Mann-Whitney U Test: Beeswarm & Permutation Null (2D)

This 2D companion builds the same Mann-Whitney U test from the ground up as its 3D counterpart, but encodes the samples geometrically instead of by fixed rank slots: two samples are drawn from a chosen distribution shape and placed on a shared real-valued x-axis, then a collision-avoiding beeswarm dodge algorithm spreads overlapping points vertically so local density is visible at a glance — Group A swarming above the axis line, Group B below it. The pooled values are still ranked jointly and the U statistic computed directly from those ranks. Beside the swarm, 2,000 label-shuffling permutations build the true null distribution of U live in a 2D histogram, and the bins at least as extreme as the observed split light up red — that fraction is the permutation p-value, earned the same way a nonparametric test earns significance without ever assuming normality.