I'm comparing two sets of measurements using a paired difference test (specifically the Wilcoxon signed-rank test). I would like to find out whether there is a significant difference between these sets of measurements, and also the direction of this inequality.
Since I do not have a theoretical reason to suspect this direction before I look at the data, I'm using a two-tailed test. This means my null hypothesis is that both sets of measurements belong to the same distribution; the alternative hypothesis is that the two measurements belong to different distributions.
The test has a sufficiently small p-value for me to reject the null hypothesis. From this, I conclude that there is a difference between these measurements. To find out the direction of the inequality, I looked at the mean of both samples. My supervisor commented that this was strange, since the Wilcoxon test evaluates rank as opposed to mean. He suggested that I use a one-tailed test instead.
My question is: is it appropriate to use a one-tailed test to find out the direction of the inequality? Or should I simply use a different aggregation technique (instead of the mean) to compare my distributions? The reason I'm hesitant is because a one-tailed test is normally used when I know the direction of the inequality a priori, which is not the case here.