## 5 ICT Fluency and the Digital Divide
### Measuring Digital Inequality  
The objective is to study the digital inequalities related to internal and external factors. We define the observation _matrix_\(M\) for internal variables where a row is formed by the values \(m_{p1}=gender_{p}\), \(m_{p2}=age_{p}\), \(m_{p3}=location_{p}\), \(m_{p4}=familySize_{p}\), \(m_{p5}=parentElducation_{p}\), \(m_{p5}=income_{p}\), and \(m_{p7}=employmentStatus_{p}\). Similarly, we define another _matrix_\(W\) for external variables where a row is formed by the values \(w_{p1}=computerProperty_{p}\), \(w_{p2}=EnglishLanguage_{p}\), \(w_{p3}=major_{p}\), and \(w_{p4}=studentsClassYear_{p}\). The splitting of variables into two groups makes explicit personal features and other features in the model. Unless the personal and external variables are completely independent, the effect of the internals on digital inequality indexes can be correlated to the effect of the external and vice-versa. We carry out the independence test between the two groups of variables and we find a no significant correlation between them as presented in Tables 3-6. Following Equation (2), using the above described variables, a linear regression model can be rewritten as Equation (9):  
\[y_{p}=\alpha+\sum_{q=1}^{a}\phi_{q}M_{pq}+\sum_{q=1}^{b}\theta_{q}W_{pq}+ \varepsilon_{p} \tag{9}\]  
where p = 1,..., n; \(\phi=(\phi_{1},\ldots,\phi_{a})^{\prime}\), \(\theta=(\theta_{1},\ldots,\theta_{b})^{\prime}\), and \(\alpha\) is the regression coefficient; \(y=(y_{1},\ldots,y_{n})^{\prime}\), and \(\varepsilon=(\varepsilon_{1},\ldots,\varepsilon_{n})^{\prime}\) is the error term. In the computation, \(\phi\), \(\theta\), and \(\alpha\) estimated using a = 7, b = 4, and n = 1237. Let us measure the Concentration Index (CI) in order to analyze the online resources access inequality patterns among students in every year. We assumed that online resources access inequality could be described by computer property and internet access. Rewriting Equation (4), we obtained Equation (10):  
\[CI=\sum_{q=1}^{a}\left(\frac{\varpi_{q}\overline{M}_{q}}{\mu}\right)\mathbb{ C}_{q}+\sum_{q=1}^{b}\left(\frac{\theta_{q}\overline{W}_{q}}{\mu}\right) \mathbb{B}_{q}+\frac{G_{\varepsilon}}{\mu} \tag{10}\]  
where \(\overline{M}_{q}\) and \(\overline{W}_{q}\) are the arithmetic means of the entries of matrix \(M\) and \(W\); that is \(\overline{M}_{q}=\left(m_{1q}+m_{2q}+\ldots+m_{aq}\right)/a\) and \(\overline{W}_{q}=\left(w_{1q}+w_{2q}+\ldots+w_{bq}\right)/b\), respectively; \(Cq\) and \(Bq\) are the concentration indexes related to the first a students and the last b students; and \(G_{\varepsilon}/\mu\) is the generalized concentration index for \(\varepsilon_{p}\). It compensates the case where the \(CI\) cannot be explained by the observation \(M\) and \(W\) such as [-1,0]. According to Equation (6), Horizontal Inequality (HI) becomes Equations (11) and (12):  
\[HI =CI-C_{N} \tag{11}\] \[=\sum_{q=1}^{a}\left(\frac{\varpi_{q}\overline{M}_{q}}{\mu} \right)\mathbb{C}_{q}+\sum_{q=1}^{b}\left(\frac{\varpi_{q}\overline{W}_{q}}{ \mu}\right)\mathbb{B}_{q}+\frac{G_{\varepsilon}}{\mu}-\frac{2}{n\mu}\sum_{p=1 }^{n}y_{p}R_{p}+1 \tag{12}\]