P-Value Calculator

The p-value computation depends on the test statistic and its sampling distribution:

For a Z-test, the test statistic follows the standard normal distribution. The p-value is computed using the cumulative distribution function (CDF) of the standard normal:

$$P(Z > z) = 1 - \Phi(z)$$

For a T-test, the test statistic follows a Student's t-distribution with ν degrees of freedom. An approximation converts the t-statistic to a z-equivalent for CDF computation:

$$z_{\text{approx}} = t \cdot \frac{1 - \frac{1}{4\nu}}{\sqrt{1 + \frac{t^2}{2\nu}}}$$

The tail type determines how the p-value is extracted:

• Two-tailed: $p = 2 \cdot P(Z > |z|)$ — tests for any difference from the null
• Left-tailed: $p = P(Z < z)$ — tests if the parameter is less than the null value
• Right-tailed: $p = P(Z > z)$ — tests if the parameter is greater than the null value

This calculator uses the Abramowitz and Stegun polynomial approximation for the standard normal CDF, which provides accuracy to about 5 decimal places — sufficient for most practical applications.

The p-value represents the probability of observing data as extreme as (or more extreme than) the actual results if the null hypothesis were true. Common interpretation thresholds:

• p ≤ 0.01: Very strong evidence against H₀
• p ≤ 0.05: Strong evidence against H₀ (conventional threshold)
• p ≤ 0.10: Moderate evidence against H₀
• p > 0.10: Weak or no evidence against H₀

A result showing 'Significant at α=0.05? = 1' means the p-value is below 0.05 and the null hypothesis would be rejected at the 5% significance level. Remember, statistical significance does not imply practical significance — always consider effect sizes and confidence intervals alongside p-values.