The numerator and denominator degrees of freedom are used in the computation…

Concept

A test statistic’s reference distribution can depend on degrees of freedom, which count the independent pieces of information remaining after constraints are imposed.

Some reference distributions use no degrees-of-freedom parameter, some use one, and a ratio of two independently scaled variance estimates requires two.

Application

An F statistic is formed as a ratio of two variance estimates after each has been scaled by its own degrees of freedom. Its reference distribution is therefore written F(d1, d2), where d₁ belongs to the numerator and d₂ to the denominator.

Contrast

• Z is referenced to the standard normal distribution and has no degrees-of-freedom shape parameter.

• Chi-square is indexed by one degrees-of-freedom parameter.

• Student’s t is indexed by one degrees-of-freedom parameter.

• F is indexed by the ordered pair of numerator and denominator degrees of freedom.

Cross-check

In an analysis-of-variance table, an F value is reported alongside one degrees-of-freedom entry for the numerator source and another for the denominator or error source.

Result

Therefore, the test statistic that uses numerator and denominator degrees of freedom is F.