Probability distribution functions (PDFs) of the projected offsets between the BCG and the weak-lensing-determined halo center for different redshift and mass bins. All error bars are estimated from 1000 bootstrap resamplings. Red and blue points represent the low- and high-redshift subsamples, respectively, with solid lines showing the best-fitting two-component 2D Gaussian models. Corresponding vertical lines indicate the values of σ₁ and σ₂. Pink and light blue points show the low- and high-mass subsamples, respectively, with dashed lines and vertical markers indicating the best fits and offset scales.