Energy storage systems (ESSs) operating in conjunction with renewable energy are commonly seen as a solution for decarbonizing the power sector. Ultimately, the use of an ESS allows power generation plants to run more consistently, resulting in less peaker plant cycling and a reduction in carbon emissions. As such, optimally sized ESSs for applications in residential, commercial, and industrial buildings is a high priority. In this work, the optimal sizing of an ESS in an Arizona single-family household is investigated using an energy storage (ES) dispatch optimization model. This optimization model is developed for residential buildings and considers both cost savings for the household and carbon emissions from power generation plants. Due to extreme Arizona summer climates, this residential building has significantly different summer and winter electrical power demand. Therefore, the optimally sized ESS is determined using one month of electrical demand in the summer, one month of electrical demand in the winter, and the entire year of electrical power demand. The results show the system is sized with a higher ES capacity to shift on-peak usage to off-peak usage hours in the summertime and smaller ES capacity in the wintertime (due to the smaller demand and difference in on-peak and off-peak usage rates). Ultimately, it is found that sizing based on a month of summer data better matches the optimal over the entire year than sizing based on a month of winter data.