In this paper we present a theoretical analysis of the deterministic on-line Sum of Squares algorithm (SS) for bin packing, introduced and studied experimentally in , along with several new variants. SS is applicable to any instance of bin packing in which the bin capacity B and item sizes s(a) are integral (or can be scaled to be so), and runs in time O(nB). It performs remarkably well from an average case point of view: For any discrete distribution in which the optimal expected waste is sublinear, SS also has sublinear expected waste. For any discrete distribution where the optimal expected waste is bounded, SS has expected waste at most O(log n). In addition, we present a randomized O(nB log B)-time on-line algorithm SS*, based on SS, whose expected behavior is essentially optimal for all discrete distributions. Algorithm SS* also depends on a new linear-programming- based pseudopolynomial-time algorithm for solving the NP-hard problem of determining, given a discrete distribution F, just what is the growth rate for the optimal expected waste. An off-line randomized variant SS** performs well in a worst-case sense: For any list L of integer-sized items to be packed into bins of a fixed size B, the expected number of bins used by SS** is at most OPT(L) + √OPT(L).