Where does degrees of freedom collide?

In classical mechanics, the degrees of freedom of a system are the number of independent coordinates required to specify its configuration completely: a single particle in space has three, a rigid body has six, and each constraint removes one. Lagrange's 1788 Mécanique analytique built mechanics on exactly such generalized coordinates, writing the equations of motion in terms of the minimal independent variables a system possesses. The referent is the dimensionality of a physical system's configuration space, which sets how many equations of motion are needed, and it is a different quantity from the free values left in a statistical estimate or the free intensive variables of a chemical mixture.[1]

Unit

The count of independent coordinates fixing a mechanical system's configuration

Recovery means

Not applicable

Measured by

Determined by counting independent coordinates and subtracting the holonomic constraints acting on the system

In physical chemistry, the degrees of freedom of a system at equilibrium, also called its variance, are the number of intensive variables such as temperature, pressure, and composition that can be changed independently without altering the number of phases present. Gibbs's 1876 On the equilibrium of heterogeneous substances established the phase rule, F = C − P + 2, that fixes this number from the counts of components and coexisting phases. The referent is how much freedom an equilibrium mixture has before a phase must appear or disappear, and although it too is a count of independent variables, it concerns intensive thermodynamic state rather than a body's position or a data set's residuals.[2]

Unit

The number of intensive variables freely adjustable at equilibrium (F = C − P + 2)

Recovery means

Not applicable

Measured by

Computed from the phase rule using the numbers of independent components and coexisting phases

In statistics, the degrees of freedom of an estimate or a test statistic are the number of values in the calculation that are free to vary once the constraints imposed by the estimated parameters are imposed, typically the sample size minus the number of parameters fitted. Fisher's 1922 On the mathematical foundations of theoretical statistics placed the concept at the centre of estimation theory and of the sampling distributions, such as chi-square and later t and F, whose shapes depend on it. The referent is the amount of independent information left in a sample after fitting, which indexes a family of reference distributions, and it is unrelated to the coordinates of a moving body or the free variables of a chemical equilibrium beyond the shared idea of counting what is free.[3]

Unit

The number of independent residual quantities after fitting parameters (often n − k)

Recovery means

Not applicable

Measured by

Computed as observations minus fitted parameters, and used to index the chi-square, t, and F reference distributions