Turn a z-score or t-statistic into a p-value — one-tailed or two-tailed — with a plain-English read on statistical significance.
A p-value is the probability of seeing a result at least this extreme if the null hypothesis were true. It is not the probability that the null is true.
p-value
—How to use this tool
z-test for large samples or known variance; t-test when variance comes from the sample.
The z-score or t-statistic your analysis produced.
Needed for the t-test only; for a one-sample t-test it is n − 1.
Two-tailed is the default — it tests for a difference in either direction.
Formula & worked example
z = 1.96 gives a two-tailed p of 0.0500 — the famous 5% boundary. z = 2.58 gives 0.0099. With a t-statistic of 2.064 on 24 degrees of freedom, the two-tailed p is also about 0.0500, because the t critical value at 5% for 24 df is 2.064.
Limitations — read this before you rely on it
A p-value assumes the test's conditions hold (random sampling, roughly normal data or a large sample, independent observations). It is not the probability that the null hypothesis is true, and a non-significant result is not proof of no effect. The normal tail uses a standard approximation accurate to about 1.5 × 10⁻⁷.
Frequently asked questions
If the null hypothesis were true, you would see a result at least this extreme about 5% of the time by chance alone. It is not a 5% chance the null is true.
Two-tailed tests for a difference in either direction and is the safer default. One-tailed only fits when you genuinely only care about one direction, decided before seeing the data.
The t-distribution is wider for small samples, reflecting the extra uncertainty of estimating variance from the sample. As degrees of freedom grow, it converges to the normal curve.
It is barely significant at 5% — treat it as weak evidence, not a discovery. Report the exact p-value, not just 'significant'.
No. A large p-value means the data are compatible with the null; it does not prove the null, especially with a small sample.