One-Sided and Two-Sided p-Values

A p-value answers: if the null hypothesis is true, how likely is a result this extreme or more extreme? What counts as extreme depends on the alternative hypothesis. A one-sided alternative, like p > 0.5, only cares about results far in one direction, so the p-value is just the area in that one tail of the sampling distribution. A two-sided alternative, like p ≠ 0.5, treats a result far above or far below the null value as equally surprising, so you shade both tails and add their areas.

Because the normal curve is symmetric, the two tails beyond a given z-score have equal area. That means the two-sided p-value is simply double the one-sided p-value for the same test statistic. A z of 1.85 has a one-sided p-value of about 0.0322 (the area above z = 1.85). Double it and the two-sided p-value is about 0.0643. Same data, same statistic, different question being asked, different answer.

This matters because the choice between one-sided and two-sided has to be locked in before looking at the sample, as part of stating the hypotheses. If someone peeks at the data, sees the sample proportion leaned high, and then decides to test only p > p0, they are quietly giving themselves extra chances to find significance that the stated significance level does not account for. That connects directly to Type I error: a test's real false-positive rate only matches its stated alpha when the direction was fixed in advance, not chosen to match what the data already showed.