Formula
Two-tailed: p = 2 × P(T ≥ |t|)Right-tailed: p = P(T ≥ t)Left-tailed: p = P(T ≤ t)Z ~ N(0, 1); T ~ Student’s t with df degrees of freedom How it works
A p-value is 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 your data would be surprising if there were really no effect, which counts as evidence against the null hypothesis.
Use a z statistic when the population standard deviation is known or the sample is large; use a t statistic with n − 1 degrees of freedom (for a one-sample test) when you estimated the standard deviation from a small sample. The t distribution has fatter tails, so the same statistic gives a larger p-value, and it approaches the normal distribution as df grows.
Choose the tails to match your alternative hypothesis. If you are testing for any difference (≠), use two tails — the classic cut-off z = 1.96 gives p ≈ 0.05. If you predicted a direction in advance (> or <), use one tail, which halves the p-value for a result in that direction.
Frequently asked questions
Does p < 0.05 mean my result is important?
No. It means the result is unlikely under the null hypothesis, not that the effect is large or useful. With huge samples, trivially small effects can be “significant”. Report an effect size or confidence interval alongside the p-value.
Is the p-value the probability that the null hypothesis is true?
No — a common misunderstanding. It is the probability of data this extreme if the null were true. It doesn’t give the probability that either hypothesis is correct.
What degrees of freedom should I use?
For a one-sample or paired t-test, df = n − 1. For a two-sample test with equal variances, df = n₁ + n₂ − 2. Welch’s test uses a more complicated formula that usually gives a non-integer df — that’s fine to enter here.