# International Encyclopedia of Statistical Science

2011 Edition
| Editors: Miodrag Lovric

# Degrees of Freedom

• Chong Ho Yu
Reference work entry
DOI: https://doi.org/10.1007/978-3-642-04898-2_204

Many elementary statistics textbooks introduce the concept of degrees of freedom (df) in terms of the number scores that are “free to vary.” However, this explanation cannot clearly show the purpose of df. There are many other approaches to present the concept of degrees of freedom. Two of the most meaningful ways are to illustrate df in terms of sample size and dimensionality. Both represent the number of pieces of useful information.

## DF in Terms of Sample Size

Toothaker (1986) explained df as the number of independent components minus the number of parameters estimated. This approach is based upon the definition provided by Walker (1940): the number of observations minus the number of necessary relations, which is obtainable from the observations (df = nr). Although Good (1973) criticized that Walker’s approach is not obvious in the meaning of necessary relations, the number of necessary relationships is indeed intuitive when there are just a few variables. “Necessary...

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