I am comparing the median values of "A" between two conditions? The samples are taken from different groups, for the variable A. How can I test for significance for this median values?
Mood's median test could be used for this, but if this is not good enough you could try quantile regression, where you can specify exactly which quantile you want. Below is an example in R for some made up data.
library(quantreg) A=c(1,2,3,4,5,6,6,6,7,8,9,10) B=c(1,1,1,1,6,6,6,6,6,10,10,10) df=data.frame(value=c(A,B), group=c(rep("A",length(A)),rep("B",length(A)))) summary(rq(value~group,data=df),se="boot") Call: rq(formula = value ~ group, data = df) tau:  0.5 Coefficients: Value Std. Error t value Pr(>|t|) (Intercept) 6.00000 0.81290 7.38100 0.00000 groupB 0.00000 1.75195 0.00000 1.00000 Warning message: In rq.fit.br(x, y, tau = tau, ...) : Solution may be nonunique