Degrees of Freedom (statistics) - Degrees of Freedom Parameters in Probability Distributions

Degrees of Freedom Parameters in Probability Distributions

Several commonly encountered statistical distributions (Student's t, Chi-Squared, F) have parameters that are commonly referred to as degrees of freedom. This terminology simply reflects that in many applications where these distributions occur, the parameter corresponds to the degrees of freedom of an underlying random vector, as in the preceding ANOVA example. Another simple example is: if are independent normal random variables, the statistic

follows a chi-squared distribution with n−1 degrees of freedom. Here, the degrees of freedom arises from the residual sum-of-squares in the numerator, and in turn the n−1 degrees of freedom of the underlying residual vector .

In the application of these distributions to linear models, the degrees of freedom parameters can take only integer values. The underlying families of distributions allow fractional values for the degrees-of-freedom parameters, which can arise in more sophisticated uses. One set of examples is problems where chi-squared approximations based on effective degrees of freedom are used. In other applications, such as modelling heavy-tailed data, a t or F distribution may be used as an empirical model. In these cases, there is no particular degrees of freedom interpretation to the distribution parameters, even though the terminology may continue to be used.