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In the Simulations and Constructions of the Reionization of Cosmic Hydrogen (SCORCH) project, we compare analytical models of the hydrogen ionization fraction with radiation-hydrodynamic simulations. We derive analytical models of the mass-weighted hydrogen ionization fraction from the local ionization balance equations as a more accurate alternative to the widely adopted model based on the volume filling factor. In particular, our model has a recombination term quadratic in the ionization fraction, which is consistent with the two-body interaction nature of recombination. Then, we use the radiation-hydrodynamic simulations to study the clumping factors needed to solve the analytical equations, and provide accurate fitting functions. We find that the ionized hydrogen clumping factors from our radiative transfer simulations are significantly different than those from other simulations that use a uniform photoionization background. In addition to redshift dependence, we also see the dependence of ionized hydrogen clumping factor on ionization fraction, and we incorporate this into our fits. We calculate the reionization histories using our analytical models and clumping factors and compare with widely adopted models, and all of our models achieve $<7\%$ difference from simulation results while the other models have $>20\%$ deviations. The Thomson optical depths from reionization calculated from our analytical models result in $<5\%$ deviation from simulations, while the previous analytical models have $>20\%$ difference in and could result in biased conclusions of the IGM reionization.