of squares for an application of this broad principle to inferential statistics. Partitioning the sum of squares in linear regression Theorem. Given a linear regression model including a constant , based on a sample containing n observations, the total sum of squares can be partitioned as follows into the explained sum of squares (ESS) and the residual sum of squares (RSS): where this equation is equivalent to each of the following forms: where is the value estimated by the regression line having , , ..., as the estimated coefficients. Proof The requirement that the model include a constant or equivalently that the design matrix contain a column of ones ensures that , i.e. . The proof can also be expressed in vector form, as follows: The elimination of terms in the last line, used the fact that Further partitioning Note that the residual sum of squares can be further partitioned as the lack-of-fit sum of squares plus the sum of squares due to pure error. See also Inner-product space Hilbert space Euclidean space Expected mean squares Orthogonality Orthonormal basis Orthogonal complement, the closed subspace orthogonal to a set (especially a subspace) Orthomodular lattice of the subspaces of an inner-product space Orthogonal projection Pythagorean theorem that the sum of the squared norms of orthogonal summands equals the squared norm of the sum. Least squares Mean squared error Squared deviations References Pre-publication chapters are available on-line. Republished as: Analysis of variance Least squares