P-Value Calculator
Calculate the exact p-value from a test statistic for z-tests, t-tests, chi-square tests, or F-tests. Determine statistical significance at multiple alpha levels. See also our Hypothesis Testing Calculator, T-Test Calculator, and Z-Test Calculator.
How to Use the P-Value Calculator
The p-value is the probability of obtaining a test statistic as extreme as (or more extreme than) the observed value, assuming the null hypothesis is true. This calculator converts any test statistic into its corresponding p-value using the appropriate distribution (normal, t, chi-square, or F).
Select the type of test you performed, enter the test statistic value, provide degrees of freedom if required, and choose the tail type. The calculator returns the exact p-value and indicates significance at common alpha levels (0.10, 0.05, 0.01, 0.001). For chi-square and F-tests, only right-tailed p-values are computed as these tests are inherently one-directional.
Remember that the p-value alone does not measure effect size or practical importance. A statistically significant result (small p-value) with a large sample may reflect a trivially small effect. Always report confidence intervals and effect sizes alongside p-values for complete statistical reporting.
Formula
Z-test p-value:
Two-tailed: p = 2 × P(Z > |z|) = 2 × (1 - Φ(|z|))
Left-tailed: p = Φ(z)
Right-tailed: p = 1 - Φ(z)
t-test p-value:
Two-tailed: p = 2 × P(T > |t|) with df degrees of freedom
Chi-square p-value:
p = P(χ² > observed) = 1 - F(χ²; df)
F-test p-value:
p = P(F > observed) = 1 - F_CDF(f; df₁, df₂)
Example Calculation
Find the two-tailed p-value for z = 1.96:
Given: z = 1.96, two-tailed test
Φ(1.96) = P(Z ≤ 1.96) = 0.97500
P(Z > 1.96) = 1 - 0.97500 = 0.02500
p-value (two-tailed) = 2 × 0.02500 = 0.05000
Significance checks:
α = 0.10: Yes (0.05 < 0.10) ✓
α = 0.05: Borderline (0.05 = 0.05)
α = 0.01: No (0.05 > 0.01) ✗
Z-Statistic to P-Value Reference Table
| |z| or |t| | p (two-tailed) | p (one-tailed) | Corresponds to |
|---|---|---|---|
| 1.282 | 0.2000 | 0.1000 | 90% confidence |
| 1.645 | 0.1000 | 0.0500 | 95% one-tail |
| 1.960 | 0.0500 | 0.0250 | 95% confidence |
| 2.326 | 0.0200 | 0.0100 | 99% one-tail |
| 2.576 | 0.0100 | 0.0050 | 99% confidence |
| 3.090 | 0.0020 | 0.0010 | 99.9% one-tail |
| 3.291 | 0.0010 | 0.0005 | 99.9% confidence |
| 3.891 | 0.0001 | 0.00005 | 99.99% confidence |
Frequently Asked Questions
What is a p-value?
A p-value is the probability of observing data as extreme as (or more extreme than) what was actually observed, assuming the null hypothesis is true. It quantifies the evidence against H₀. A small p-value (typically < 0.05) indicates that the observed result would be unlikely under H₀, leading to rejection of the null hypothesis. It is NOT the probability that H₀ is true.
How do I interpret the p-value?
Compare the p-value to your chosen significance level α. If p < α, reject H₀ (result is statistically significant). If p ≥ α, fail to reject H₀ (insufficient evidence). Common thresholds: p < 0.05 (significant), p < 0.01 (highly significant), p < 0.001 (very highly significant). Always consider effect size and practical significance alongside statistical significance.
What is the difference between one-tailed and two-tailed p-values?
A two-tailed p-value tests for any difference from H₀ (in either direction) and equals twice the one-tailed p-value. A one-tailed p-value tests for a difference in a specific direction only. The two-tailed p-value is more conservative. Use one-tailed only when you have a strong directional hypothesis specified before seeing the data and would not act on a difference in the opposite direction.
Why is p = 0.05 used as the significance threshold?
The 0.05 threshold is a convention introduced by Ronald Fisher, not a universal truth. It means accepting a 5% chance of Type I error (false positive). Different fields use different thresholds: particle physics uses 5σ (p ≈ 0.0000003), genomics uses p < 5×10⁻⁸, and exploratory research may use p < 0.10. The threshold should be chosen based on the consequences of errors in your specific context.
Can a p-value be exactly 0?
Theoretically, a p-value is never exactly 0 for continuous distributions — it can be extremely small but not zero. When software reports p = 0.000, it means the value is below the display precision (e.g., p < 0.0001). Report such values as "p < 0.001" rather than "p = 0." Very small p-values indicate strong evidence against H₀ but do not prove H₁ is true.
What are common misinterpretations of p-values?
Common errors: (1) p-value is NOT the probability H₀ is true, (2) 1-p is NOT the probability H₁ is true, (3) a non-significant result does NOT prove H₀, (4) a significant result does NOT prove practical importance, (5) p-values are NOT comparable across different studies without considering sample size and effect size. The ASA statement (2016) emphasizes that p-values do not measure effect size or the importance of a result.