P-Value Calculator
Enter your z or t statistic (and degrees of freedom for t), pick the tail and a significance level, and get the p-value, the critical value and a reject or do-not-reject verdict.
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Formula
With F the CDF of the standard normal (z) or of Student t with the given degrees of freedom (t):
- Left-tailed:
p = F(statistic) - Right-tailed:
p = 1 - F(statistic) - Two-tailed:
p = 2 × (1 - F(|statistic|)) - Reject the null hypothesis when
p < α. - The critical value is
F⁻¹(1 - α/2)(two-tailed),F⁻¹(1 - α)(right) orF⁻¹(α)(left).
Worked example
z = 1.96, two-tailed, α = 0.05:
p = 2 × (1 - Φ(1.96)) = 0.049996, shown as 0.049996- Critical value
Φ⁻¹(0.975) = 1.959964 - p = 0.049996 is just below 0.05, so the verdict is reject the null hypothesis.
Same test with t = 2.228 and df = 10: p = 0.050012, critical value 2.228139, so it is just not significant.
Frequently asked questions
What is a p-value?
A p-value is the probability of seeing a test statistic at least as extreme as yours if the null hypothesis were true. For z = 1.96 the two-tailed p-value is 0.0500 (more precisely 0.049996).
How do I get a p-value from a z-score?
For a right-tailed test it is 1 - Φ(z); for a left-tailed test it is Φ(z); for a two-tailed test it is 2 × (1 - Φ(|z|)). With z = 1.96 the right-tailed p-value is 0.0250 and the two-tailed one is 0.0500.
How is a t-test p-value different?
It uses the Student t distribution, which has heavier tails that depend on the degrees of freedom. With t = 2.228 and df = 10 the two-tailed p-value is 0.0500, while z = 2.228 would give 0.0259.
What does it mean when p is below alpha?
If p < α (for example 0.0122 < 0.05) the result is called statistically significant and you reject the null hypothesis. It does not give the probability that the null is true, and it says nothing about the size or importance of the effect.