P-value Calculator

Math

Calculate exact P-values from test statistics across Z, Student’s t, Chi-Square (χ²), and F distributions. Evaluate one-tailed and two-tailed significance, compare against α thresholds, and explore step-by-step hypothesis decisions.

Presets:

✓ Statistically Significant (p ≤ α)

Two-Tailed P(|Z| > |z|)

p = 0.049996 (4.9996%)

Decision at α = 0.05: Reject H₀ at α = 0.05. The observed Z = 1.9600 has a two-tailed p-value of 0.0500, indicating statistically significant evidence against the null hypothesis.

All Tail Probabilities for Z = 1.9600

Left-Tailed: P(Z < 1.9600)

0.975002 (97.5002%)

Right-Tailed: P(Z > 1.9600)

0.024998 (2.4998%)

Two-Tailed: P(|Z| > |1.9600|)

0.049996 (4.9996%)

Central Area: P(-|Z| < X < |Z|)

0.950004 (95.0004%)

Step-by-Step Hypothesis Testing Steps

1 1. Standard Normal CDF Evaluation

Φ(z) = Φ(1.9600) = 0.975002

Evaluates the area under the standard normal density curve from -∞ to z.

2 2. One-Tailed Probabilities

Left-Tailed: P(Z < 1.9600) = 0.975002 | Right-Tailed: P(Z > 1.9600) = 1 - Φ(z) = 0.024998

Directional hypothesis tests test only one direction of the effect.

3 3. Two-Tailed P-Value

P(|Z| > |1.9600|) = 2 × [1 - Φ(|1.9600|)] = 2 × 0.024998 = 0.049996

Measures the probability of observing an extreme test statistic in either tail.

4 4. Decision at Significance Level (α)

p = 0.049996 ≤ α = 0.05 → Reject H₀ (Significant)

A p-value smaller than α indicates the observed discrepancy is unlikely under the null hypothesis.

How This Calculator Works

The p-value (probability value) is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis (H₀) is correct. A small p-value (typically ≤ 0.05) indicates that extreme sample evidence has occurred, leading researchers to reject the null hypothesis.

📐 Hypothesis Testing Framework & Distribution Selection

p-value = P(Test Statistic ≥ Observed Statistic | H₀ is true)

Z-Test: Used when population variance σ² is known, or sample size n is large (CLT)

t-Test: Used when population variance is unknown and estimated using sample standard deviation s with df = n - 1

Chi-Square (χ²): Used for categorical goodness-of-fit and contingency table tests of independence

F-Test: Used in Analysis of Variance (ANOVA) to compare variances across multiple groups

α (Alpha): Significance level: maximum allowable probability of committing a Type I error (false positive)

Key Mathematical Rules & Properties

Formulate Hypotheses

State the Null Hypothesis H₀ (no effect/difference) and Alternative Hypothesis H₁ or Hₐ.

Calculate Test Statistic

Compute the Z, t, χ², or F score from the observed sample data and standard error.

Determine Tail Direction

Use two-tailed tests for non-directional hypotheses (≠) or one-tailed tests for directional claims (> or <).

Compare p with α

If p ≤ α: Reject H₀ (statistically significant). If p > α: Fail to reject H₀.

Understanding your calculation

Method, example and limitations

How the calculation works

A p-value measures how incompatible the test statistic is with a specified null model, using the selected distribution and one-sided or two-sided alternative.

Example to check

For a standard-normal z statistic of 1.96, the two-sided p-value is approximately 0.05.

Before using the result

A p-value is not the probability that the null hypothesis is true or a measure of effect size. Choose the test and tail appropriately and consider multiple testing.

Calculation methodology · Report a result that needs review