Degrees of Freedom at the Start of the Second 100 Years: A Pedagogical Treatise

Degrees of freedom is a critical core concept within the field of statistics. Virtually every introductory statistics class treats the topic, though textbooks and the statistical literature show mostly superficial treatment, weak pedagogy, and substantial confusion. Fisher first defined degrees of freedom in 1915, and Walker provided technical treatment of the concept in 1940. In this article, the history of degrees of freedom is reviewed, and the pedagogical challenges are discussed. The core of the article is a simple reconceptualization of the degrees-of-freedom concept that is easier to teach and to learn than the traditional treatment. This reconceptualization defines a statistical bank, into which are deposited data points. These data points are used to estimate statistical models; some data are used up in estimating a model, and some data remain in the bank. The several types of degrees of freedom define an accounting process that simply counts the flow of data from the statistical bank into the model. The overall reconceptualization is based on basic economic principles, including treating data as statistical capital and data exchangeability (fungibility). The goal is to stimulate discussion of degrees of freedom that will improve its use and understanding in pedagogical and applied settings.