Definition:Degrees of Freedom (Statistics)

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Definition

The number of degrees of freedom is essentially the number of independent units of information in a sample relevant to the estimation of a parameter or calculation of a statistic.

One approach is to regard the $n$ observations as the initial data, one of which is used to determine the total or mean.

As the mean must be known before we can determine deviations from it, there are $n - 1$ degrees of freedom left to estimate the variance.

Hence, in the sense that the total is fixed, only $n - 1$ values can be assigned arbitrarily, as the remaining one is then fixed to ensure the correct total.


Examples

Arbitrary Example

Consider a $2 \times 2$ contingency table with fixed marginal totals.

In this context there is only $1$ degree of freedom.

This is because once a total has been assigned to any one of the $4$ category cells, the remaining values are determined by the constraint that they must add up to the fixed marginal totals.


Take as an example the contingency table below:

$\begin{array}{r|cc|c}

& \text {Column 1} & \text {Column 2} & \text {Row totals} \\ \hline \text {Row 1} & a & b & 12 \\ \text {Row 2} & c & d & 13 \\ \hline \text {Column totals} & 15 & 10 \end{array}$


If we arbitrarily assign $a = 10$, it follows that:

\(\ds b\) \(=\) \(\ds 2\)
\(\ds c\) \(=\) \(\ds 5\)
\(\ds d\) \(=\) \(\ds 8\)


Similarly, if we set $a = 5$, then:

\(\ds b\) \(=\) \(\ds 7\)
\(\ds c\) \(=\) \(\ds 10\)
\(\ds d\) \(=\) \(\ds 3\)


Also see

  • Results about degrees of freedom in the context of statistics can be found here.


Sources

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