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The within-group and between-group scatter decomposition separates noise from group separation: $W$ measures variation about each group sample mean, and $B$ measures separation of those means about the overall sample mean. In linear discriminant analysis, seek a projection $a^{T}x$ maximizing between-group relative to within-group variation,Normalize by $a^{T}Wa=1$; a Lagrange multiplier calculation gives the generalized eigenvalue problem $Ba=\lambda Wa$. The leading canonical discriminant directions have the largest eigenvalues, and subsequent directions are orthogonal in the $W$ inner product. There are at most $\min (p,g-1)$ nonzero separating directions. Projecting observations onto these directions gives a low-dimensional display of group separation.

For classification, estimate the common covariance matrix by $S_p=W/(n-g)$ and use the linear discriminant analysis scoreswhere ${\pi }_{\nu }$ are class prior probabilities. Assign to the largest score. The same within-group scatter sets the noise scale, while between-group scatter selects the directions in which the group means separate most strongly.