I’m having an argument with another numerically literate person. We both saw a news site state that under no circumstances should you compare the median of one set with the mean of another.
It’s important to note what we mean by compare: no one is talking about constructing an hypothesis and test statistic to compare with a distribution. This is just a sort of layman, back-of-the-envelope rest of two distributions measured in dollars. One has a long trail sprinkled with upper outliers, the other does not and is presumably bell shaped. Also, I think we both agree that there are better ways to compare the two; the question is really about whether the news reporter should be chided for doing this mean-to-median comparison.
Best Answer
"Under no circumstances" is too strong.
If (!) a distribution $F$ is symmetric, then the mean and the median coincide. This is the case, e.g., for the normal distribution.
Thus, you could compare a symmetric $F$'s mean to some other distribution $G$'s median (since you would in fact be comparing $F$'s median to $G$'s). Or you could compare $F$'s median to $G$'s mean (same thing).
Note that only one of the two distributions needs to be symmetric. And of course, symmetry is sufficient but not necessary for the mean and the median coincide; there are asymmetric distributions with equal means and medians.
However, I find it rather hard to imagine a situation where we would want to compare the central tendencies of a symmetric and an asymmetric distribution. In general, it's better to follow the advice not to compare apples and oranges (or means and medians).