I noticed an error in a graded exam during office hours. Should I give the student the lower grade? You are a professor. A student comes to you with a question on an exam they have taken and received back. The student wrote 5 as the answer to a question, and they want to defend why 5 is a correct answer and they should receive full marks. Their argument is invalid, but you notice that the answer was misread as 9 and the student was given partial credit (5 is entirely incorrect and they should have received zero credit). Do you regrade the question, giving the student zero points, or do you leave it alone?

Arguments for regrading the question:


It would be unfair to the other students to give this student marks they did not earn.
Pursuing accurate grading is what is important and is the objective of the course.


Arguments for leaving the question alone:


The student came forward with a good-faith intention to learn why their argument wasn't correct, and shouldn't be penalized - this would break an implicit contract, and encourage students to not seek further feedback.
The unfairness actually stems from the misgrade itself, and there isn't anything wrong with leaving the misgrade alone now that it already exists. After all, if the student hadn't come forward, the misgrade would still have been left alone.
Let the grade stand, but point out your error.

The principle I try to follow is that students shouldn't be negatively affected by faculty mistakes (or that such effects should be minimized).  

I wouldn't suggest adding an additional assignment to maintain the grade, since this is effectively another kind of penalty.  But I would suggest being transparent with the student, explaining your mistake, and showing them where they went wrong (otherwise they are deprived of the feedback that the correct grade would have provided).

If the exam were graded on a curve and re-grading this one question on Student A's test would materially improve other students' grades, I would give other students the boost (i.e. the grade they would have received had Student A's exam been correctly marked the first time) but leave Student A's grade intact.