P-Value Calculator

A p-value answers one narrow question: if the null hypothesis were true, how often would random sampling produce a result at least as extreme as the one you got? A p of 0.03 means such a result would appear 3% of the time by chance alone. It is a statement about data given a hypothesis, never about the hypothesis given the data — that reversal is the single most common misreading in applied statistics.

For a standard normal distribution, the area beyond ±1.96 is 0.025 in each tail, so the two-tailed p-value is exactly 0.05 and the one-tailed p is 0.025. This is why 1.96 turns up everywhere: it is the critical z for a 95% confidence interval. A t-statistic of 1.96 on 20 degrees of freedom gives a larger p — about 0.064 — because the t-distribution has heavier tails when the sample is small.

Use a two-tailed test when any difference matters, in either direction — the default for almost all research. Use a one-tailed test only when a difference in the opposite direction would be meaningless or impossible to act on, and only when that was decided before seeing the data. Switching to one-tailed after the fact to halve a p-value from 0.08 to 0.04 is a recognised form of p-hacking.

The 0.05 threshold is an arbitrary inheritance from Fisher, and nothing physical changes between p = 0.049 and p = 0.051. Report the actual value rather than just crossing a line, and pair it with an effect size and a confidence interval — a tiny, useless difference will reach significance in a large enough sample, and a large, important one can miss it in a small sample.

P-values assume the test's conditions hold and that you ran one prespecified analysis. Testing many outcomes and reporting the best inflates false positives — twenty independent tests at 0.05 yield roughly one significant result from pure noise, which is why corrections such as Bonferroni exist. Stopping data collection when the result looks good has the same effect. A p-value from an analysis chosen after seeing the data no longer means what it claims.