Relationship between MSE and RSS

Formulas of MSE and RSS

Residual Sum of Squares (RSS) is the numerator in the formula of Mean Squared Error (MSE). That is, RSS is part of MSE formula.

\[ SSR=\sum_{i=1}^{n} (\hat{y_i}-y_i)^2 \]

\[ MSE=\frac{SSR}{n-p-1}=\frac{\sum_{i=1}^{n} (\hat{y_i}-y_i)^2 }{n-p-1}\]


\( n \) is the number of observations.

\( \hat{y_i} \) is is estimated value.

\( y_i \) is observed value.

\( p \) is the is the number of estimated parameters (excluding the intercept).

Relationship between MSE and RSS

Thus, RSS is the sum of the squares of residuals. Building on RSS, MSE takes the number of observation (i.e., \( n\) ) into consideration. That is, MSE is “mean” of squared errors (estimated by squared residuals).

Why do we use \( n-p-1 \) rather than \( n\) as the denominator in the formula of MSE? It is because \( n\) leads to a biased estimate of MSE, whereas \( n-p-1 \) leads to an unbiased estimate of MSE. (See the discussion here on Wikipedia.)

Further Reading