# Degrees of freedom

Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or other physical processes are possible. This relates to the philosophical concept to the extent that people may be considered to have as much freedom as they are physically able to exercise.[citation needed]

## Applications

### Statistics and mathematics

Degrees of freedom (statistics) refers to the number of variables in a statistic calculation that can vary. It can be calculated by subtracting the number of estimated parameters from the total number of values in the sample. For example, a sample variance calculation based on n samples will have n-1 degrees of freedom, because sample variance is calculated using the sample mean as an estimate of the actual mean.

In mathematics, this notion is formalized as the dimension of a manifold or an algebraic variety. When degrees of freedom is used instead of dimension, this usually means that the manifold or variety that models the system is only implicitly defined.

### Mechanics

Degrees of freedom (mechanics) describes the number of independent motions that are allowed to a body. In the case of a mechanism made of several bodies, it refers to the number of possible independent relative motions between the pieces of the mechanism. Ball-and-socket joints, such as the human wrist or hip, can freely rotate and have three degrees of freedom. A free, rigid object, such as a ship at sea, has six degrees of freedom: three rotations and three translations about each perpendicular axis.

In the study of complex motor control, there may be so many degrees of freedom that a given action can be achieved in many different ways. The brain must select and combine movements with different degrees of freedom. This issue is sometimes called the degrees of freedom problem.

### Physics and chemistry

Degrees of freedom (physics and chemistry) refers to the parameters used to describe a physical system, such as chemical composition and pressure. The minimum number of fixed parameters required to define a system, or the dimension of its phase space, is known as its degrees of freedom.