P-value Calculator

A p-value (probability value) measures the strength of evidence against the null hypothesis. Specifically, it is the probability of obtaining test results at least as extreme as the results actually observed, assuming that the null hypothesis is true. A lower p-value means stronger evidence against the null hypothesis.

In most scientific research, a significance level (alpha) of 0.05 is used as a threshold. If the p-value is less than or equal to 0.05, the result is considered statistically significant, and the null hypothesis is rejected. This indicates a 5% or lower chance that the observed difference occurred by random coincidence.

A two-tailed p-value tests for a difference in either direction (positive or negative). A one-tailed p-value tests for a difference in a specific direction (only greater than, or only less than). A two-tailed test is more conservative and is standard in most scientific studies unless there is a strong reason to predict directionality.

Degrees of freedom (df), which depend on your sample size (n - 1), dictate the shape of the t-distribution curve. For small sample sizes (low df), the t-distribution has fatter tails, meaning a higher t-score is needed to achieve the same p-value compared to Z-scores. As df increases, the t-distribution approaches the standard normal distribution.

Choose Z-test if you are testing sample means with a known population variance or if your sample size is very large (usually n > 30). Choose T-test if your sample size is small and the population variance is unknown, requiring estimation via sample standard deviation.