Variant: df
A parameter that appears in some probability distributions used in statistical inference, particularly the t-distribution, the chi-squared distribution, and the F-distribution. The phrase 'degrees of freedom' was introduced by Sir Ronald Fisher in 1922.
In the case of the t-distribution, the term usually reflects the fact that the population variance has been estimated. The number of degrees of freedom is equal to the number of independent pieces of information concerning the variance. In the most familiar case of n observations, x1, x2,..., xn, from a population with unknown mean and variance, there are (n − 1) independent deviations from the mean, since
,
.
In the case of a random variable with a chi-squared distribution, if it can be expressed as the sum of squares of m independent standard normal variables, then the distribution has m degrees of freedom.
In the case where the chi-squared distribution is used as a goodness-of-fit test, each independent parameter estimated from the data represents another constraint, so the number of degrees of freedom is (c−1−p), where c is the number of cells, p is the number of parameters estimated from the sample data, and there is the constraint that the sum of the observed frequencies is the sum of the expected frequencies.
In the case where the chi-squared distribution is used to test the null hypothesis of independence in a J×K contingency table, there are (J−1) (K−1) degrees of freedom.
