We explore the distance fluctuation criterion ("Lindemann criterion") for melting transitions. Distances from average positions in accord with Lindemann, or interparticle distances, in accord with Jellinek and Berry or Etters and Kaelberer, are examined. The primary goal is to determine which of these offers the more useful criterion. The choice of origin can sometimes effect the significance of the index. We study three systems with two kinds of potentials. They are all composed of 64 particles: (a) and (b), a homopolymer and a cluster that consist of beads interacting pairwise through square-well potentials, and (c) a cluster of particles interacting pairwise through Morse potentials. For each of the noncrystalline structures, in contrast to the crystals originally studied by Lindemann, the fluctuation parameter based on interparticle distances gives a clearer separability of liquid and solid phases than that based on fluctuations from average positions. The solid-like forms of the Morse cluster, the square-well cluster, and the square-well homopolymer have similar behavior, indicating that a broad class of systems can be evaluated with this index. In these systems, relative fluctuation parameters provide a suitable criterion for the melting transition. The critical values for the interparticle distance criterion, which are in the range of 0.03-0.05, are smaller than those for the Lindemann criterion (0.1-0.15).