If we run linear regression in R, then we will see a term in the report “degrees of freedom” and that value equals n-p where n is the number of observations (data points) and p is the number of parameters (including intercept).

But what does it mean? How is it useful?

`Short Answer: the degree of freedom represents the number of "free" or independent $N(0,1)^2$ in the residuals from linear regression. The following three parts will explain this further.`

Latexed version can be found on https://jiayangnie.com/post/lm1/ , which I explained how it relates to the measurement of variance, the hypothesis testing for beta and an extension to 2-sample t-test.