Quick answer
A p-value is the probability of a result at least as extreme as the observed statistic if the null hypothesis is true. A two-tailed z of 1.96 gives p = 0.04999.
Choose the test statistic type, enter its value and the tails (and degrees of freedom where needed). The calculator returns the p-value.
Related tools: Critical Value Calculator: Critical values for z, t, chi-square and F distributions. Expected Value Calculator: Expected value and variance of a discrete distribution. T Statistic Calculator: t statistic from summary statistics.
How to use the P-Value Calculator
- Choose test statistic: z, t, chi-square and F.
- Enter statistic value.
- Optionally, enter degrees of freedom (df₁) (above 0).
- Optionally, enter df₂ (F only) (above 0).
- Choose test type: Two-tailed, Left-tailed and Right-tailed.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Test statistic | Choose one: z; t; chi-square; F | Starts at z |
| Statistic value | A number | — |
| Degrees of freedom (df₁) (optional) | A number | > 0 |
| df₂ (F only) (optional) | A number | > 0 |
| Test type | Choose one: Two-tailed; Left-tailed; Right-tailed | Starts at Two-tailed |
Formula and method
Formula
p = tail area of the reference distribution beyond the statistic
IEEE-754 double precision; shown to at most 6 decimal places.
Chi-square and F are right-tailed statistics: left/right options apply the corresponding tail; two-tailed doubles the smaller tail for symmetric z and t, and uses right-tail p for chi-square and F.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Test statistic = z; Statistic value = 1.96; Test type = Two-tailed.
| Result | Value |
|---|---|
| p-value | 0.049996 |
Example 2
Inputs: Test statistic = z; Statistic value = 1.96; Test type = Right-tailed.
| Result | Value |
|---|---|
| p-value | 0.024998 |
Example 3
Inputs: Test statistic = t; Statistic value = 2; Degrees of freedom (df₁) = 2; Test type = Two-tailed.
| Result | Value |
|---|---|
| p-value | 0.183503 |
The first line is the headline result (p-value). IEEE-754 double precision; shown to at most 6 decimal places.
Interpretation
A p-value below 0.05 is conventionally 'statistically significant'. It's not the probability that the null is true, nor the size of the effect. A z of 1.96 is just on the 5% boundary.
Details
The t, chi-square and F distributions need degrees of freedom, and chi-square needs a non-negative statistic. One-tailed p-values are half the two-tailed ones for symmetric distributions: 0.025 here.
How the result changes with statistic value
Holding every other input at the first worked example, this table changes only statistic value. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Statistic value | p-value |
|---|---|
| 0.98 | 0.327086 |
| 1.96 | 0.049996 |
| 3.92 | 0.000089 |
| 7.84 | 0 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Degrees of freedom (df₁): left empty shows "Enter the degrees of freedom." beside that field and no result.
- df₂ (F only): left empty shows "Enter df₂." beside that field and no result.
- Statistic value: entering "-1" shows "Chi-square cannot be negative." beside that field and no result.
Choosing between the P-Value Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Critical Value Calculator](/statistics/critical-value/) | distribution, significance level and tails | Critical value |
| [Expected Value Calculator](/statistics/expected-value/) | values and probabilities | Expected value E[X] |
| [T Statistic Calculator](/statistics/t-statistic/) | sample mean, hypothesised mean and sample standard deviation | t statistic |
| [Degrees of Freedom Calculator](/statistics/degrees-of-freedom/) | test and n₁ | Degrees of freedom |
| [One-Sample T-Test Calculator](/statistics/t-test-calculator/) | sample mean, hypothesised mean and sample standard deviation | t statistic |
Background: hypothesis testing and inference
A hypothesis test compares a test statistic with what the null hypothesis predicts. The decision rule is to reject the null hypothesis when the p-value is at or below the significance level α, which you should choose before seeing the data (0.05 is conventional, not special). For a two-tailed z test the p-value is 2 × (1 − Φ(|z|)); for a one-sample t test the statistic is (x̄ − μ₀) ÷ (s ÷ √n) with n − 1 degrees of freedom.
Failing to reject the null hypothesis does not prove it true, and a statistically significant result can still be too small to matter in practice.
Assumptions and limitations
- df required for t, chi-square and F
- Chi-square needs a non-negative statistic
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for degrees of freedom (df₁), df₂ (F only) and statistic value to confirm this.
Frequently asked questions
What p-value is significant?
A p-value below 0.05 is the common significance threshold, though some fields use 0.01 or 0.005.
Is p = 0.05 proof?
No. p = 0.05 means data this extreme would occur 5% of the time if the null were true; it is not proof either way.
What if p is large?
There's not enough evidence against the null.
What does the P-Value Calculator calculate?
P-value from a z, t, chi-square or F statistic. It returns p-value.
What formula does the P-Value Calculator use?
p = tail area of the reference distribution beyond the statistic. Chi-square and F are right-tailed statistics: left/right options apply the corresponding tail; two-tailed doubles the smaller tail for symmetric z and t, and uses right-tail p for chi-square and F.
What do I need to enter in the p-Value Calculator?
You enter Test statistic, Statistic value and Test type. Optional: Degrees of freedom (df₁) and df₂ (F only).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. df required for t, chi-square and F.
What are the limits of this calculator?
df required for t, chi-square and F. Chi-square needs a non-negative statistic.
What does the “Test statistic” option change?
It sets how the p-Value is worked out. The choices are z, t, chi-square and F.
What is an example of a p-Value result?
With test statistic z, statistic value 1.96 and test type Two-tailed, the p-value is 0.049996.
Related calculators
- Critical Value Calculator — Critical values for z, t, chi-square and F distributions.
- Expected Value Calculator — Expected value and variance of a discrete distribution.
- T Statistic Calculator — t statistic from summary statistics.
- Degrees of Freedom Calculator — Degrees of freedom for one- and two-sample t-tests.
- One-Sample T-Test Calculator — One-sample t-test from summary statistics.
- One-Sample Z-Test Calculator — Z-test for a mean with known population standard deviation.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
- Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.