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As far as I'm concerned, the degree of freedom is simply the number of linear equations need to be satisfied. However, it seems closely related to the statistical deduction. For example

  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.

Is there an intuitive explanation of the ubiquitousness of degree of freedom?

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marked as duplicate by Xi'an, mdewey, Michael Chernick, Stefan, user158565 Apr 19 at 4:08

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