Degree of Freedom

Degrees of Freedom is a property of a Statistic or Point Estimator that describes how many variables are free to vary while maintaining the feature’s validity.

• For statistic/PE $T$ that is a Linear Transformation, dof = [Nullity|nullity]

Example

For example, a sample mean is a linear function of its inputs, because it is a sum of the RVs, all divided by a scalar. This is a linear transformation, as well.

$$\text{mean}(X_{\mathrm{1..}n})=\frac{\sum X_{\mathrm{1..}n}}{n}=\overline{X}$$

Due to this linearity, all of the inputs except for the last can be any value. If you allow the first $n-1$ RV to vary, then the $n^{\text{th}}$ RV is forced to a specific value.

$$X_n=n\overline{X}-\sum X_{\mathrm{1..}n-1}$$

Therefore, for sample mean, $\text{dof}=n-1$