Does A Small P-Value Prove The Alternative Hypothesis?

No. It measures how incompatible the observed statistic is with the stated null model and assumptions. It is not the probability that either hypothesis is true.

When Is A Two-Tailed Chi-Square Or F Test Appropriate?

Two-sided lower and upper rejection regions can be used for variance-type tests. Chi-square goodness-of-fit is normally right-tailed, so the sample builder defaults to that direction.

P-Value Calculator

Calculate exact tail probabilities, critical values and significance decisions for z, Student t, chi-square and F statistics.

Distribution Workbench

P-Value Analysis

Evaluate an observed test statistic, inspect its tail area and compare the exact analytical probability with your alpha threshold.

p = 0.0499957903

Test Setup

Enter a reported statistic directly or calculate one from the supported sample summary.

Test Statistic

Tail DirectionAlpha Threshold

Common Alpha

Analysis Result

The p-value is the tail probability at least as extreme as the observed statistic under the selected null distribution.

Exact P-Value

0.0499957903

Statistically Significant At α = 0.05

Test Statistic: 1.96
Degrees Of Freedom: Not Used
Critical Value: -1.95996398 and 1.95996398
Tail Area: Two-Tailed

Distribution View

The solid marker is the observed statistic. Dashed markers show the rejection boundary at the selected alpha.

Distribution DensityP-Value AreaTest StatisticCritical Value

Method & Interpretation

Every probability is calculated locally with deterministic numerical routines. No simulation, lookup service or uploaded data is used.

Statistic Formula

The test statistic was entered directly.

Probability Formula

p = 2 × min[Φ(z), 1 − Φ(z)]

Numerical Method

Standard normal cumulative distribution evaluated with the regularized gamma function. Critical values are found by deterministic monotonic bisection against the same cumulative distribution.

A small p-value measures incompatibility with the selected null model; it does not measure effect size, practical importance, the probability that the null hypothesis is true or whether the test assumptions are appropriate.

EDUCATIONAL USE: This calculator supports statistical checking and learning. It is not a substitute for study design expertise, assumption checks, multiple-testing controls or domain review.

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What Is the P-Value Calculator?

The P-Value Calculator evaluates standard normal, Student t, chi-square and F distributions locally. It supports left-, right- and two-tailed tests, alpha thresholds, exact cumulative and survival probabilities, critical values, an annotated distribution view and direct or sample-derived test statistics.

How It Works

Choose a distribution and enter a test statistic, or build one from one-sample z or t data, chi-square goodness-of-fit counts or an F variance ratio. Set the tail and alpha, then review the p-value, rejection threshold, formula and method. Copy the result or export PDF, CSV or text.

When to Use It

Use it to check a correctly specified hypothesis test, study worked examples or verify statistical software output. Choose the test, tail and assumptions before interpreting the p-value; this calculator does not establish effect size, practical importance or causality.

Frequently Asked Questions

Does A Small P-Value Prove The Alternative Hypothesis?: No. It measures how incompatible the observed statistic is with the stated null model and assumptions. It is not the probability that either hypothesis is true.
When Is A Two-Tailed Chi-Square Or F Test Appropriate?: Two-sided lower and upper rejection regions can be used for variance-type tests. Chi-square goodness-of-fit is normally right-tailed, so the sample builder defaults to that direction.

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