# Why is degree of freedom so important? [duplicate]

1. Dividing by the degree of freedom of a statistic gives an unbiased estimator. $$E(\overline{\sigma^2}) = E\left(\frac{\sum_{i}^{n}(X_i-\overline{X})^2}{\color{red}{n-1}}\right) = \sigma^2.$$ Similar conclusions can be found in regression. The denominator of unbiased estimators of SSE, SSR, and SST is their degree of freedom respectively.
2. In Pearson's chi-squared test, the degree of freedom of the $$\chi^2$$ is essentially the number of unfitted parameters.