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How do I work out degrees of freedom for terms in an ANOVA?

For N subjects and g group variables (factors) with respective levels L1, L2, ... , Lg and a within subjects factors, W, the table below gives the degrees of freedom for various types of sources variation in an analysis of variance.

Source of variation




2-way interaction


K-way interaction of factors

$$\prod_text{k} (L_text{k}-1)$$

Between subjects error terms

Error (one-way anova between subjects)

N - L1

Error (between subjects)

N - df of terms involving between subjects factors - 1

Within subjects error terms

Error (subjects x W1), no between subjects factor


Error (subjects x W1 x W2, no between subjects factor )


Error (subjects x W1 x W2, 1 between subjects factor)


Error (subjects x Within subjects interaction)

df of Error (between subjects) x df(Within subjects interaction term)


Boniface DR (1995) Experiment design and statistical methods for behavioural and social research. Chapman and Hall:London. (This book contains further details about computing degrees of freedom and also SS in balanced designs for terms in an ANOVA).

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