## 3 Results
### Variation Ranges for the \(Madm_{m}\), \(Ci_{m}\), and \(\beta_{m}\) Indicators  
From the perspective above, we proposed the \(CI_{M}\) index, defined by Equation (100), as a device for comparing the consistency of the formal structure supporting the presently addressed grading methods. Concurrent exploration of its variability begins by noting that, according to Equation (95), the involved term \(CP_{M}\) is constrained as set by Inequality (99). Concerning its second determining factor \(MAD_{M}\), we require achieving a few algebraic steps to obtain its variation range. Moreover, let us denote using the symbol \(GM_{q}\) the \(q\)-th simulated student grade for the method \(M\). Then, since we are dealing with normalized scores, we have  
\[0<GM_{q}\leq 1. \tag{102}\]  
Let \(\overline{GO_{M}}\) stand for the average objective grade that \(M\) assigns. Then, since usually the joining score lies in the interval \((0,1.0]\), we may set  
\[\overline{GO_{M}}=1/2. \tag{103}\]  
Now, let us consider the function \(U(GM_{q})\) defined through  
\[U\big{(}GM_{q}\big{)}=GM_{q}-1/2. \tag{104}\]  
Then, for \(0<GM_{q}\leq 1/2\), we have that \(U\big{(}GM_{q}\big{)}\) lies in the interval \((-1/2,0]\), that is, \(U\big{(}GM_{q}\big{)}\) remains negative and attains a minimum value of \(-1/2\). Correspondingly, in the interval \(1/2<GM_{q}\leq 1\), \(U\big{(}GM_{q}\big{)}\) takes values within \((0,1/2]\), that is, \(U\big{(}GM_{q}\big{)}\) remains positive and attains a maximum value of \(\
ot{\!\!\lambda}\). Therefore, altogether, for \(0<GM_{q}\leq 1\), we have  
\[0\leq\big{|}GM_{q}-\overline{GO_{M}}\big{|}\leq 1/2. \tag{105}\]  
Now, for all the simulation procedures addressed here, we set \(q=1,2,\ldots,100\), and \(\overline{GO_{M}}=1/2\), and we also define  
\[MAD_{M}=\frac{\sum_{q=1}^{100}\big{|}GM_{q}-\overline{GO_{M}}\big{|}}{100}. \tag{106}\]  
Thus, we have  
\[0<MAD_{M}\leq 1/2. \tag{107}\]  
Considering this last inequality and that the term \(CP_{M}\) takes values as indicated by Inequality (99), we have  
\[2/\left(\rho+2\right)\leq(1+MAD_{M}\cdot CP_{M})^{-1}<1. \tag{108}\]This result, along with Inequality (99), and Equation (100) set the ratio \(CI_{M}\) staying in the range  
\[((\rho+2)\theta+2)/(\rho+2)\leq CI_{M}<\rho+1. \tag{109}\]  
Then, considering Inequalities (99) and (107), we obtain  
\[2/3(\rho+1)\leq(CP_{M}+1)^{-1}(MAD_{M}+1)^{-1}\leq 1/(\theta+1). \tag{110}\]  
In the other hand, using Inequalities (99) and (109), we obtain  
\[(\rho+2)\theta^{2}+2\theta\Big{)}/(\rho+2)\leq CI_{M}CP_{M}<\rho(\rho+1). \tag{111}\]  
Then, Inequalities (110) and (111) along with Equation (101) determine for \(\beta_{M}\) the variation range:  
\[\overline{GO_{M}}=1/2. \tag{103}\]0  
\(CI_{M}\) and \(\beta_{M}\)-Based Method Performance Comparison Criteria  
To judge the mathematical consistency of two grading methods, we suggest the following general \(CI_{M}\)-based comparison criterion: for two grading methods \(A\) and \(B\), with their respective Consistency Indices \(CI_{A}\) and \(CI_{B}\), we say that method \(A\) is more consistent than method \(B\) if \(CI_{A}>CI_{B}\). We can interpret this by stating that the method with the higher Consistency Index (\(CI_{M}\)) achieves a better balance between low grading dispersion (\(MAD_{M}\)) and reasonable grading complexity (\(CP_{M}\)). If \(CI_{A}\) and \(CI_{B}\) values are close, the methods \(A\) and \(B\) have similar consistency levels, and additional considerations (e.g., fairness and efficiency) may be needed to determine the preferable approach. If \(CI_{M}\) values are very different, the method with the higher \(CI_{M}\) is preferable in terms of grading consistency. We can further elaborate on the presently offered general comparison criteria by adding a threshold-based ranking approach. This provides a structured way to further classify methods. In brief, we can propose benchmark thresholds: (1) One for low consistency, enforced whenever \(0<CI_{M}\leq 1\) (\(M\) produces unstable grading or excessive complexity).