant at that alpha level.
| Z-Score | Two-Tailed P | Significance | Confidence |
|---|---|---|---|
| ±1.645 | 0.10 | α = 0.10 | 90% |
| ±1.96 | 0.05 | α = 0.05 ★ | 95% |
| ±2.576 | 0.01 | α = 0.01 | 99% |
| ±3.291 | 0.001 | α = 0.001 | 99.9% |
| ±5.0 | <3×10⁻⁷ | 5-sigma | Physics discovery |
★ The 1.96 / α = 0.05 threshold is the standard in most social and medical sciences.
Step-by-Step Hypothesis Testing
State your hypotheses
H₀ (null): No effect exists. H₁ (alternative): An effect exists (or is in a specific direction).
Choose significance level (α)
Typical choices: 0.05 for social science, 0.01 for medicine, 0.001 for stringent research. Do this BEFORE collecting data.
Calculate the test statistic
Compute z = (x̄ − μ₀) / (σ/√n) for z-test, or t = (x̄ − μ₀) / (s/√n) for t-test.
Find the p-value
Use this calculator! Enter your z- or t-score and tail type to get the exact p-value.
Make a decision
If p ≤ α: reject H₀ (statistically significant). If p > α: fail to reject H₀. Never "accept" H₀ — absence of evidence ≠ evidence of absence.
Common P-Value Misconceptions
❌ Myth: P-value = probability that H₀ is true
The p-value assumes H₀ IS true and measures how likely the data would be. It's not the probability that the null hypothesis is correct.
❌ Myth: p < 0.05 means the effect is large or important
With large samples, tiny and practically meaningless effects become statistically significant. Always report effect sizes (Cohen's d, R²) alongside p-values.
✅ Reality: P-value is just one piece of evidence
The American Statistical Association (2016) recommends interpreting p-values alongside confidence intervals, effect sizes, study design quality, and domain knowledge.