**Degrees of freedom
(d.f.)** are the total number of independent pieces of information
contributing to the component of variation, minus the number of pieces required
to measure it. Analysis of variance is always reported with two values of
degrees of freedom. The first informs on the number of test samples, and the
second informs on the number of independent and random replicates available for
calibrating the test effect against the background 'error' variation.

For example, a result *F*_{2,12}
= 3.98, *P *< 0.05 indicates a
significant effect at a threshold Type-I error
of *α* = 0.05. This outcome
applies to a design with *a* = 3
samples or treatment levels, giving 2 test d.f. (= *a* - 1, since one grand mean is required to test variation of *a* sample means). The *a* samples were allocated amongst a total
of *N* = 15 sampling units, giving 12
error d.f. (= *N* - *a*, since *a* sample means are required to test within-sample variation of *N* observations).

Doncaster, C. P. & Davey, A. J. H. (2007) *Analysis of Variance and Covariance: How to
Choose and Construct Models for the Life Sciences*. Cambridge: Cambridge
University Press.

http://www.southampton.ac.uk/~cpd/anovas/datasets/