P-Value Calculator

Select Distribution / Test Type

Test Statistic (t)

Please enter a valid numeric test statistic.

Degrees of Freedom (df)

Degrees of freedom must be an integer ≥ 1.

Denominator Degrees of Freedom (df₂)

Denominator df must be an integer ≥ 1.

Hypothesis Direction / Tail Type

Significance Level (α) Default threshold is 0.05 (5%)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Hypothesis Testing Reference

What Is a P-Value?

A p-value (probability value) is the probability of obtaining test results at least as extreme as the observed data, assuming the null hypothesis ($H_0$) is strictly true. It measures how consistent your empirical sample data is with the null hypothesis.

A smaller p-value provides stronger evidence against $H_0$. When the p-value is less than or equal to your chosen significance level ($\alpha$, commonly 0.05), you reject the null hypothesis in favor of the alternative hypothesis ($H_a$).

When to Use It

t-Tests & Z-Tests: Evaluating if sample means differ significantly from a known population mean or between two independent groups.
ANOVA (F-Tests): Determining whether variance between multiple group means exceeds random sampling variance.
Chi-Square Tests (χ²): Testing categorical independence in contingency tables or goodness-of-fit distributions.
Regression Analysis: Assessing if individual predictor slopes ($\beta_i$) or overall models ($R^2$) are statistically significant.

The Cumulative Probability Formulas

$$\text{Two-Tailed t-test: } p = 2 \times \left(1 - F_t(|t|; df)\right)$$ $$\text{Right-Tailed Z-test: } p = 1 - \Phi(z)$$ $$\text{F-Test / Chi-Square: } p = 1 - F(\text{statistic}; df)$$

Where $\Phi(z)$ is the standard normal cumulative distribution function and $F(\cdot)$ represents the respective cumulative distribution function (CDF).

Worked Example

Scenario: You run an independent two-sample t-test yielding $t = 2.35$ with $df = 24$ and $\alpha = 0.05$ (two-tailed).

Step 1: Find the upper cumulative probability: $F_t(2.35; 24) \approx 0.9863$.

Step 2: Calculate one-tail area: $P(T \ge 2.35) = 1 - 0.9863 = 0.0137$.

Step 3: Multiply by 2 for a two-tailed test: $p = 2 \times 0.0137 = 0.0274$.

Conclusion: Since $p = 0.0274 \le 0.05$, reject $H_0$. The observed difference is statistically significant at the 5% level.

Computation Complete

Your P-Value Results

Computed P-Value

0.0274

Reject Null Hypothesis (Statistically Significant at α = 0.05)

Distribution Student's t

Test Statistic 2.3500

Degrees of Freedom df = 24

Tail Type Two-Tailed

Alpha (α) 0.05

P-Value Calculator

P-Value Calculator

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Interpretation

Distribution Curve & Tail Probability Area

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