Test Statistic

A test statistic is a quantity computed from sample data and used to evaluate a statistical hypothesis. It reduces the observed data to a number whose magnitude can be compared against a reference distribution under the null hypothesis.

What it does

In a hypothesis test, the test statistic is the bridge between raw observations and a decision rule. Instead of comparing whole datasets directly, the procedure asks whether the observed statistic would be unusual if the null hypothesis were true.

Common examples

• A z statistic compares an estimate to a hypothesized value using a known or assumed standard error.
• A t statistic is used when the variance must be estimated from the sample.
• A chi^2 statistic is used for variance tests, contingency tables, and goodness-of-fit problems.
• An F statistic compares explained and unexplained variance across models.

Interpretation

Large or extreme values of a test statistic indicate stronger disagreement with the null model, but the interpretation depends on the reference distribution and the test design. A test statistic by itself is not a conclusion; it becomes meaningful only together with a sampling model, assumptions, and a decision rule or uncertainty statement.

Relation to model comparison

Test statistics are central to classical significance testing, while other frameworks such as information criteria and probabilistic model comparison focus more directly on predictive adequacy or relative support among models.

See also: Hypothesis test, Hypothesis testing, information criteria, Statistical inference