🧪 P-Value Calculator
Convert a test statistic into a p-value for z, t, chi-square and F tests – one- or two-tailed – and see whether it is significant.
Quick answer: For a z statistic of 2.1 the two-tailed p-value is 2 × P(Z > 2.1) = 2 × 0.017864 = 0.035729, which is significant at the 0.05 level but not at the 0.01 level.
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P-Value Calculator inputs
Result
Two-tailed p-value for z = 2.1
0.035729
Significant at the 0.05 level, not at 0.01
- Significant at α = 0.05?
- Yes
- Significant at α = 0.01?
- No
- Distribution
- N(0, 1)
Step by step
- Distribution under the null hypothesis: N(0, 1)
- Left tail: P(Z < 2.1) = 0.982136
- Right tail: P(Z > 2.1) = 0.017864
- Two-tailed test: p = 2 × P(Z > |2.1|) = 2 × 0.017864 = 0.035729
- Compare with α: p < 0.05 but p ≥ 0.01
Critical values (two-tailed, N(0, 1))
| Significance level α | Reject H₀ when the statistic is | Your result |
|---|---|---|
| 0.1 | |stat| > 1.6449 | Significant |
| 0.05 | |stat| > 1.96 | Significant |
| 0.01 | |stat| > 2.5758 | Not significant |
| 0.001 | |stat| > 3.2905 | Not significant |
How to use the p-value calculator
- Choose the distribution of your test statistic: z, t, chi-square (χ²) or F.
- Choose the tail that matches your alternative hypothesis.
- Enter the statistic and, for t, χ² and F, the degrees of freedom.
Formulas
Right-tailed p = P(T ≥ t) · Left-tailed p = P(T ≤ t) · Two-tailed p = 2 × P(T ≥ |t|)
For χ² and F, which are not symmetric, the two-tailed p-value is twice the smaller tail. Most chi-square and F tests (goodness of fit, independence, ANOVA) are right-tailed.
Worked example
A z-test gives z = 2.1 and the alternative is two-sided. The right-tail area is P(Z > 2.1) = 0.017864, so p = 2 × 0.017864 = 0.035729. That is below 0.05 (reject H₀ at the 5% level) but above 0.01.
A t-test with t = 2.0 and 10 degrees of freedom gives a two-tailed p of about 0.0734 – larger than the z result because the t distribution has heavier tails.
Which test gives which statistic?
| Test | Statistic | Degrees of freedom |
|---|---|---|
| z-test (σ known, large n) | z | – |
| One-sample / paired t-test | t | n − 1 |
| Two-sample t-test (pooled) | t | n₁ + n₂ − 2 |
| Chi-square independence | χ² | (r − 1)(c − 1) |
| One-way ANOVA | F | k − 1, N − k |
Reading the result
- p < α: statistically significant – reject the null hypothesis.
- p ≥ α: not significant – the data is consistent with H₀ (which is not proof that H₀ is true).
- Statistical significance says nothing about the size or importance of an effect; report an effect size or confidence interval too.
Frequently asked questions
What is a p-value?
The probability of getting a test statistic at least as extreme as the one you observed, assuming the null hypothesis is true. A small p-value means the data would be surprising if H₀ were true, which is evidence against it. It is not the probability that H₀ is true.
Should I use a one-tailed or two-tailed test?
Use two-tailed when any difference matters (the mean could be higher or lower). Use one-tailed only if you decided in advance that only one direction is of interest. The two-tailed p-value for z and t is double the one-tailed value.
How many degrees of freedom do I use?
One-sample t-test: n − 1. Two-sample pooled t-test: n₁ + n₂ − 2. Chi-square goodness of fit: categories − 1; test of independence: (rows − 1)(columns − 1). ANOVA F: df₁ = groups − 1, df₂ = total N − groups.
Is p = 0.05 significant?
By the usual rule you reject H₀ when p < α, so p = 0.05 exactly is borderline and not below 0.05. Report the exact p-value and remember that 0.05 is a convention, not a law: p = 0.049 and p = 0.051 carry almost the same evidence.