# Degrees of Freedom

### Gyroscope

Number of orthogonal axes about which the spin axis is free to rotate.

### Mechanics

The number of degrees-of-freedom of a mechanical system is equal to the minimum number of independent co-ordinates required to define completely the positions of all parts of the system at any instant of time. In general, it is equal to the number of independent displacements that are possible. A spring-mass which cannot rotate or move horizontally is said to have one degree of freedom (in the vertical direction). An unconstrained mass has six degrees of freedom (3 translations and 3 rotations). A continuous system such as a beam has an infinite number of degrees of freedom. A system of particles could have up to 6 degrees of freedom, but would normally have less because of constraints (e.g. some of them might be joined by rigid links).

### Statistics

Degrees of freedom is a term used in statistics to characterize the number of independent pieces of information contained in a statistic. For example, if we begin with a random sample of n observations and estimate the mean by the sample average, we are left with only (n-1) independent measurements from which to estimate the variance or deviations around the mean. In a simple regression, where we estimate both an intercept and a slope, only (n-2) degrees of freedom remain to measure variability around the fitted line.

**See also: **Complexity, Connection, Fixed Connection, Single Degree of Freedom, Six Degrees of Freedom, Support.

**Subjects: ** Mechanical Engineering