Approximate computing is an emerging design paradigm that leverages the inherent error tolerance present in many applications to improve their power consumption and performance. Due to the forgiving nature of these error-resilient applications, precise input data is not always necessary for them to produce outputs of acceptable quality. This makes the memory subsystem (i.e., the place where data is stored), a suitable component for introducing approximations in return for substantial energy savings. Towards this end, this paper proposes a systematic methodology for constructing a quality configurable approximate DRAM system. Our design is based upon an extensive experimental characterization of memory errors as a function of the DRAM refresh-rate. Leveraging the insights gathered from this characterization, we propose four novel strategies for partitioning the DRAM in a system into a number of quality bins based on the frequency, location, and nature of bit errors in each of the physical pages, while also taking into account the property of variable retention time exhibited by DRAM cells. During data allocation, critical data is placed in the highest quality bin (that contains only accurate pages) and approximate data is allocated to bins sorted in descending order of quality, with the refresh rate serving as the quality control knob. We validate our proposed scheme on several error-resilient applications implemented using an Altera Stratix IV GX FPGA based Terasic TR4-230 development board containing a 1GB DDR3 DRAM module. Experimental results demonstrate a significant improvement in the energy-quality trade-off compared to previous work and show a reduction in DRAM refresh power of up to 73 percent on average with minimal loss in output quality.