wilcoxon signed-rank test to compare 2 pseudo-randomized distributions?

I need help to find the good statistic to use to compare two distributions of data coming from pseudo randomized trials.
I have recorded some activity of different entities while a subject was performing a task with 4 conditions: A (start on the right, stay on the right), B (start on the right, go on the left), C (start on the left, stay in the left) and D (start on the left go on the right). I have recorded N independent entities, with variables number of trials for each conditions. I want to test if each of my N entities are discriminating the start position and/or the final position. To do so, and for each of N, I picked a random trial of each conditions (A, B, C and D), and I combined them so that I have the activity a 4 pseudo-trials corresponding to: - SR: star on right, mean(A+B), -SL: start on left, mean(C+D), -FR: finish on right, mean(A+D) and FL:finish on the left, mean(B+C).
For those 4 pseudo-trials, I have computed 2 indexes: an origin index--abs(SR-SL)/(SR+SL)--and a destination index--abs(FR-FL)/(FR+FL)--so that I have a measure of how well this entity discriminates the original position paired to a measure of how well it discriminates the final position.
To create a pseudo population of trials, I perform this randomization 10 000 times, so that I have 10 000 origin indexes paired to 10 000 destination indexes.
I want to compare those 10 000 position indexes to those 10 000 destination indexes. None of them have normal distributions. I've been using a wilcoxon signed-rank test but I wonder if there is not a size effect related to the 10 000 pseudo randomization procedure.
Anyone has an idea if the wilcoxon signed-rank test is valid in this condition or what test I should use?
Thank you, and sorry about the complexity of my description.


TS Contributor
just a quick thought, without having understood the details: maybe you could split one data set to groups and count the number of points in each group, then count the number of points in each of the groups from the second distribution and do a chi-squared test?