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

A bunch of definitions: Image File history File links Broom_icon. ... The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ... Look up degree in Wiktionary, the free dictionary. ...

In statistics, the term degrees of freedom (df) is a measure of the number of independent pieces of information on which the precision of a parameter estimate is based. The degrees of freedom for an estimate equals the number of observations (values) minus the number of additional parameters estimated for that calculation. As we have to estimate more parameters, the degrees of freedom available decreases. It can also be thought of as the number of observations (values) which are freely available to vary given the additional parameters estimated. It can be thought of two ways: in terms of sample size and in terms of dimensions and parameters. A graph of a normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... The factual accuracy of this article is disputed. ... Estimation is the calculated approximation of a result which is usable even if input data may be incomplete, uncertain, or noisy. ...

The maximum numbers of quantities or directions, whose values are free to vary before the remainders of the quantities are determined, or an estimate of the number of independent categories in a particular statistical test or experiment. Degrees of freedom (df) for a sample is defined as: df = n - 1 Where n is the number of scores in the sample.

Essentially, degrees of freedom are a count of the number of pieces of independent information contained within a particular analysis.

The intuitive way to understand the degrees of freedom

There's a really good visual demonstration of degrees of freedom in "Statistics: An Introduction using R" by Michael J. Crawley (Wiley, ISBN 13:978-0-470-02298-6) p36-37. To paraphrase: Suppose we had a sample of 6 numbers with an average was 5. The sum of these numbers must be 30 otherwise the mean would not be 5. |_| |_| |_| |_| |_| |_| Fill each box in turn with a positive or negative real number. The first could be any number, for example 3. |3| |_| |_| |_| |_| |_| The next could be anything, say 9. |3| |9| |_| |_| |_| |_| The next could also be anything, say 4, 0 and 6. |3| |9| |4| |0| |6| |_| However, the last value can't be any number, it has to be 8 because the numbers must add to 30. There is total choice in selecting the first five numbers but none in selecting the sixth. There are five degrees of freedom when selecting six numbers. In general there are (N-1) degrees of freedom when estimating the mean from a sample of size N.

For contingency table, df=(row-1)(column-1). For example consider a 2x2 table. Suppose we have 100 respondents with 40 men and 60 women. We ask them their attitude towards a social event, e.g. "do you like Bush?" Of all the 100 respondents, 50 say 'Yes' and 50 say 'No' (table 1).

Table 1.

Now for the 2x2 table, if any one of the cell is determined, the remain cells are determined as well. For example, if we know cell(1,1)=10, then for males, cell(2,1)=40-10=30; for those answer 'Yes', cell(1,2)=50-10=40; and for females, cell(2,2)=60-40=20 (see table 2). Also, cell(2,2) can be calculated by 50-30=20 for those who answer 'No'.

Table 2.

Any way, for a 2x2 table, df=(2-1)*(2-1)=1. For any contingency table, df is always (row-1)(column-1).

See also

• True variance
• Normal curve
• Sample (statistics)
• Fisher's method for combining independent tests of significance