I presume you want to say residual variance instead of residual deviance.
For regression analysis, you can have as many predictor variables as you like, at least you can have $X,X^2,X^3,...,X^n$.
For each coefficient, there is a sacrification of 1 degree of freedom.
Suppose you have $n$ observations, and $p-1$ predictor variables, you've got $n-p$ degree of freedom for your regression model. And you have $n-1$ degree of freedom for the null variance, since you have to sacrifice 1 degree of freedom for the mean.
Coefficient of determination is $R^2=1-\frac{SSE}{SSTO}$
The adjusted coefficient of determination $R^2_{adj}=1-\frac{SSE/n-p}{SSTO/n-1}$
The coefficient of determination $R^2$ measures how well your regression fits the observation.
If there is a perfect matching, you have $SSE=0$.
For the worst case of null variance, you have $SSE=SSTO=(n-1)\sigma^2=S^2$.
The purpose of your regression model is to reduce the variance of estimation.
$SSR=SSTO-SSE$ measures the reduction of the variance of estimation from the null variance $SSTO$ to the current variance $SSE$.
Thus $R^2=\frac{SSR}{SSTO}=1-\frac{SSE}{SSTO}$ measures the proportion of variance reduction, and it measures how well your regression estimation fits the observation.
You can also use adjusted coefficient of determination $R_{adj}^2$ to include the effect of sacrification of degree of freedom.