The phrase "degrees of freedom" is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. The three usages are linked historically and through the underlying mathematics through the concept of dimensionality, but they are not identical.

Physics and chemistry

In physics and chemistry, each independent mode in which a particle or system may move or be oriented is one degree of freedom. For a roughly dumbbell-shaped hydrogen molecule, three such modes would be rotation (twirling), translation (hurtling through space) and vibration (the two dumbbell "balls" bouncing together and apart). According to the Equipartition Theorem of thermodynamics, in case of thermal equilibrium each degree of freedom in every particle of a system will contain the same energy on average (equal to kT, the temperature of the system multiplied by the fundamental Boltzmann constant). However, thermal equilibrium can only be reached among interacting particles, a process called thermalization . According to quantum mechanics and more specifically Heisenberg's uncertainty principle, the amount of energy within any degree of freedom is never zero, but is always at least equal to the zero-point energy for that mode.

Engineering

In mechanical engineering, aeronautical engineering and robotics, degrees of freedom (DOF) describes flexibility of motion. A mechanism or linkage that has complete freedom of motion (even if only in a limited area, or envelope) has six degrees of freedom. Three modes are translation - the ability to move in each of three dimensions. Three are rotation, or the ability to change angle around three perpendicular axes.

To put it in simpler terms, each of the following is one degree of freedom:

1. Moving up and down (heaving );
2. moving left and right (swaying );
3. moving forward and back (surging);
4. tilting up and down (pitching);
5. turning left and right (yawing);
6. tilting side to side (rolling).

See also: Euler angles. In robotics Degrees of Freedom is the number of directions that a robot can pivot or move a joint. A human arm is considered to have 7 DOF. A shoulder gives pitch, yaw and roll, an elbow allows for pitch, and a wrist allows for pitch yaw and roll. Only 3 of those movements would be necessary to move the hand to any point in space, but people would lack the ability to grasp things from different angles or directions.

Statistics

In statistics, degrees of freedom are the number of values in probability distributions that are free to be varied. Examples of this statistical parameter include the chi-square distribution, the F-distribution, Student's t-distribution, and the beta distribution that underlies them. See Pearson's chi-square test and analysis of variance for more information.

In the familiar uses of these distributions, the degrees of freedom takes only integer values (usually low ones). The underlying mathematics do allow for fractional degrees of freedom, which can arise in more sophisticated uses.