P-value calculator
A p-value is a tail area: how much of the null distribution is at least as extreme as your result. Here it is, shaded.
If the null hypothesis were true, a z statistic at least as extreme as 2.1 would turn up in about 3.57% of samples. That is rare enough, by your chosen α, to count as evidence against H₀.
A p-value is a shaded area
Assume the null hypothesis is true. Then your test statistic should follow the curve above. The p-value is the area in the tail or tails beyond your statistic: the chance of a result at least as extreme as yours if nothing but chance were at work.
- Right-tailed (H₁: μ > μ₀): p = P(Z ≥ z)
- Left-tailed (H₁: μ < μ₀): p = P(Z ≤ z)
- Two-tailed (H₁: μ ≠ μ₀): p = 2 × P(Z ≥ |z|)
What a p-value is not
It is not the probability that the null hypothesis is true, and it does not measure how big or important an effect is. A tiny, useless difference can give p < 0.001 with a huge sample. The American Statistical Association's 2016 statement on p-values makes these points and recommends reporting effect sizes and confidence intervals alongside p.
Getting the test statistic
For a one-sample mean test, z = (x̄ − μ₀) / (σ/√n), or t = (x̄ − μ₀) / (s/√n) with n − 1 degrees of freedom when σ is unknown. See the t distribution calculator for critical values.
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Questions students ask
How do you find a p-value from a z-score?
For a right-tailed test, p = 1 − Φ(z). For left-tailed, p = Φ(z). For two-tailed, p = 2 × (1 − Φ(|z|)). Φ is the standard normal cumulative area (the z table).
What does a p-value actually mean?
The probability of getting a result at least as extreme as yours if the null hypothesis were true. It is not the probability that the null hypothesis is true.
What is a significant p-value?
By convention, p < 0.05 is called statistically significant, but the cut-off is a choice. The American Statistical Association's 2016 statement warns against treating 0.05 as a bright line.
One-tailed or two-tailed?
Use one-tailed only if you decided the direction before seeing the data and a result in the other direction would mean nothing to you. Otherwise use two-tailed.