P-Value Calculator

A p-value is the probability of observing a test statistic at least as extreme as the one computed from your sample data, assuming the null hypothesis is true. It does not tell you the probability that the null hypothesis is true or false — it is purely a measure of how surprising your data would be if nothing were happening.

When you run a hypothesis test, you collect data and calculate a single number — the test statistic — that summarises how far your sample is from what the null hypothesis predicts. The p-value then answers: if the null hypothesis were correct, how often would random chance produce a result this extreme or more extreme? A small p-value means your data would rarely arise by chance alone, which is taken as evidence against the null.

The conventional threshold is p < 0.05, often called the α (alpha) level or significance level. If your p-value falls below this threshold, the result is labelled statistically significant. However, this threshold is a social convention, not a physical law. A p-value of 0.049 and 0.051 are functionally identical in practice — the difference is noise in your sample, not a categorical change in what the data mean.

A z-score arises when you know the population standard deviation (or have a large enough sample that the standard error is estimated precisely). The test statistic follows the standard normal distribution N(0, 1).

The p-value is computed from the cumulative distribution function Φ(z). for a right-tailed test you want P(Z ≥ z) = 1 − Φ(z). for a left-tailed test you want P(Z ≤ z) = Φ(z). for a two-tailed test you want P(|Z| ≥ |z|) = 2 × min(Φ(z), 1 − Φ(z)).

Φ(z) is computed via the complementary error function: Φ(z) = ½ × erfc(−z / √2). As an example: z = 1.96 gives Φ(1.96) ≈ 0.9750 and a two-tailed p-value of 2 × (1 − 0.9750) = 0.0500. This is why z = 1.96 is the classic critical value at the 5% level.

The Student's t-distribution is used when the population standard deviation is unknown and must be estimated from the sample. The shape of the distribution depends on the degrees of freedom (df), which equals n − 1 for a one-sample test or n₁ + n₂ − 2 for an independent two-sample test.

The two-tailed p-value for a t-statistic uses the regularised incomplete beta function:

As an example: t = 2.064 with df = 24 gives a two-tailed p-value of approximately 0.0500. The critical t-value at 5% with 24 df is exactly t₀.₀₅ = 2.064. With more degrees of freedom the t-distribution approaches the normal, so the critical value approaches z = 1.96.

The F-distribution arises in ANOVA and regression analysis when comparing variance ratios. It has two degrees of freedom parameters: df₁ (numerator) and df₂ (denominator). Because the F-statistic is always non-negative and the distribution is right-skewed, hypothesis tests on F are always right-tailed.

The p-value uses the regularised incomplete beta function with the transformation:

As an example: F = 3.89 with df₁ = 2 and df₂ = 27 gives a right-tail p-value of approximately 0.033, indicating the between-group variance is significantly larger than within-group variance at the 5% level.

The chi-square (χ²) distribution is used in goodness-of-fit tests, tests of independence in contingency tables, and tests on variance. Like F, the statistic is always non-negative, so the p-value is typically taken from the right tail.

The chi-square CDF equals the lower regularised incomplete gamma function P(a, y), where a = df/2 and y = χ²/2:

As an example: χ² = 7.815 with df = 3 gives a right-tail p-value of approximately 0.050, which is the critical value for a goodness-of-fit or independence test at the 5% level with 3 degrees of freedom.