# Degrees of Freedom

**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* = *n* − *r*). 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...

## References and Further Reading

- Eisenhauer JG (2008) Degrees of freedom. Teach Stat 30(3):75–78Google Scholar
- Good IJ (1973) What are degrees of freedom? Am Stat 27:227–228Google Scholar
- Saville D, Wood GR (1991) Statistical methods: the geometric approach. Springer, New YorkMATHGoogle Scholar
- Toothaker LE, Miller L (1996) Introductory statistics for the behavioral sciences, 2nd edn. Brooks/Cole, Pacific GroveGoogle Scholar
- Walker HW (1940) Degrees of freedom. J Educ Psychol 31:253–269Google Scholar
- Wickens T (1995) The geometry of multivariate statistics. Lawrence Erlbaum, HillsdaleGoogle Scholar