Degrees of freedom (statistics)

In statistics, the unknown parameters of a population can be estimated from a sample. The number of independent observation values ​​minus the number of estimable parameters is called the number of degrees of freedom:

The degree of freedom can be a number of " unnecessary " measurements indicate, that are not needed to determine the parameters.

The degrees of freedom are required in the estimation of variances. In addition, different probability distributions, which hypothesis tests are performed on the sample, depending on the degrees of freedom.

A sample consists of more measured values ​​( ) as ( estimable ) parameters () are available, so that deviations ( residuals ) are made between the measured values ​​and those derived from the estimated parameters setpoints. These residuals are to be estimated in addition to the parameters. Without the observations to know, can be set up conditions for the residuals already, so that only independent residuals remain. From observations therefore can be estimated parameters and independent residuals.

Example

In order to estimate the expected value and the variance of a population are observed values ​​. First, the expected value is to be estimated, it is here in front of parameters. Using the least squares method to obtain the estimate of the expected value as the arithmetic mean of the observations:

And for the residuals, the condition ( ), that their sum is zero:

For the estimation of the variance of the sum of squares is

Needed. This sum of squares has degrees of freedom equal to the number of independent residuals. The expected value of the sum of squares is generally

For an unbiased estimate of the variance of the residual sum of squares is therefore divided by the number of degrees of freedom:

If the expectation value of the population is known, there is no parameter to be estimated; it is so. In this case, the estimator of the variance of degrees of freedom:

Degrees of freedom as parameters of distributions

The number of degrees of freedom is also parameters of several distributions. If the observations are normally distributed, the quotient of the sum of squares of the residuals and the variance has the distribution with degrees of freedom:

More dependent on the number of degrees of freedom distributions are the distribution and the distribution. These distributions are required for the estimation of confidence intervals for the parameters and hypothesis testing.