P-value Calculator Compute p-values for Z, T, Chi-Square, and F tests — with distribution visuals, real-world benchmarks,...
P-value Calculator
Compute p-values for Z, T, Chi-Square, and F tests — with distribution visuals, real-world benchmarks, and statistical best practices.
Z-test (two-tailed)**:
$$ p = 2 \left(1 - \Phi(|z|)\right) $$ where $\Phi$ = standard normal CDF
T-test**:
$$ p = 2 \left(1 - T_{df}(|t|)\right) $$ where $T_{df}$ = Student’s t CDF
Example** (z = 2.1, two-tailed): $$ p = 2(1 - \Phi(2.1)) = 2(1 - 0.9821) = \mathbf{0.0358} $$
✅ Pro Tip**: P-value is the probability of seeing data *at least as extreme* as observed, **assuming H₀ is true** — *not* the probability H₀ is true.
⚠️ Avoid these misinterpretations:
- “p = 0.051 is not significant, p = 0.049 is”** — Arbitrary threshold; report exact p
- Ignoring effect size** — p < 0.001 for +0.1% conversion lift = statistically significant, but trivial
- Multiple comparisons** — 20 tests at α=0.05 → ~64% chance of ≥1 false positive
- p-hacking** — Trying transformations until p < 0.05
✅ Best Practices**:
- Pre-register analysis plan
- Report **confidence intervals** (e.g., “+2.1%, 95% CI [0.4%, 3.8%]”)
- Use **Bonferroni** or **FDR** for multiple tests
| Context | Typical p | Interpretation |
|---|---|---|
| Clinical Trial (FDA) | < 0.05 | Required for approval |
| A/B Test (Web) | < 0.05 | Statistically reliable lift |
| Social Science (Journals) | < 0.005 | Stricter standard post-replication crisis |
| Exploratory Research | < 0.10 | Hypothesis-generating only |
📉 Critical Values (α = 0.05)**:
- Z: ±1.96 (two-tailed)
- t (df=30): ±2.04
- χ² (df=1): 3.84
- F (2,27): 3.35
➡️ Z-test
“z = 2.1, two-tailed → p = ?” → **0.0358** (significant at α=0.05)
➡️ T-test
“t = 2.5, df = 15, two-tailed” → **0.0248**
➡️ Chi-Square
“χ² = 9.49, df = 4” → **0.0500** (critical value)
➡️ F-test
“F = 3.2, df₁=2, df₂=27” → **0.0542**
Note: Uses high-accuracy rational approximations (Abramowitz & Stegun). All p-values two-tailed unless specified.