Mean BCG offset as a function of merger score under three definitions: past-merger (left), future-merger (middle), and full-merger (right) scores. Each point represents the average offset within a merger score bin, with vertical bars indicating the standard error of the mean. The total sample size is 325 clusters. To ensure roughly equal sample counts in each bin, we apply quantile-based binning: full- and past-merger scores are divided at [0.0, 0.2, 0.4, 0.6, 0.8, 1.0]; future-merger scores are divided at [0.0, 0.3, 0.48, 0.66, 0.84, 1.0] due to the heavy concentration near zero. A clear positive trend is visible in the first panel, suggesting that BCG displacement increases with merger activity based on past events.