in mathematics, any of the number of independent quantities necessary to express the values of all the variable properties of a system. A system composed of a point moving without constraints in space, for example, has three degrees of freedom because three coordinates are needed to determine the position of the point.

The number of degrees of freedom is reduced by constraints such as the requirement that a point move along a particular path. Thus, a simple pendulum has only one degree of freedom because its angle of inclination is specified by a single number. In a chemical system, the condition of equilibrium imposes constraints: properties such as temperature and composition of coexisting phases cannot all vary independently (see phase rule).

If, in a statistical sample distribution, there are n variables and m constraints on the distribution, there are n−m degrees of freedom.