Let \(X\) be a complex space and \(Z\) a closed complex subspace defined by a sheaf \(I\). For \(x\in Z\) the authors consider the analytic spread \(s(I_ x)\), the height \(ht(I_ x)\) and the analytic deviation \(s(I_ x)- ht(I_ x)\). They show that \(Z\) is stratified by the analytic subsets \[ {\mathcal G}_ n(Z,X):= \{x\in Z: s(I_ x)- ht(I_ x)\geq n\}. \] The stratification is used to define embedded intersection components for an intersection of complex subspaces. The authors make a construction which enables them to define an intersection multiplicity for such components. In the case of projective spaces one gets relations to investigations of Fulton and Stückrad-Vogel. The technics of the authors are these of commutative algebra and semianalytic Stein compacta.