Total Sum of Squares

What is the Total Sum of Squares?

The coefficient of determination measures how well a regression line explains the relationship between a dependent variable (Y) and an independent variable (X). The coefficient of determination is computed from the sums of squares.

Example of Total Sum of Squares (TSS):

We can write the deviation from mean for Y

$ Y_{i}\, -\, \overline{Y}\, = (\widehat{Y_{i}}\, -\, \overline{Y})\, +\, (Y_{i}\, -\, \widehat{Y_{i}}) $

Therefore,

$ \sum \, (Y_{i}\, -\, \overline{Y})^{2}\, = \, \sum (\widehat{Y_{i}}\, -\, \overline{Y})^{2}\, +\, \sum (Y_{i}\, -\, \widehat{Y_{i}})^{2} $

Or TSS = ESS + RSS

where:
TSS = total sum of squares (total variation in Y)
ESS = explained sum of squares (variation in Y explained by the regression model)
RSS = residual sum of squares (unexplained variation in Y)