Is it possible to find two data sets A and B such that A has a greater median and mode, while B has a greater average and range? If you think the answer is yes, provide an example. If you think the answer is no, explain why not.

Since median and mode are both forms of average, I'm assuming that where you used the word average you were referring to the mean. In which case:

No, it's not possible, but I can't write a whole proof right now.

are you sure?