Averaging exam scores with very different means/distributions I'm teaching a course that has two exams which together form a large part of the students' grades.

On the first exam, the mean score was very high (~90%) with the majority of students scoring over 80%.

On the second exam, the mean score was much lower (~70%), with many students scoring below 50%. The distribution was also much "flatter". (This is not too surprising, since the second exam covered more material and more difficult material.) 

On both exams, there were some students who scored ~100%.  There were even some students (that I feel should be rewarded) who scored much better on the harder second exam than  on the first easier exam.

When computing final grades, is it fair to simply average (with equal weight) the scores on these two exams?

Other possible options, none of which feel completely satisfactory to me:


Scale the second exam up in some way so that it has a higher average, then combine the scores. This seems unsatisfactory to me, since it will actually hurt students (relative to their classmates) who did better on the second exam than the first.
Weight each students' higher exam score more heavily. This seems problematic for the same reason as before.
Simply weight more heavily the more difficult second exam. This is nice in that it helps students who did well on a more difficult exam and rewards improvement over time. But is it fair to do without telling the students in advance?


What do you usually do in these situations? Am I overthinking it?
Follow what you defined previously. The 'average policy' needs to be done beforehand, so nobody can argue that you defined/changed it to specifically screw/help someone. If you didn't define it previously, the safer route is the arithmetic average, which is kinda standard (or something equivalent for your institution, if different).

More than that, IMHO, you are overthinking. Grades aren't a reward that you give, they are a measure of how much the student learnt the content. A professor of mine says that he doesn't give grades, the students earn it. If they performed badly, they performed badly. You have a duty to fail under performing students.