Degrees of Freedom - short version

The number of values that are free to be varied given information, such as the sample mean, is known.

Degrees of Freedom - long version

In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.

Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom (df). In general, the degrees of freedom of an estimate of a parameter is equal to the number of independent scores that go into the estimate minus the number of parameters used as intermediate steps in the estimation of the parameter itself (which, in sample variance, is one, since the sample mean is the only intermediate step).

Mathematically, degrees of freedom is the dimension of the domain of a random vector, or essentially the number of 'free' components: how many components need to be known before the vector is fully determined.

The term is most often used in the context of linear models (linear regression, analysis of variance), where certain random vectors are constrained to lie in linear subspaces, and the number of degrees of freedom is the dimension of the subspace. The degrees-of-freedom are also commonly associated with the squared lengths (or "Sum of Squares") of such vectors, and the parameters of chi-squared and other distributions that arise in associated statistical testing problems.

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