P-Value Result
P-Value
0.0124
Significant!
Test Configuration
Test Statistic
Result
P-Value
0.0124
p = 0.0124 ≤ α = 0.05 → Reject H₀
Test Statistic
2.5000
P-Value
0.012419
Alpha (α)
0.05
Interpretation
Strong evidence against H₀
Distribution Visualization
P-Value Interpretation Guide
Critical Values Reference
| α | Two-tailed z | One-tailed z | Two-tailed t (df=30) |
|---|---|---|---|
| 0.10 | ±1.645 | 1.282 | ±1.697 |
| 0.05 | ±1.960 | 1.645 | ±2.042 |
| 0.01 | ±2.576 | 2.326 | ±2.750 |
| 0.001 | ±3.291 | 3.090 | ±3.646 |
?How Do You Calculate a P-Value?
A p-value is the probability of obtaining results at least as extreme as observed, assuming the null hypothesis is true. If p < 0.05 (common threshold), results are statistically significant - unlikely due to chance. P-value does NOT indicate effect size or practical importance, only statistical significance.
What is a P-Value?
The p-value is the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. It quantifies the strength of evidence against the null hypothesis in statistical hypothesis testing. A small p-value suggests the observed data is unlikely under the null hypothesis.
Key Facts About P-Values
- P-value: probability of results as extreme as observed if null is true
- Common significance level (alpha): 0.05 (5%)
- p < alpha: reject null hypothesis (statistically significant)
- p >= alpha: fail to reject null hypothesis
- Smaller p-value = stronger evidence against null hypothesis
- P-value does NOT measure effect size or practical importance
- One-tailed vs two-tailed tests affect p-value
- Multiple testing requires p-value correction (Bonferroni, etc.)
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Quick Answer
A p-value is the probability of obtaining results at least as extreme as observed, assuming the null hypothesis is true. If p < 0.05 (common threshold), results are statistically significant - unlikely due to chance. P-value does NOT indicate effect size or practical importance, only statistical significance.
Frequently Asked Questions
What is a p-value?
A p-value is the probability of obtaining results at least as extreme as observed, assuming the null hypothesis is true. It measures evidence against the null hypothesis. Lower p-values = stronger evidence against H₀.
What does statistical significance mean?
A result is statistically significant when p-value ≤ α (significance level, usually 0.05). This means we reject the null hypothesis. Significance indicates the effect is unlikely due to chance alone, but doesn't measure importance or size of effect.
When should I use one-tailed vs two-tailed tests?
Use two-tailed when you're testing for any difference (H₁: μ ≠ μ₀). Use one-tailed when you have a specific directional hypothesis (H₁: μ > μ₀ or μ < μ₀). Two-tailed is more conservative and generally preferred.
What is the difference between z-test and t-test?
Z-test assumes known population standard deviation and normal distribution (large samples, n>30). T-test is used when σ is unknown (estimated from sample) or small samples. T-test accounts for extra uncertainty in small samples.
What does "fail to reject H₀" mean?
Failing to reject H₀ means there's insufficient evidence against it, NOT that H₀ is true. It's like a "not guilty" verdict - we can't prove guilt (reject H₀), but that doesn't prove innocence (accept H₀).
Last updated: 2026-01-15
How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
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