P-Value Calculator
Convert a z, t, chi-square, or F statistic into an exact distribution-tail probability and compare it with a selected significance level.
P-value formulas
A p-value is a tail probability under a specified null distribution. For a right-tailed test it is P(T≥t observed); for a left-tailed test it is P(T≤t observed). A two-sided symmetric test uses 2×min(left tail, right tail).
Normal probabilities use the standard normal CDF. Student’s t and F probabilities use the regularized incomplete beta function. Chi-square probabilities use the regularized upper incomplete gamma function.
Checked examples
| Distribution | Inputs | p-value |
|---|---|---|
| z, two-sided | z=1.96 | ≈0.049996 |
| z, right-tailed | z=1.645 | ≈0.049985 |
| t, two-sided | t=2.228, df=10 | ≈0.0500 |
| t at zero | t=0, df=4 | 1 two-sided |
| Chi-square | χ²=10, df=2 | ≈0.00673795 |
| F distribution | F=1, df₁=5, df₂=5 | 0.5 right-tailed |
How to use the calculator
- Select the distribution matching the test statistic.
- Enter the statistic and required degrees of freedom.
- For z or t, choose the alternative hypothesis specified before analyzing the data.
- Select α and calculate.
- Compare p with α while also considering effect size, interval estimates, assumptions, and study design.
Choosing the correct distribution
- Use z when the statistic follows a standard normal distribution under the null.
- Use t for a Student’s t statistic with known degrees of freedom.
- Use chi-square for χ² statistics such as goodness-of-fit or independence tests.
- Use F for variance-ratio statistics, including many ANOVA and regression tests.
- Do not choose a distribution only because it produces a smaller p-value.
Limits and interpretation
- This calculator converts an already-computed statistic; it does not verify how that statistic was obtained.
- z and t support two-sided and directional alternatives. Chi-square and F use their conventional right tails.
- Degrees of freedom may be positive decimals where an approximation such as Welch’s test produces them.
- A p-value is not the probability that the null hypothesis is true and does not measure effect size.
- Repeated testing, selective reporting, dependence, invalid assumptions, or biased samples can make the interpretation misleading.
Frequently asked questions
Is p≤0.05 always significant?
Only relative to a preselected α of 0.05. The threshold should be chosen before examining results, and statistical significance does not guarantee practical importance.
When should I use a two-sided test?
Use it when departures in either direction matter. A one-sided alternative should be justified before observing the data.
Why do chi-square and F use the right tail?
Their common tests use nonnegative statistics where larger values indicate a greater departure from the null model.
Can I calculate p from raw sample data here?
Use a test-specific calculator for raw data. This tool starts from a z, t, χ², or F statistic and its degrees of freedom.
Related calculators
Use the T-Test Calculator for raw mean comparisons, the Chi-Square Calculator for categorical counts, the Normal Distribution Calculator for z-scores and normal probabilities, or the Sample Size Calculator to plan a study.