Madam empirical χ² distribution functions from 60 residual noise maps compared with the theoretical cumulative probability density. For a smoothed map, we count the degrees of freedom as the number of included eigenmodes in the inverse NCM. Top: three sets of low-resolutions maps using just one of the low-resolution methods at a time. The high-resolution maps for INW and smoothing methods had N_side = 1024. Bottom: since it is suboptimal to compute a low-resolution spherical harmonic expansion from a noisy high-resolution map, we test how well the smoothing approach works in conjunction with the two pixel-based downgrading methods.