I've got a big database with many (10000s) numerical variables and a categorical variable. The categorical variable stratifies samples into 2 groups, A and B. I want to find the numerical variable that best aligns with the categorical group. The intended use would be to, based on the value of the chosen numerical value, assign the correct group to a sample. The ideal result would be to find the numerical variable with the least overlap in distributions, or even better, 0 overlap. See images to better illustrate my point.
So far I have done a t-test, anova and KS test, which have found statistically significant different distributions. However the top ones have a lot of overlap still. Is there a better test for this purpose? And what would be the methods used to predict the categorical group given a numerical value from the chosen numerical variable?