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In statistics, the **residual sum of squares (RSS)** is the sum of squares of residuals. It is also known as the **sum of squared residuals (SSR)** or the **sum of squared errors of prediction (SSE)**. It is a measure of the discrepancy between the data and an estimation model. A small RSS indicates a tight fit of the model to the data.

In general, total sum of squares = explained sum of squares + **residual sum of squares**. For a proof of this in the multivariate ordinary least squares (OLS) case, see partitioning in the general OLS model.

In a model with a single explanatory variable, RSS is given by

where *y*_{i} is the *i* ^{th} value of the variable to be predicted, *x*_{i} is the *i* ^{th} value of the explanatory variable, and is the predicted value of *y*_{i} (also termed ). In a standard linear simple regression model, , where *a* and *b* are coefficients, *y* and *x* are the regressand and the regressor, respectively, and ε is the error term. The sum of squares of residuals is the sum of squares of estimates of ε_{i}; that is

where is the estimated value of the constant term and is the estimated value of the slope coefficient *b*.

The general regression model with *n* observations and *k* explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is

where *y* is an *n* × 1 vector of dependent variable observations, each column of the *n* × *k* matrix *X* is a vector of observations on one of the *k* explanators, is a *k* × 1 vector of true coefficients, and *e* is an *n*× 1 vector of the true underlying errors. The ordinary least squares estimator for is

The residual vector is , so the residual sum of squares is, after simplification,

- , where H is the hat matrix, or the prediction matrix in linear regression.

- Sum of squares (statistics)
- Squared deviations
- Errors and residuals in statistics
- Lack-of-fit sum of squares
- Degrees of freedom (statistics)#Sum of squares and degrees of freedom
- Chi-squared distribution#Applications

- Draper, N.R.; Smith, H. (1998).
*Applied Regression Analysis*(3rd ed.). John Wiley. ISBN 0-471-17082-8.