Variance and CI for the difference in medians One way to obtain a (1-alpha)% confidence interval for the difference in medians for two samples is to use bootstrapping. Generate 10,000 bootstrap samples of both group's samples and find the difference in medians of each, forming a bootstrap distribution for the difference in medians. However, I am looking for a formula that would not require bootstrapping.
For example, the confidence interval for the difference in means (under appropriate assumptions) is (xbar1-xbar2) +/- t_0.025 * se , is there a similar formula for the difference in medians? Maybe a non-parametric test, but there has to be something, no?
 A: There is a median test, based on a chi-square test. It breaks each group up into cases above and below the individual medians, and determines whether the frequencies of those groups differ from what would be predicted if all the data came from the same distribution. "[I]t only considers the position of each observation relative to the overall median," so it doesn't have much power to distinguish true differences in medians.
You will sometimes find the Wilcoxon-Mann-Whitney (WMW) test cited as a test for medians, but that is not a test for medians if the distributions being compared differ in symmetry or scale parameters. So although it's not a parametric test per se, you have to make some assumptions about the underlying distributions to use it as a test on medians.
And neither of those tests can provide confidence intervals (CI) for the difference in medians. With the WMW test, for example, what you can get with a 2-sample test, according to the manual page for wilcox.test, is:

a nonparametric confidence interval and an estimator... for the difference of the location parameters x - y ... the estimator for the difference in location parameters does not estimate the difference in medians (a common misconception) but rather the median of the difference between a sample from x and a sample from y.

That's why bootstrapping is so useful in situations like this: it doesn't require any assumptions about the underlying distributions, although in some cases it does require some care in finding the most reliable estimate of the CI. I understand some reluctance to approach bootstrapping if you haven't had any experience with it, but it's well worth your while to become familiar with it as it's useful in so many applications.
