Interpreting P-Values
A p-value is the probability of getting evidence at least as extreme as your sample if the null hypothesis were true — a small p-value means your data would be surprising under the null.
Definition
The p-value answers one precise conditional question: assuming the null hypothesis is exactly true, how likely is a result as extreme as (or more extreme than) the one you observed, in the direction the alternative predicts? It is read off the sampling distribution of the test statistic under H₀. A small p-value says the observed data would be unusual if H₀ held, which counts as evidence against H₀.
The formula & procedure
Decide on a significance level α (commonly 0.05) before seeing the data. If the p-value ≤ α, reject H₀ and call the result "statistically significant"; if the p-value > α, fail to reject H₀. For a one-sided test the p-value is a single tail of the null distribution; for a two-sided test it is both tails (often twice the one-tail area). In symbols the p-value is P(data this extreme | H₀ true) — never P(H₀ true | data).
Worked example
A test returns a p-value of 0.03 for the alternative Hₐ: p > 0.5. Read correctly: if the true proportion really were 0.5, samples as favorable as (or more favorable than) the one observed would occur only about 3% of the time. At α = 0.05, because 0.03 ≤ 0.05, you reject H₀ and call the result statistically significant. But note what 0.03 does not mean — it is not "a 3% chance H₀ is true," and it says nothing about how large or important the effect is, only how surprising the data are under H₀.
The classic mistake
Reading the p-value backward as the probability the null hypothesis is true (or that the alternative is true). Two more: conflating statistical significance with practical importance — a tiny, meaningless effect can be "significant" with a big enough sample, and a real effect can miss significance with a small one — and choosing α after seeing the p-value to make a result come out significant.
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