P-Value Calculator

A p-value is the probability of obtaining a test statistic at least as extreme as the observed value, assuming the null hypothesis H₀ is true. It does not measure the probability that H₀ is true — it measures how surprising the data would be if H₀ were true. Our p-value calculator computes exact p-values using the standard normal distribution (for Z-tests) and the Student t-distribution (for t-tests), returning two-tailed, left-tailed, and right-tailed p-values simultaneously so you always have the right value for your hypothesis.

1. 1. Select Test Type and Enter Your Statistic: Choose Z-test (for known σ or large samples) or t-test (for unknown σ or small samples). Enter your test statistic and, for a t-test, the degrees of freedom (df = n − 1 for one-sample tests).
2. 2. Instant Browser-Based Calculation: Click Calculate P-Value and the tool instantly computes all three p-values using the standard normal CDF (Z-test) or the Student t-distribution CDF (t-test). All processing happens locally in your browser — your data is never sent to any server.
3. 3. Interpret Results and Check Significance: View the highlighted p-value for your chosen tail type, the significance table at four α levels, and a plain-English interpretation of your result. Use the two-tailed p-value for most research questions.

• What is a P-Value?: A p-value is the probability of observing a test statistic at least as extreme as the one computed, assuming the null hypothesis (H₀) is true. A small p-value (typically < 0.05) means the observed result is unlikely under H₀, providing evidence to reject it. A large p-value means the data is consistent with H₀.
• Z-Test vs. t-Test: Use a Z-test when the population standard deviation (σ) is known or the sample size is large (n > 30) — the test statistic follows a standard normal distribution. Use a t-test when σ is unknown and the sample is small — the statistic follows a Student t-distribution with df degrees of freedom, which has heavier tails than the normal distribution.
• Two-Tailed vs. One-Tailed Tests: A two-tailed test checks whether the parameter differs from the null value in either direction (H₁: μ ≠ μ₀). A right-tailed test checks for an increase (H₁: μ > μ₀); a left-tailed test checks for a decrease (H₁: μ < μ₀). The two-tailed p-value is always twice the smaller one-tailed p-value.
• Significance Level (α) and Type I Error: The significance level α is the threshold below which you reject H₀. Common choices are α = 0.05 (5% chance of a false positive) and α = 0.01 (1% chance). Choosing a smaller α reduces Type I errors (false positives) but increases Type II errors (false negatives). Our p-value calculator shows significance at α = 0.10, 0.05, 0.01, and 0.001.

A statistically significant p-value does notmean the effect is practically important — it only means the result is unlikely under H₀. Always consider effect size (Cohen's d, r², etc.) alongside the p-value. Our p-value calculator uses the Abramowitz & Stegun error function approximation for the normal CDF (max error ≈ 1.5 × 10⁻⁷) and Lentz's continued fraction algorithm for the regularised incomplete beta function used in the t-distribution CDF, providing accuracy sufficient for all practical statistical work.