P-Value Calculator

A p-value is the probability of obtaining test results at least as extreme as the observed results, assuming the null hypothesis is completely true. A smaller p-value indicates stronger evidence against the null hypothesis.

The standard threshold is 0.05 (5%). A p-value less than 0.05 indicates strong evidence against the null hypothesis, allowing you to reject it. Some fields use stricter thresholds like 0.01 or 0.001.

Two-tailed tests check for differences in both positive and negative directions, effectively doubling the tail probability compared to a one-tailed test. This makes them more conservative but also more comprehensive.

Enter the z-score value into the calculator, select the tail direction, and the tool computes the area under the standard normal curve beyond your z-score. The result is the p-value.

Enter the t-score and the degrees of freedom. The calculator integrates the Student t-distribution to find the tail probability corresponding to your t-statistic.

A one-tailed test checks for a difference in a specific direction (greater than or less than), while a two-tailed test checks for any difference in either direction. Two-tailed tests are more conservative.

Use a one-tailed test when you have a specific directional hypothesis (e.g., "is the treatment better?"). Use a two-tailed test when you only care whether there is any difference, regardless of direction.

It means there is less than a 5% probability that the observed results occurred by random chance under the null hypothesis. This is the conventional threshold for statistical significance.

A p-value above 0.05 means the results are not statistically significant at the 5% level. You fail to reject the null hypothesis — the observed difference could plausibly be due to random variation.

No. A p-value is a probability and must always fall between 0 and 1. If you get a p-value above 1, there is likely an error in your test statistic calculation.

Chi-Square p-values are used in tests of independence and goodness-of-fit to determine whether observed categorical data differs significantly from expected distributions.

F-distribution p-values are used in ANOVA (Analysis of Variance) tests to determine whether the means of three or more groups differ significantly from each other.

Larger sample sizes make it easier to detect small effects, producing smaller p-values. A tiny difference can be statistically significant with a very large sample, even if it is not practically meaningful.

No. A low p-value only provides evidence against the null hypothesis. It does not prove the alternative hypothesis is true, and it says nothing about the size or practical importance of the effect.

Statistical power is the probability of correctly rejecting a false null hypothesis (detecting a real effect). It depends on sample size, effect size, and the chosen significance level.

No. The significance level should be chosen based on the context and consequences of errors. Medical research often uses 0.01 or 0.001, while exploratory studies may use 0.10.

When performing multiple hypothesis tests, the Bonferroni correction adjusts the alpha threshold by dividing it by the number of tests to reduce the chance of false positives.

APA style reports p-values as "p = .032" (with a leading zero omitted for values less than 1). For very small values, report "p < .001". Always include the test statistic and degrees of freedom.

A 95% confidence interval excludes the null hypothesis value if and only if the two-tailed p-value is less than 0.05. They are mathematically equivalent approaches to significance testing.

A p-value alone cannot be converted to a confidence interval because a CI requires the standard error and sample mean. However, knowing both the p-value and the test statistic allows you to reconstruct the CI.

Effect size measures the magnitude of the difference or relationship, independent of sample size. A statistically significant p-value does not guarantee a large or meaningful effect size — always report both.

Yes. This calculator uses precise numerical integration algorithms to compute p-values from standard statistical distributions. Results match those from established statistical software packages.

The P-Value Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The P-Value Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.