P-value calculator

A p-value measures how surprising your data would be if the null hypothesis were true. A small value says the observed result would rarely occur by chance alone under that assumption, which counts as evidence against the null. It does not give the probability that the null is true, nor the probability that your finding is real; both are frequent misreadings. The p-value speaks only to the data given the hypothesis, never the other way round, and it says nothing on its own about how large or important the effect is.

The 0.05 significance threshold is a convention, not a natural boundary. Calling p below 0.05 significant simply means you have agreed to risk a 5% false-positive rate when the null is in fact true. A result at 0.049 is not meaningfully different from one at 0.051, so treat the cut-off as a guide rather than a verdict. Set the threshold, often written as alpha, before you collect data, and read a borderline value alongside the effect size and the study design. For the fuller picture of what the number can and cannot tell you, see what a p-value is and what it is not.

A two-tailed p-value tests for a difference in either direction, counting extreme results in both tails of the sampling distribution. A one-tailed p-value looks only one way, asking whether the effect runs in a single pre-specified direction, and for a symmetric distribution it is exactly half the two-tailed value. That halving is tempting, because it makes a result easier to call significant, which is precisely why the choice has to be principled rather than convenient.

Use a two-tailed test as the default. It is the more conservative option and the one reviewers expect unless you have a strong directional hypothesis stated before seeing the data and you genuinely have no interest in an effect in the opposite direction. Switching to one-tailed after the analysis to push a finding under 0.05 is a recognised form of p-hacking. Decide the tails when you choose the test, a decision that belongs with the rest of your design and is worth settling using the decision guide for statistical tests.

The same effect produces a smaller p-value in a larger sample. As the number of observations grows, the standard error shrinks, the test statistic gets bigger, and the tail area falls. This is why a trivial difference can become statistically significant once the sample is large enough, and why a real difference can stay non-significant in a study that is simply too small. Significance is therefore as much a statement about your sample size as about the phenomenon itself.

Two practical consequences follow. First, never treat a non-significant result as proof of no effect; it may only reflect low statistical power. Plan the sample in advance so the test can detect an effect worth finding, which is the job of a prospective power analysis. Second, in a very large study, report the effect size and its interval so readers can see whether a significant result is also a meaningful one, rather than letting the small p-value carry the whole conclusion.

A p-value tells you whether an effect is detectable; an effect size tells you how big it is. They answer different questions, and a small p-value does not imply a large effect. Because significance depends on sample size, a tiny, practically meaningless difference can return p below 0.001 in a big enough study, while a substantial effect can miss significance in a small one. Reporting only the p-value leaves the reader unable to judge whether the finding matters in the real world.

That is why current reporting standards expect an effect size beside every significance test, often with a confidence interval around it. Quote the magnitude in interpretable terms, a standardised Cohen's d, a correlation, or a difference in the original units, and read it against what is typical in your field. Pairing the two stops you from overselling a significant but trivial result or dismissing a meaningful one that fell just short, a balance set out in effect sizes explained.

APA style asks for the exact p-value to two or three decimal places rather than a vague p < .05, because the precise figure lets readers judge the strength of evidence for themselves. Drop the leading zero, since a p-value cannot exceed one, and write values such as p = .03 or p = .278. The one exception is a value too small to print meaningfully, which is reported as p < .001 by convention rather than padded with leading zeros.

Place the p-value in the same sentence as the test statistic, its degrees of freedom, and the effect size, so the full result reads as one unit. For example, the test statistic, then the exact p-value, then the standardised effect. Keeping that order consistent across the chapter makes the results easy to scan and hard to fault. The complete set of spacing, italics, and decimal rules lives in our APA style guide for statistics, and the APA results formatter assembles the sentence for you.