Calculation of relative performance gain I want to measure the improvement of students using exam marks of semester 1 to semester 3. The following detail should be considered when calculating the measure;


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*If Student A get 30 (sem 1) and 50 (sem2), he has obtained 20 marks improvement. If student B obtained 70 (sem1) and 90 (sem2) he has also obtained a 20 marks improvement.However, Student B should get a better performance measure than student A, as reaching 90 from 70 is difficult than reaching 50 from 30.


I want to know, if there is any standard measurement I can use to calculate this?
 A: If going from $10$ to $30$ is as good as going from $70$ to $90$, and if marks are out of $100$, then you might consider using something related to the the log odds ratio, so in this case the improvement from $m_1/100$ to $m_2/100$ could be measured as
$$\log\left(\dfrac{\tfrac{m_2}{100}}{1-\tfrac{m_2}{100}}\right) -\log\left(\dfrac{\tfrac{m_1}{100}}{1-\tfrac{m_1}{100}}\right)$$
Using natural logarithms, this would give:


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*the change from $30/100$ to $50/100$ measured as about $+0.847$

*the change from $70/100$ to $90/100$ measured as about $+1.350$

*the change from $10/100$ to $30/100$ also measured as about $+1.350$


This have the advantage of being 


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*symmetric measures of change, so $50/100$ to $30/100$ measured as about $-0.847$)  

*additive, so the measure of change from $10/100$ to $50/100$ is about $+2.197$ which is equal to the sum of the two measures of changes $1.350+0.847$ going from $10/100$ to $30/100$ and then to $50/100$


If you think the measures are too small, you can multiply them all by a constant
One disadvantage is that moving up from $0/100$ or moving up to $100/100$ is measured as infinite improvement  
