P-Value Calculator

Calculate p-values from test statistics for various statistical tests. Determine if results are statistically significant. Supports z-tests, t-tests, chi-square tests, and F-tests with comprehensive interpretation.

Two-tailed is most common

Standard normal test statistic

T-test statistic value

Usually n - 1 or n - 2

rho�^2 test statistic

Number of categories - 1

F-test statistic (ANOVA)

Between-group df

Within-group df

P-Value Definition: P-value = Probability of obtaining test results at least as extreme as observed, assuming H₀ is true For Different Tests: Z-Test (Standard Normal): p = P(Z >= |z|) for two-tailed p = P(Z >= z) for right-tailed p = P(Z <= z) for left-tailed T-Test (t-distribution): p = P(T >= |t|) for two-tailed Uses t-distribution with df = n - 1 or n - 2 Chi-Square Test: p = P(rho�^2 >= observed rho�^2) Always right-tailed F-Test (ANOVA): p = P(F >= observed F) Always right-tailed Decision Rules: If p < alpha → Reject H₀ (significant) If p >= alpha → Fail to reject H₀ (not significant) Standard alpha levels: alpha = 0.05 (most common) alpha = 0.01 (more conservative) alpha = 0.10 (more liberal)
Example 1 (Z-Test, Two-Tailed): Z-score = 2.5 alpha = 0.05 P-value = 2 * P(Z > 2.5) = 2 * 0.0062 = 0.0124 Result: p = 0.0124 < 0.05 → SIGNIFICANT Reject H₀ at 5% level Example 2 (T-Test): t-statistic = 2.1 df = 20 Two-tailed, alpha = 0.05 P-value ≈ 0.048 Result: p = 0.048 < 0.05 → SIGNIFICANT Marginally significant Example 3 (Chi-Square): rho�^2 = 7.82 df = 3 alpha = 0.05 P-value ≈ 0.050 Result: p ≈ 0.05 → BORDERLINE Very close to significance threshold Example 4 (F-Test): F = 3.5 df1 = 2, df2 = 27 alpha = 0.05 P-value ≈ 0.044 Result: p < 0.05 → SIGNIFICANT At least one group mean differs Interpretation Guide: p < 0.001: *** (highly significant) p < 0.01: ** (very significant) p < 0.05: * (significant) p < 0.10: † (marginally significant) p >= 0.10: ns (not significant)

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. Low p-values (< 0.05) suggest data is inconsistent with null hypothesis. Example: p=0.03 means 3% chance of seeing this result if null hypothesis were true.

How do you interpret a p-value?

Standard interpretation: p < 0.05 = statistically significant (reject null hypothesis), p < 0.01 = highly significant, p >= 0.05 = not significant (fail to reject null). However, p-values are continuous - p=0.051 and p=0.049 are not fundamentally different. Consider effect size and context, not just significance threshold.

What is the difference between one-tailed and two-tailed tests?

One-tailed (directional): tests for effect in one specific direction (greater than OR less than). Two-tailed (non-directional): tests for effect in either direction (different from). Two-tailed p-value = 2 * one-tailed p-value. Use two-tailed unless you have strong prior reason for directional hypothesis. Most research uses two-tailed tests.

How is p-value calculated from a test statistic?

P-value is the area under the probability distribution curve beyond the test statistic. For z-test: find area beyond z-score in standard normal distribution. For t-test: use t-distribution with appropriate degrees of freedom. For rho�^2: use chi-square distribution. Statistical software or tables provide these probabilities.

What does p-value NOT tell you?

P-value does NOT indicate: (1) Size or importance of effect, (2) Probability that null hypothesis is true, (3) Probability that alternative hypothesis is true, (4) Clinical or practical significance. Small p-value with large sample might show trivial effect. Always report effect sizes alongside p-values.

Why is 0.05 the standard threshold?

The alpha = 0.05 threshold is convention, not law. It means 5% chance of Type I error (false positive). Fisher proposed it as reasonable balance. Some fields use alpha = 0.01 (more conservative) or alpha = 0.10 (more liberal). Recent movement toward lower thresholds (0.005) in some sciences. Choose alpha before analyzing data.

What is the relationship between p-value and confidence intervals?

If 95% confidence interval excludes null value, then p < 0.05 (two-tailed). If 99% CI excludes null, then p < 0.01. Confidence intervals provide more information: they show effect size range, not just significance. Example: Mean difference 5.2, 95% CI [2.1, 8.3] → significant (doesn't include 0) and effect size visible.

Can you have a p-value greater than 1?

No, p-values range from 0 to 1 (or 0% to 100%). P-value is a probability, so it cannot exceed 1. Very small p-values may be reported as "< 0.001" or scientific notation (p = 2.3 * 10⁻⁵). P = 0 is theoretically possible but usually reported as "< 0.0001" due to computational precision.

What is p-hacking and why is it a problem?

P-hacking is manipulating data or analysis to achieve p < 0.05: testing multiple hypotheses, removing outliers selectively, stopping data collection when p < 0.05 reached. This inflates Type I error rate far above 5%. Solution: pre-register analysis plan, report all tests performed, adjust for multiple comparisons (Bonferroni, FDR).

How do you report p-values in research?

Report exact p-values (e.g., p = 0.032) rather than just "p < 0.05". Exception: very small values can be "p < 0.001". Include test statistic, degrees of freedom, and effect size. Example: "t(28) = 2.45, p = 0.021, d = 0.52" provides complete information. Never report "p = 0.00" - use "p < 0.001".