Calibrated standard platinum resistance thermometers (SPRTs) are used to realize the International Temperature Scale of 1990 (ITS-90) from 13.8033 K to 1234.93 K. The SPRTs are calibrated at a series of fixed points, each assigned a temperature on the ITS90, by measuring the ratios of the SPRT resistances at those temperatures to that at the triple point of water (TPW). For realizing the scale with a calibrated SPRT, a user measures the resistance ratio at the unknown temperature and uses ITS-90-defined equations to interpolate between fixed points. The uncertainty of the SPRT temperature is therefore largely influenced by the propagation of fixed-point resistance-ratio uncertainties. In this paper, we rigorously derive the equations for calculating these uncertainties for a variety of circumstances and we use software tools written by us to perform these calculations using realistic uncertainties for fixed points and other input parameters. For properly calculating the standard uncertainty for SPRT realization of the ITS-90, correlations between the input quantities must be considered, in particular those involving measurement of the TPW resistance. The proper calculation depends on three factors involving SPRT use and calibration. The different combinations of these factors result in six different equations for calculating the realization uncertainty. We derive these six equations, specify the conditions of their use and discuss the relevant uncertainty components for each of them. We also compare the results of these equations with those of two approximations that may be used for calculating the standard uncertainty and explain the conditions under which the simpler approximations agree with the more detailed calculations. Because these calculations are complicated, we are making our software tools available upon request to the user community.