degreeoffreedom

Degrees of freedom (DoF) is a measure of the number of independent parameters required to specify a system’s state or the number of independent pieces of information available in a dataset. In mechanics, DoF refers to how many independent coordinates are needed to describe a configuration. Each generalized coordinate contributes one DoF.

In spatial mechanics, a freely moving rigid body has six DoF: three translational coordinates (x, y, z) and three rotational coordinates (pitch, yaw, roll). Constraints such as joints, contacts, or grounding reduce the DoF, shaping how the system can move. For planar mechanisms, DoF reflects the number of independent motions, and there are standard rules (such as Gruebler’s criterion) to estimate mobility from the number of links and joints; these rules account for joints that permit one or more independent movements.

In statistics, degrees of freedom denote the number of independent values that can vary in an analysis. If a dataset has n observations and p parameters are estimated, the residual degrees of freedom are typically n − p. Degrees of freedom determine the exact sampling distributions used in hypothesis tests and confidence intervals, including chi-square, t, and F distributions. They also influence the precision of estimates and the calculation of standard errors.

Common examples: a simple pendulum has one DoF, a free rigid body in space has six, and a particle constrained to move on a plane has two. DoF are context-dependent and can change with constraints or parameterization. They are distinct from, though related to, sample size or data quantity.