p-value

p-value is the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true. It is not the probability that the null is correct.

Also known asprobability value

ByHoang TruongUpdated June 30, 2026

FrameworkHypothesis testing

See it move

Loading infographic...

The infographic is a probability-density curve of the test statistic, with a shaded tail area beyond t = 2.10 at 28 degrees of freedom representing the p-value. The annotated p = 0.045 clears the α = 0.05 threshold so the null would be rejected at that level, but falls above α = 0.01 so it would not be rejected there, demonstrating that the same sample can yield opposite decisions depending on the significance level chosen.

Where it fits

SubjectData Analysis & StatisticsCoreTopicHypothesis TestingCore

The formula

LaTeX

p = 2 \cdot P(|T| \geq |t_{obs}| \mid H_0)

Variables

$T$: Test statistic random variable under H₀
$t_{o} b s$: Observed value of the test statistic computed from the sample

Probability of obtaining a result as extreme as the observed one in either tail, if H₀ were true; for a one-tailed test drop the factor of 2 and remove the absolute values.

LaTeX

\text{Reject } H_0 \iff p \leq \alpha

Variables

$α$: Pre-specified significance level (decimal)

Decision rule: a p-value at or below the significance threshold provides sufficient evidence to reject the null hypothesis.

Check yourself

PracticeCORE

A researcher tests whether a new production process reduces the mean defect rate. The one-tailed test yields a p-value of 0.03 at significance level α = 0.05. Which interpretation is correct?

Select an answer to check your understanding.