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Degrees of freedom

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

Table of contents
1 Physics and Chemistry
2 Engineering
3 Statistics

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 dumbell-shaped hydrogen molecule, three such modes would be rotation (twirling), translation (hurling through space) and vibration (the two dumbbell "balls" bouncing together and apart). According to thermodynamics, 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). 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.


In mechanical and aeronautical engineering, degrees of freedom (DOF) describes flexibility of motion. A mechanism 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).

A mechanism that can (for instance) be raised and lowered, which has a pivoting head that can tilt forward or back, left or right, can be described as having 3 degrees of freedom (colloquially, 3DOF).


In statistics, degrees of freedom is a statistical parameter in many important probability distributions. Examples 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 take only integer values, usually low ones, though the underlying mathematics allow for fractional degrees of freedom and these do arise in some more sophisticated uses.