Degrees of Freedom (Statistics)/Examples

Examples of Degrees of Freedom in context of Statistics

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:

\(\ds b\)

\(=\)

\(\ds 7\)

\(\ds c\)

\(=\)

\(\ds 10\)

\(\ds d\)

\(=\)

\(\ds 3\)