dfwithin

dfwithin is a term used in statistics, often appearing in notation or software outputs for analysis of variance (ANOVA). It denotes the degrees of freedom associated with the within-group or residual variation in a model. These degrees of freedom quantify how much independent information remains to estimate within-group variability after accounting for the estimated group effects.

In a standard one-way ANOVA with k groups and a total of N observations, dfwithin equals N minus k. This arises because the total degrees of freedom are N − 1, and the between-group degrees of freedom are k − 1, leaving dfwithin = (N − 1) − (k − 1) = N − k. The within-groups or error degrees of freedom are used to compute the mean square error (MSE) as SSwithin divided by dfwithin.

dfbetween and dftotal are related measures: dfbetween = k − 1 and dftotal = N − 1. Together, they partition the total variability into between-group and within-group components.

In more complex designs, such as factorial ANOVA or repeated-measures (within-subjects) designs, dfwithin may refer to the degrees of freedom for within-subjects effects or for residual error terms. The exact calculation depends on the model structure, replication, and whether the design is between-subjects, within-subjects, or mixed. Software packages may label these components as dfwithin, dferror, or similar, and the interpretation follows the specific model terms being tested.

In practice, dfwithin is used to compute MSwithin (or MSE) and to form F-statistics for tests of hypotheses about group means or within-subject effects. Always verify the design and the corresponding df values in the analysis output.