BH formation efficiency as a function of the ratio of initial cluster mass to critical mass. The figure has been reproduced from Vergara et al. (2024); we fit an asymmetric sigmoid function, represented by the dashed black line. The shaded gray bands represent the confidence intervals estimated by bootstrap resampling: the darker, middle, and lighter gray bands correspond to the 1σ, 2σ, and 3σ uncertainty regions. These intervals quantify the uncertainty in the fit curve, arising from data variability. We added the simulation from Vergara et al. (2025) and the simulations from this work.