Hypothesis Test and p Value

From a test statistic to a probability

This calculator finds a tail probability under the null hypothesis from a test statistic you have already calculated. The z mode uses the standard normal distribution. The t mode uses the specified degrees of freedom. Chi square and F calculations use the upper tail only. Keep numerator and denominator degrees of freedom in the correct order for F.

Interpretation

A two sided z or t test uses p = 2P(T ≥ |t|). For t = 1 with one degree of freedom the upper tail is 0.25 and the two sided p value is 0.5. Reject the null hypothesis when p is at most α. Choose α and the alternative direction before analysis.

Limits

A small p value does not guarantee a large effect. Valid inference depends on sampling, independence and distribution assumptions. There is no multiple comparison adjustment. Statistics are limited to a magnitude of one million and degrees of freedom to one hundred thousand.

Separate the statistic from the decision threshold

The statistic comes from data, while the significance level is part of the decision rule. Changing α at the same statistic and degrees of freedom does not change p. Only the comparison's outcome may change. Record the rule before analysis instead of changing the threshold afterwards to obtain significance. The distribution must match the method used to calculate the statistic.

In a special example that permits a manual solution, choose F = 3 with numerator and denominator degrees of freedom both 2. For this particular distribution, upper tail probability is 1/(1 + F), so p = 1/(1 + 3) = 0.25. With a previously selected illustrative threshold α = 0.2, p remains larger and the null hypothesis cannot be rejected. This threshold is chosen to explain the comparison, not as a recommendation.

Failure to reject does not prove the model or equality of two groups. Data quantity and variability also affect the test's ability to distinguish effects. Assess an effect estimate and suitable uncertainty interval alongside the result. This tool calculates a tail area from a supplied statistic. It does not select tests from raw data, investigate independence or repair experimental design problems.

Frequently asked questions

How do you calculate a p value from a z score?

For an example standard normal test statistic of 1.96, the two sided p value is 0.04999579, with a selected significance level of 0.05. The upper tail beyond the absolute statistic is doubled, and a small p value does not establish a large effect or correct for multiple comparisons.

Calculate with these values