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In developing coarse-grained (CG) polymer models it is important to reproduce both local and molecule-scale structure. We develop a procedure for fast calculation of the bond-orientation correlation and the internal squared distance 〈R²(M)〉 through evaluation of the probability distribution functions that represent a CG model. Different CG models inherently contain or omit correlations between CG variables. Here, we construct CG models that contain specific correlations between CG variables. The importance of different correlations is tested on CG models of polyethylene, polytetrafluoroethylene, and poly-L-lactic acid. The chain stiffness and 〈R²(M)〉 are calculated using both analytic evaluation and Monte Carlo sampling, and approximate model results are compared with exact results from all-atom simulations. For polymers with an exponential correlation decay, the bond-orientation correlation and 〈R²(M)〉 indicate which CG variable correlations are most important to reproduce molecule-scale structure. Analysis of the bond-orientation correlation and internal-squared distance indicates that for poly-L-lactic acid the bond-orientation correlation requires qualitatively different additional terms in CG models and quantifies the error in neglecting this important behavior. [ABSTRACT FROM AUTHOR]