How to regrade an exam that was too difficult? Our multiple choice exam was clearly too difficult.


N questions.
N + 10 total duration (in minutes).
All questions have the same value (100 / N).
Wrong answers have a penalty of 25% (25 / N).
Only one correct answer per question.


What strategies do you recommend to consider reviewing the grades?

What we though of:


Remove the penalty for wrong answers.
For each student, increase the value of each correct answer.
Add X (e.g. 10 points) to each final result.
Add X% (e.g. 10%) to each final result.
Remove the questions with worst performance.


We can combine options and each one has advantages and drawbacks. Note that we use absolute grading (not grading on the curve).

What is your experience?

Follow-up: Thank you all for the feedback. In the end, we decided to review each question. In some cases (four in total), we decided to consider as correct some of the incorrect options that were not 100% clear (and had a significant number of students choosing them). Plus, we added a bonus of +1 points to all students. We kept the penalty for wrong answers since it would be unfair to remove it.
There isn't enough information here to really give good advice. You suggest the exam was too difficult. It may also be that it was invalid.  It may be that some of the questions were stated in such a way as to be misleading. There are measures of the validity of such exams, by the way. They measure the validity of a question by the proportion of students who answered it incorrectly compared to how those same students did overall. Some questions are negatively correlated with overall performance. 

Another issue is what your institution permits. Some have very strict rules about this. They are IMO unwarranted and counterproductive, but they may bind you. 

But absent such rules, you should care more about fairness than you do about numbers. To achieve fairness you may need to drop the exam or give an alternate. You may even want to rethink your overall grading scheme. 

One simple modification overall is to give course grades based on the, say, 8 best of 10 assignments/exams/whatever. Then the question of a poorly designed quiz never arises. Another way is to have quizzes every day so that no individual quiz is determinative of much of anything. 

However, some of the things that you might try will leave some students unsatisfied; especially the best students who worked the hardest. You can be kind to the strugglers, but not at the expense of the superstars. The situation is worst if the system itself puts the students in competition with one another for grades. Strict curve grading is IMO immoral as it makes it into a zero sum game where I can only advance at someone else's expense. If the system doesn't permit top marks for everyone (assuming it is deserved) then it is fatally flawed. 

Your purpose, I hope, is teaching, not grading. Use the exam as a teaching moment. Even have a class discussion about the questions that caused difficulty. Try to learn why people did poorly. Even permit different students to have different sorts of adjustments as needed. 

I used to have fairly strict rules about such things, but the understanding was always "This is the standard and you will do no worse than X if you do Y". And I always tried to make it an advantage to learn something even if was after the deadline or the exam. Make people want to learn, not just want to maximize points.