From Wikipedia, the free encyclopedia

This article or section is in need of attention from an expert on the subject. Please help recruit one or improve this article yourself. See the talk page for details. Please consider using {{Expert-subject}} to associate this request with a WikiProject

For other senses of these terms, see degrees of freedom or degree.

Degrees of freedom is the number of categories or classes being tested minus 1.

In statistics, the term degrees of freedom has two distinct senses.

Contents 1 Residuals 2 Parameters in probability distributions 3 See also 4 External links

In fitting statistical models to data, the vectors of residuals are often constrained to lie in a space of smaller dimension than the number of components in the vector. That smaller dimension is the number of degrees of freedom for error.

Perhaps the simplest example is this. Suppose

are random variables each with expected value μ, and let

be the "sample mean". Then the quantities

are residuals that may be considered estimates of the errors X_i − μ. The sum of the residuals (unlike the sum of the errors) is necessarily 0. That means they are constrained to lie in a space of dimension n − 1. If one knows the values of any n − 1 of the residuals, one can thus find the last one. One says that "there are n − 1 degrees of freedom for error."

An only slightly less simple example is that of least squares estimation of a and b in the model

where ε_i, and hence Y_i are random. Let and be the least-squares estimates of a and b. Then the residuals

are constrained to lie within the space defined by the two equations

One says that there are n − 2 degrees of freedom for error.

The capital Y is used in specifying the model, and lower-case y in the definition of the residuals. That is because the former are hypothesized random variables and the latter are data.

Another simple and frequently seen example arises in multiple comparisons.

Interpretation degree of freedom (df) in regression when n is sample size

1. df = n-2 for linear regression, because two points are needed to draw a straight line, this means 2 out of n points are lying on the straight line, the rest n-2 are lying around the straight line, this means standard error or aggregate fluctuations around the line are because of n-2 points, so df = n -2 for linear regression

2. df = n-3 for cubic regression, because three points are needed to draw a cubic curve, this means 3 out of n points are lying on the curve, the rest n-3 are lying around the curve, this means standard error or aggregate fluctuations around the curve are because of n-2 points, so df = n -3 for cubic regression

The probability distributions of residuals are often parametrized by these numbers of degrees of freedom. Thus one speaks of a chi-square distribution with a specified number of degrees of freedom, an F-distribution, a Student's t-distribution, or a Wishart distribution with specified numbers of degrees of freedom in the numerator and the denominator respectively.

In the familiar uses of these distributions, the number of degrees of freedom takes only integer values. The underlying mathematics in most cases allows for fractional degrees of freedom, which can arise in more sophisticated uses.

• True variance

• http://web.archive.org/web/20060706151302/http://seamonkey.ed.asu.edu/~alex/computer/sas/df.html