Top panel: black points show for all clusters the estimated total number of background-subtracted cluster members at m140 <  M^* + 1.3 and within r <  0.7r₅₀₀ divided by the total projected halo mass within the same region (N_gal/M_halo). The solid gray line shows the median value for this cluster sample. The dashed lines show an estimate of the intrinsic scatter of N_gal/M_halo. See text for details. Bottom panel: estimate of the effect of differences in the background density along the line of sight of the clusters on the estimated red fraction of cluster galaxies. Black points show the red fraction for all clusters as estimated adopting the default control field for the statistical residual background subtraction (Sect. 3.2). Dark gray bars show the effect on the estimated red fraction of adopting control sub-fields with densities in the 10th to 90th percentile range across the default control field. Light gray bars show the effect of considering the background-subtracted cluster member samples contaminated or incomplete according to the face-value N_gal/M_halo of each cluster with respect to the median N_gal/M_halo of the sample. See text for details. For comparison, light gray empty circles show the red fraction in field galaxies in the same m140 magnitude range, and with a photo-z within ±0.15 of the cluster redshift.