# same average with different medians

The following is an exercise in statistics:

Let $A$ and $B$ be two finite sets of datas with possible values $0,1,2,3,4$. The median of $A$ is 2 while the median of $B$ is $3$. Also, $A$ and $B$ have the same average. Can we say that the numbers in $B$ is generally bigger than the numbers in $A$?

I know nothing but the definitions of median and average in this exercise. Also, I'm confused with the meaning of "generally bigger". Could anyone help?

[Added:] This question is motivated by the following problem from a practice exam of one of my friends:

• I don't think there is any technical definition of 'generally bigger.' Maybe if you can construct samples A and B with the required properties, you will have something to say. (BTW: 'data' is already the plural of 'datum', and 'datas' is not (yet) an English word.) – BruceET Sep 27 '15 at 7:12
• @Bruce Trumbo: Thanks for your comment. The question has been edited. An example I can come up with is $A=\{1,2,4\}$ and $B=\{1,3,3\}$. I was wondering if the problem itself is rather ill-defined. – Jack Sep 27 '15 at 22:34
• The problem is rather ill-defined, but given your own example, I would say the answer would be $\textrm{no}$. As only one number in $B$ is bigger than the 'corresponding' number in $A$. – Hetebrij Sep 27 '15 at 22:43
• This problem has been answered with (A) as the correct answer (I could provide the link if you want). I don't see how can this be true. We can say that the marks of all students in Section II are higher but we can't say that Manuel's score was higher since this would contradict the first few words in the problem beginning. Answer (B) makes sense but it does not use the concept of mean! What a "mean" question :) – NoChance Sep 27 '15 at 23:01

I cannot think of any reasonable definition of "generally higher" (much less an obvious one that could be presupposed without providing it) that would make any of the first three answers correct. You can extend your example by an arbitrary number of scores of $1$ and $4$, e.g. $(1,1,1,2,4,4,4)$ and $(1,1,1,3,3,4,4)$, and it makes no sense by any stretch of the expression "generally higher" to apply it to that case (and the expression "no discernable difference" is even worse – what would prevent a difference from being discerned?).