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I hope this question is appropriate for this site, if not sorry in advanced.

I'm trying to come up with a formula to generate a grade, which will take into consideration the diffculty level of the questions.

The person taking this quiz can determine the diffculty of the questions so I want the result to give more credit to the harder questions, because if someone answered 8/8 questions correctly in level 8, it's much harder than answering 8/8 correctly in level 1.

Each one is asked X questions in 1-8 levels, but not necessarily all of them.

For example lets say X was 60 and these are the results:

5/9 - level1

4/9 - level2

3/8 - level3

3/7 - level4

4/8 - level5

0/0 - level6

2/10 - level7

1/9 - level8

Does anyone have an idea for such a formula?

Udi Idan
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2 Answers2

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Try giving different number of points for different level correct answers: a correct answer for level 1 is worth 1 point, for level 2 is worth 2 points, and so on. Then one would need more correct answers on a low level to get a high score than if they attempted high level questions.

user141592
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  • I thought of that, but as X in not a constatnt (the user can determine it), I think the precentage of correct answers is of importance, isn't it? – Udi Idan Dec 28 '14 at 22:10
  • @UdiI Maybe if you combine this by calculating the average score for each level, and weight it to calculate a weighted average over all the levels. – user141592 Dec 28 '14 at 22:13
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There are many functions that will work for what you want. I'll describe one group of such functions.

You could use increasing "weights" $w_1, w_2, \dots, w_8$ with $0 < w_8 < 1$ and, for all $n$ between $1$ and $7$ $$0 < w_n < 1 \qquad \text{and}\qquad w_n<w_{n+1}$$

$$\text{and} \qquad \sum\limits_{k=1}^8 w_k=1$$

If the fraction (number answered correctly over the number of questions for that level - not the number that they answered otherwise you could get $0/0$) that a person gets right on a level $n$ is $f_n$, then their total score will be $$\sum\limits_{k=1}^{8} w_kf_k$$

This score will have a maximum of $1$ and a minimum of $0$.

Depending on how you assign the weights, this score could change quite a bit, but because the weights satisfy the restrictions I've stated in the first part above, the higher levels are more important.

Zubin Mukerjee
  • 17,832
  • I think I will have a problem with "number answered correctly over the number of questions for that level - not the number that they answered otherwise you could get 0/0" because it's not a one time quiz, so over time a user can answer more correct answers then total questions for a level. – Udi Idan Dec 28 '14 at 22:27
  • @UdiI Okay then define $f_n$ for each level $n$ as "number answered correctly over number answered" unless "number answered" is zero, in which case you can let $f_n=0$ (instead of the problematic $0/0$) – Zubin Mukerjee Dec 28 '14 at 22:42
  • This makes sense because if they have done no questions for a level then they should get no points for it. – Zubin Mukerjee Dec 28 '14 at 22:42