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DEGREES OF FREEDOM
It can be defined as the number of values in the final calculation of a statistic that are free to vary and the number of independent ways by which a dynamical system can move without violating any constraint imposed on it.
In other words, it can also be described as the minimum number of independent coordinates that is used to specify the position of the system completely and the estimates of statistical parameters which can be based upon different amounts of information or data
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It is also described as the number of independent pieces of information that go into the estimate of a parameter. It is also considered as an estimate of a parameter which is equal to the number of independent scores that go into the estimate minus the number of parameters used as intermediate steps in the estimation of the parameter itself.
Mathematically, it can be defined either as the number of dimension of the domain of a random vector or essentially the number of 'free' components. It reveals that how many components that are required to be known before the vector is fully determined.
This is most often used in the context of linear models and linear regression as well as analysis of variance where certain random vectors are constrained to lie in linear subspaces.
This also gives the value of the dimension of the subspace and so this is commonly associated with the squared lengths or "sum of squares" of the coordinates of such vectors. It can be taken as the parameters of Chi-squared distribution as well as other distributions that arise in associated statistical testing problems.
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It is also considered as the distribution parameters or through hypothesis testing, it is considered as the underlying geometry that defines degrees of freedom and is critical to a proper understanding of the concept.
In equations, the typical symbol that is used to represent this is ‘Nu’ and in the text as well as in tables the abbreviation "d.f." is also commonly used.Ronald A. Fisher used the letter n to symbolize this, but according to modern usage, it typically uses the notation n to represent the sample size.
