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Multiple regression uses independent variables to explain the behaviour of the dependent variable. Part of the variability in the dependent variable can be explained by its linear association with the independent variables. We will develop a measure of the proportion of the variability in the dependent variable that can be explained by the multiple regression.
Let the multiple regression model fitted by least squares be
where 
are the least squares estimates of the population regression model and 
’s are the residuals from the estimated regression model.
The model variability can be partitioned into the components
where
Total sum of squares: 
Error sum of squares: 
Regression sum of squares: 
The coefficient of determination for a multiple regression model, usually called the coefficient of determination, is denoted by 
and is defined as the proportion of the total sample sum of squares 
that is explained by the multiple regression model.
It tells us how good the multiple regression model is and how well the independent variables included in the model explain the dependent variable.
The value of the coefficient of determination always lies in the range 0 to 1, that is
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| Standard assumptions for the multiple regression models | | | Adjusted coefficient of determination |