As far as I know if the assumption for Mann Whitney for similar distribution of shape is not met the test results represent the test of the mean rank instead of median. What does this mean for the result?? I am analyzing the difference between scores of two independent sample groups (6 in Experiment Group and 5 in Control group) for Intrinsic Motivation Inventory score. I am testing motivation after using a fitness application. Experiment Group using my application and control group using other application. All subjects are university students ( 8 males , 3 Females).
If two samples have roughly the same shape, then the Mann-Whitney-Wilcoxon test (a rank sum test) can be considered a test whether the locations (often expressed as medians) differ. Consider fictitious data sampled in R.
set.seed(104) y1 = rgamma(100, 3, 1/3) y2 = rgamma(100, 3, 1/3) + 4 median(y1); median(y2)  7.684493  11.85169 stripchart(list(y1,y2), ylim=c(.5,2.5), pch="|")
Because the P-value of the Wilcoxon rank sum test is near $0.$ we can say that sample medians $7.68$ and $11.95$ are significantly different at the 1% level.
wilcox.test(y1, y2) Wilcoxon rank sum test with continuity correction data: y1 and y2 W = 2634, p-value = 7.477e-09 alternative hypothesis: true location shift is not equal to 0
However, if two samples have distinctly different shapes, rejection of the null hypothesis of the Wilcoxon rank sum test should be interpreted to mean that the population distribution of one sample 'stochastically dominates' the population distribution of the other.
set.seed(2022) x1 = rgamma(100, 3, 1/3) x2 = rnorm(100, 12, 3) summary(x1) Min. 1st Qu. Median Mean 3rd Qu. Max. 0.868 5.643 7.962 8.411 11.274 19.239 summary(x2) Min. 1st Qu. Median Mean 3rd Qu. Max. 5.654 9.625 11.456 11.756 13.877 19.377 stripchart(list(x1,x2), ylim=c(.5,2.5), pch="|")
wilcox.test(x1,x2) Wilcoxon rank sum test with continuity correction data: x1 and x2 W = 2531, p-value = 1.624e-09 alternative hypothesis: true location shift is not equal to 0
Stochastic domination means that values in the second population tend to be larger than values in the first. Perhaps this is best illustrated by showing the empirical CDFs (ECDFs) of the two samples. The dominating ECDF (blue in the figure below) plots to the right of the other ECDF, and thus below.
hdr = "ECDFs of x1 and x2 (blue)" plot(ecdf(x1), xlim=c(0, 20), main=hdr) lines(ecdf(x2), col="blue")