degree of freedom

1. Statistics. Any of the unrestricted, independent random variables that constitute a statistic.
2. Physics.
   1. Any of the minimum number of coordinates required to specify completely the motion of a mechanical system.
   2. Any of the independent thermodynamic variables, such as pressure, temperature, or composition, required to specify a system with a given number of phases and components.

Any one of the number of independent ways in which the space configuration of a mechanical system may change. A material particle confined to a line in space can be displaced only along the line, and therefore has one degree of freedom. A particle confined to a surface can be displaced in two perpendicular directions and accordingly has two degrees of freedom. A particle free in physical space has three degrees of freedom corresponding to three possible perpendicular displacements. A system composed of two free particles has six degrees of freedom, and one composed of N free particles has 3N degrees. If a system of two particles is subject to a requirement that the particles remain a constant distance apart, the number of degrees of freedom becomes five. Any requirement which diminishes by one the degrees of freedom of a system is called a holonomic constraint. See also Constraint.

Number of data items that are independent of one another. Given a sample of data and the computation of some statistic (e.g., the mean), the degrees of freedom are defined as (number of observations included in the formula) minus (number of parameters estimated using the data). For example, the mean statistic for N sample data points has n DF, but the variance formula has (n - 1) DF because one parameter (the mean X) has to be estimated before the variance formula can be used.

A statistic, based on the number of observations and groups in a study, that is necessary to determine statistical significance. One looks up the degrees of freedom and the significance level in a table of significance values to determine if the magnitude of the value obtained is significant. Used with the t-test, chi square, analysis of variance, and correlation.

statistics Effectively the count of observations, within a set of such, that could be altered independently without changing one or more statistics relating to the set. For a set of n observations with a known mean, the degrees of freedom = n - 1, since all but one could be changed with compensating change only to one other to produce the same mean. If the set is an array with m columns and n rows, and column totals are known, then the degrees of freedom = (m - 1) × n; if both column and row totals are known, the degrees of freedom = (m - 1) × (n - 1).

A number which in some way represents the size of the sample or samples used in a statistical test. In some cases, it is the sample size, in others it is a value which has to be calculated. Each test has its specific calculation, and the correct value for each test must be calculated before the result of the test can be checked for statistical significance.

Used to define statistical distributions of several tests, usually based on the number of data items less the number of parameters estimated.

Degrees of freedom can mean:

• Degrees of freedom (physics and chemistry)
• Degrees of freedom (engineering)
• Degrees of freedom (statistics)
• Degrees of freedom (numerics)

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