# Residual Sum of Squares (RSS)

## What Is the Residual Sum of Squares (RSS)?

A residual sum of squares (RSS) is a statistical technique used to measure the amount of variance in a data set that is not explained by a regression model itself. Instead, it estimates the variance in the residuals, or error term.

Linear regression is a measurement that helps determine the strength of the relationship between a dependent variable and one or more other factors, known as independent or explanatory variables.

### Key Takeaways

• A residual sum of squares (RSS) measures the level of variance in the error term, or residuals, of a regression model.
• Ideally, the sum of squared residuals should be a smaller or lower value than the sum of squares from the regression model's inputs.
• The RSS is used by financial analysts in estimating the validity of their econometric models.

## The Formula for RSS Is

where

• yi = the ith value of the variable to be predicted
• f(xi) = predicted value of yi
• n = upper limit of summation

## Understanding the Residual Sum of Squares (RSS)

In general terms, the sum of squares is a statistical technique used in regression analysis to determine the dispersion of data points. In a regression analysis, the goal is to determine how well a data series can be fitted to a function that might help to explain how the data series was generated. Sum of squares is used as a mathematical way to find the function that best fits (varies least) from the data.

The residual sum of squares (RSS) measures the amount of error remaining between the regression function and the data set after the model has been run. A smaller residual sum of squares figure represents a regression function. Residual sum of squares–also known as the sum of squared residuals–essentially determines how well a regression model explains or represents the data in the model.

Residual standard error *RSE) is another statistical term used to describe the difference in standard deviations of observed values versus predicted values as shown by points in a regression analysis. It is a goodness-of-fit measure that can be used to analyze how well a set of data points fit with the actual model.

RSE is computed by dividing the RSS by the number of observations in the sample less 2 and then taking the square root: RSE = [RSS/(n-2)]1/2