## 3 Problem Definition and Measurements
### Inequality Measurements CI and C\({}_{n}\)  
The Concentration Index (CI) is a statistical measure commonly used as a measurement of inequality. A CI of zero expresses perfect equality, where all values are the same (e.g., where every student has the same ICT fluency, that is, no digital inequality). A CI of 1 (or 100%) expresses maximal inequality among values (e.g., for a large number of students, where only one student is ICT fluent, the CI will be very nearly one). In order to formulate the CI, we assume that the change in the independent variables can explain interpersonal variations between the dependent variables. Under this assumption, the CI can be acquired using a linear regression model linking the variables \(y\) and \(x\) by developing Equation (1) as follows [63]:  
\[\left[\begin{array}{c}y_{1}\\ y_{2}\\ \vdots\\ y_{n}\end{array}\right]=\mathbf{\alpha}+Z\left[\begin{array}{c}\varnothing_{1} \\ \varnothing_{2}\\ \vdots\\ \varnothing_{l}\end{array}\right]+\left[\begin{array}{c}\varepsilon_{1}\\ \varepsilon_{2}\\ \vdots\\ \varepsilon_{n}\end{array}\right] \tag{1}\]  
where the observation matrix \(Z=\left[\begin{array}{cccc}x_{11}&x_{12}&\cdots&x_{1l}\\ x_{21}&x_{22}&\cdots&x_{2l}\\ \vdots&\vdots&\ddots&\vdots\\ x_{n1}&x_{n2}&\cdots&x_{nl}\end{array}\right]\).  
Equivalently, for the \(p^{th}\) student, the above equation can be rewritten as Equation (2):  
\[y_{p}=\alpha_{p}+\sum_{q=1}^{l}\phi_{q}x_{pq}+\varepsilon_{p} \tag{2}\]  
where p = 1,..., n; q = 1,..., \(\left[\phi_{1},\ldots,\phi_{l}\right]^{\prime}\), and \(\mathbf{\alpha}=\left(\alpha_{1},\ldots,\alpha_{n}\right)^{\prime}\) are the regression coefficients; \(\mathbf{y}=\left(y_{1},\ldots,y_{n}\right)^{\prime}\), and \(\mathbf{\varepsilon}=\left(\varepsilon_{1},\ldots,\varepsilon_{n}\right)^{\prime}\) is the error term. Given the matrix Z, \(\phi\) and \(\alpha\) can be estimated using existing algorithms such as the least square method [64].  
We will now derive the \(C_{q}\) and the concentration index for the \(q^{th}\) variable, where the related observations are \(x_{q}=\left(x_{1q},\ldots,x_{nq}\right)\) such that \(x_{1q}\leq\ldots\leq x_{nq}\). Let us consider that the components of \(\mathbf{y}\) are also ranked from lowest to highest. The \(Cq\) index is a similarity measure between \(\mathbf{y}\) and \(x_{q}\) defined as the normalized and shifted inner product as Equation (3) [65; 66]:  
\[C_{q}=\frac{2}{n\mu}\sum_{p=1}^{n}y_{p}x_{pq}-1 \tag{3}\]  
where \(\mu\) is the mean of \(y\) components and \(q\) = 1,..., \(l\). Given the regression parameters \(\phi_{p}\), the arithmetic mean \(\widetilde{\mathrm{x}}_{\mathrm{q}}\) of \(\mathrm{x}_{\mathrm{q}}\), the CI is the weighted sum of \(C_{q}\) up to some additive term as Equation (4) [63; 67].  
\[CI=\sum_{q=1}^{l}\biggl{(}\frac{\phi_{q}\widetilde{\mathrm{x}}_{q}}{\mu}\biggr{)} C_{q}+\frac{G_{\varepsilon}}{\mu} \tag{4}\]The second term on the right is the generalized concentration index for \(\varepsilon_{p}\). It compensates the case where the CI cannot be explained by the observation \(x\). It is calculated as:  
\[G_{\varepsilon}=\frac{2}{n}\sum_{p=1}^{n}\varepsilon_{p}R_{p},\]  
The CI takes its values in the interval [-1, +1], -1 means inequality favoring the worse off and +1 means inequality favoring the better off. Let us consider \(R_{p}=p/n\) the normalized rank of \(p^{th}\) sample. There are many ways to calculate the Concentration Index for need \(C_{N}\), one of them is by Equation (5) [66]:  
\[C_{N}=\frac{2}{n\mu}\sum_{p=1}^{n}y_{p}R_{p}-1 \tag{5}\]  
The CI for \(C_{N}\) takes its values in [-1, +1], -1 means inequality favoring the worse off and +1 means inequality favoring the better off.