A p-value is the probability, assuming the null hypothesis is true, of getting a test statistic at least as extreme as the one you observed. Small p-values mean your data would be surprising under H₀ and count as evidence against it. If p is at or below your significance level α, you reject the null hypothesis.
What “extreme” means depends on the alternative hypothesis. For a right-tailed test it is the area to the right of your statistic, for a left-tailed test the area to the left, and for a two-tailed test both tails are counted. For the symmetric z and t distributions the two-tailed p-value is simply twice the one-tail area beyond |x|; for chi-square and F it is twice the smaller tail.
A p-value is not the probability that H₀ is true, and it says nothing about how large or important an effect is. With huge samples, trivial differences become “significant”; with small samples, important ones may not. Always report the effect size and a confidence interval alongside it.
How to use the P-Value Calculator
- Distribution of the test statistic
- Test statistic (z, t, χ² or F)
- Degrees of freedom (df, or numerator df₁ for F)
- Denominator degrees of freedom (df₂)
- Type of p-value
- Significance level (α) — Probability of rejecting a true null hypothesis, e.g. 0.05
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Pick the distribution, type the statistic and the degrees of freedom (one value for t and χ², two for F), and choose the tail. The calculator evaluates the exact cumulative distribution function and shades the region that the p-value covers.
Worked example
A z-test gives z = 2.1. The right-tail area is 1 − Φ(2.1) = 0.0179, so the two-tailed p-value is 2 × 0.0179 = 0.0357. At α = 0.05 we reject H₀; at α = 0.01 we would not.
The same number as a t statistic with 10 df gives a two-tailed p-value of 0.062, and as χ² with 1 df (right-tailed) p = 0.147 – the distribution matters as much as the number.
Interpreting p-values
| p-value | Common wording |
|---|---|
| p > 0.10 | No evidence against H₀ |
| 0.05 < p ≤ 0.10 | Weak evidence |
| 0.01 < p ≤ 0.05 | Moderate evidence – significant at 5% |
| 0.001 < p ≤ 0.01 | Strong evidence |
| p ≤ 0.001 | Very strong evidence |
Tips
- State the null and alternative hypotheses before looking at the result.
- Choose the significance level (usually 0.05) in advance.
- Report the effect size and confidence interval, not just the p-value.
- All tests run in your browser using exact distribution functions.
Common mistakes to avoid
- Reading “not significant” as “no effect” – the test may just lack power.
- Running many tests and reporting only the significant ones (use a correction).
- Using a z-test with a small sample and an unknown population standard deviation (use t).
Formula
Left-tailed: p = cdf(x) · Right-tailed: p = 1 − cdf(x) · Two-tailed (z, t): p = 2 × (1 − cdf(|x|)) · Two-tailed (χ², F): p = 2 × min(cdf(x), 1 − cdf(x))
Worked example
Inputs — Distribution of the test statistic: Standard normal (z); Test statistic (z, t, χ² or F): 2.1; Type of p-value: Two-tailed (H₁: not equal); Significance level (α): 0.05.
Results — p-value = 0.04; Decision = Reject H₀ at α = 0.05 – p = 0.0357 ≤ 0.05 and z = 2.1 lies in the rejection region.; Left-tail area P(X ≤ x) = 0.98; Right-tail area P(X ≥ x) = 0.02; Critical value(s) at α = ±1.96.
Key terms
- p-value
- The probability under H₀ of a result at least as extreme as the observed one.
- Tail area
- The area under the density curve beyond a given value.
- Cumulative distribution function (cdf)
- The probability that the statistic is at most x.
- Significance level (α)
- The threshold for rejecting H₀, chosen in advance.
Frequently asked questions
What is the p-value for a z-score of 1.96?
0.05 two-tailed and 0.025 one-tailed. For z = 2.58 the two-tailed p-value is about 0.01.
How do I get a p-value from a t-score?
Use the t distribution with the right df. For t = 2.1 with 20 df, the two-tailed p-value is 0.0486; with 5 df it is 0.090.
Is p = 0.05 significant?
At α = 0.05, yes – the rule is reject when p ≤ α. In practice report the exact value, as results right at the threshold are weak evidence.
Why are chi-square tests usually right-tailed?
Large χ² values mean observed counts are far from expected ones, so only the right tail signals a departure from H₀.
Can a p-value be 0?
Not exactly. Very small values are shown in scientific notation, and should be reported as, for example, p < 0.001.
What is the difference between one-tailed and two-tailed p-values?
For symmetric distributions the two-tailed value is twice the one-tailed value. Choose the tail before looking at the data.
Does a large p-value prove H₀?
No. It only means the data are compatible with H₀; they may also be compatible with meaningful effects.
What formula does the P-Value Calculator use?
Left-tailed: p = cdf(x) · Right-tailed: p = 1 − cdf(x) · Two-tailed (z, t): p = 2 × (1 − cdf(|x|)) · Two-tailed (χ², F): p = 2 × min(cdf(x), 1 − cdf(x)).