Degrees of Freedom (statistics)

Degrees Of Freedom (statistics)

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.

The number of independent ways by which a dynamical system can move without violating any constraint imposed on it, is called degree of freedom. In other words, the degree of freedom can be defined as the minimum number of independent coordinates, which can specify the position of the system completely.

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 number of 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.

While introductory texts may introduce degrees of freedom as distribution parameters or through hypothesis testing, it is the underlying geometry that defines degrees of freedom, and is critical to a proper understanding of the concept. Walker (1940) has stated this succinctly:

For the person who is unfamiliar with N-dimensional geometry or who knows the contributions to modern sampling theory only from secondhand sources such as textbooks, this concept often seems almost mystical, with no practical meaning.