We present the methodology for correlation analysis of chemical bond operators (CACB) on ab initio wave functions. In CACB the wave function is analyzed in a hierarchy of quantities (charge, bond order, bond-bond correlation), where each quantity is the expectation value of an operator related to the statistical covariance of the previous quantity. CACB does not require any preconceived notion of which atoms are bonded and should be useful for reasoning about the similarity, stability, and reactivity of molecular systems. CACB does not require any special form of the wave function, but the applications here are for Hartree-Fock (HF) type wave functions. We use CACB to analyze the bonding in a number of molecules including transition states for several reactions. This analysis extracts chemically useful information without using preconcerned notations of bonding.