An exam is composed of x questions that are taken in a pool of questions (so a question can be in several exams).
Each question gives an amount of points and you can success (gain all the points) or fail (0 points) the question.
I want to compare a student score on a specific exam to the whole community based on question success rate (the whole community hasn't passed this specific exam but can have passed some questions on another exams).
- Question 1 (1pt) : success rate 75% of students who have passed this question (no matter the examen) have succeed to this question
- Q2 (10pts): 25% success rate
- Q3 (5pts): 50% success rate
The student passes the exam A and has the following result:
- Q1: success (gain 1pt)
- Q2: fail (0pts)
- Q3: success (gain 5pts)
Is there a way (or an approximate one) to say: "this student is better than XX% of the community on this exam" ?
What I've tried
I tried the following approach but I'm not a mathematician and I don't know if it's legal or not:
- compute a score for the exam by balancing the question point depending on the success rate:
- Q1: 75% (success rate) of 1pt = 0.75pts
- Q2: 25% of 10pts = 2.5pts
- Q3: 50% of 5pts = 2.5pts
- = (0.75 + 2.5 + 2.5) / (1 + 10 + 5) = 36%
- compute the score of the student (by cumulating the points of good answers): (1 + 5) / (1 + 5 + 10) = 37,5%
- try to find the student score on a distribution which is centered on the exam score (36%) and which height is calculated using the standard deviation of the question success rates of the exam