What does a p-value mean?

The p-value is the probability of observing a test statistic at least as extreme as the one you calculated, assuming the null hypothesis is true. A small p-value (typically below 0.05) suggests the observed result is unlikely under the null hypothesis, and you reject it in favour of the alternative hypothesis.

What is the difference between a one-tailed and two-tailed test?

A two-tailed test checks whether the result is extreme in either direction (higher or lower than expected). A one-tailed test checks only one direction. For the same test statistic, a one-tailed p-value is half the two-tailed p-value. Use a one-tailed test only when you had a directional hypothesis before collecting data.

When do I use a t-score instead of a z-score?

Use a t-score when your sample is small (typically n below 30) or when the population standard deviation is unknown and you are estimating it from the sample. The t-distribution has heavier tails than the normal distribution, producing larger p-values for the same test statistic, which accounts for the extra uncertainty.

P-value Calculator

The p-value is the probability of getting a test result at least as extreme as the one you observed, assuming the null hypothesis is true. It is the core output of most statistical hypothesis tests. A small p-value indicates that your result would be rare under the null hypothesis, giving you grounds to reject it. The conventional threshold is 0.05: results with p below 0.05 are called statistically significant at the 5% level. Results with p below 0.01 are significant at the 1% level. The choice between a one-tailed and a two-tailed test should be made before you collect data. A two-tailed test asks whether the result is extreme in either direction; a one-tailed test asks whether it is extreme in a specific direction. For the same test statistic, the one-tailed p-value is exactly half the two-tailed p-value. This calculator supports two types of test statistic. The z-score mode uses the standard normal distribution and is appropriate when you know the population standard deviation or have a large sample. The t-score mode uses the t-distribution with your specified degrees of freedom, which is appropriate for smaller samples or when the population SD is unknown. The z = 1.96 default corresponds to a two-tailed p-value of 0.0500, which is the exact boundary of the 5% significance level. Enter your test statistic, choose the test type and tail option, and add degrees of freedom if you are using a t-test.

Test type

Tails

Test statistic (z or t)

Degrees of freedom

0.0500

p-value

Significance at 0.05Significant

Significance at 0.01Not significant

Test typeZ two-tailed

How it works

For the z-test, the calculator uses the complementary error function (erfc) to compute the tail area of the standard normal distribution. The one-tailed p-value is the area in one tail beyond |z|; the two-tailed value doubles it. For the t-test, the calculator uses an accurate numerical approximation of the regularised incomplete beta function to compute tail probabilities for the t-distribution with the specified degrees of freedom. Results are significant at 0.05 if p is below 0.05, and significant at 0.01 if p is below 0.01.

Worked example

With a z-score of 1.96 and a two-tailed test, the area in both tails of the standard normal distribution equals exactly 0.0500. This is the boundary of the 5% significance level: a z-score of 1.96 gives a p-value of exactly 0.05 two-tailed, so results with |z| greater than 1.96 are significant at the 5% level. These match the default values pre-filled above.

Related calculators

Margin of Error Calculator: use z-scores to build confidence intervals for survey proportions.
Standard Error Calculator: find the SE needed to construct a test statistic.
Bayes Theorem Calculator: update probabilities using prior knowledge and new evidence.
Correlation Coefficient Calculator: test whether two variables are linearly related.

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