Simulations of Type Ia Supernovae ( SN Ia ) surveys are a critical tool for correcting biases in the analysis of SN Ia to infer cosmological parameters . Large scale Monte Carlo simulations include a thorough treatment of observation history , measurement noise , intrinsic scatter models and selection effects . In this paper , we improve simulations with a robust technique to evaluate the underlying populations of SN Ia color and stretch that correlate with luminosity . In typical analyses , the standardized SNIa brightness is determined from linear ‘ Tripp ’ relations between the light curve color and luminosity and between stretch and luminosity . However , this solution produces Hubble residual biases because intrinsic scatter and measurement noise result in measured color and stretch values that do not follow the Tripp relation . We find a 10 \sigma bias ( up to 0.3 mag ) in Hubble residuals versus color and 5 \sigma bias ( up to 0.2 mag ) in Hubble residuals versus stretch in a joint sample of 920 spectroscopically confirmed SN Ia from PS1 , SNLS , SDSS and several low-z surveys . After we determine the underlying color and stretch distributions , we use simulations to predict and correct the biases in the data . We show that removing these biases has a small impact on the low-z sample , but reduces the intrinsic scatter \sigma _ { \textrm { int } } from 0.101 to 0.083 in the combined PS1 , SNLS and SDSS sample . Past estimates of the underlying populations were too broad , leading to a small bias in the equation-of-state of dark energy w of \Delta w = 0.005 .