## 5 Calculating the Average Rank, the Consistency of Expert Estimates and the Criteria Weights  
All 15 sub-criteria presented in the questionnaire, which determined the choice of passengers to travel by train rather than by bus, were divided into three groups and named key criteria (Figure 2b,c). The group of safety and environmental protection included two sub-criteria (A and C), the economy group embraced five sub-criteria (B, D, H, L, M) and the ride comfort group consisted of eight sub-criteria (E, F, G, I, J, K, N, O). The ranks of the significance of sub-criteria, determining the choice by passengers of a trip by train rather than traveling by bus, were used for calculating the average values of the ranks \(\overline{R}_{j}\), the concordance coefficient \(W\), Pearson's chi-square statistic \(\chi^{2}\) and sub-criteria weights \(\omega_{j}\) and \(\omega^{\prime}_{j}\) (Table 2). The following table also presents the mean values \(\overline{\omega}_{j}\) of the sub-criteria weights \(\omega_{j}\) and \(\omega^{\prime}_{j}\) and the standard deviations of the ranks \(R_{ij}\). The total of sub-criteria ranks was \(\sum\limits_{j=1}^{15}R_{j}=6240\), while the sum of average ranks \(\overline{R}_{j}\) of all \(j\) sub-criteria was \(\sum\limits_{j=1}^{15}\overline{R}_{j}=120.0\). The average value \(\overline{R}=416\) of sub-criteria ranks was calculated by Equation (6) or \(\overline{R}=6240/15=416\). The sum of squared deviations \(S=147\),\(172\) (Equation (7)). The concordance coefficient \(W=0.194\), showing the consistency of the estimates of respondents (52 passengers), was calculated by Equation (6). Based on the data from the passengers' survey and using Equation (11), \(\chi^{2}=141.5\) was obtained. The critical value \(\chi^{2}_{\alpha,\upsilon}\) taken from the table of chi-squared distribution with \(\upsilon=15-1=14\) degrees of freedom and the significance level \(\alpha=0.01\) was equal to \(29.1413\). The empirical value \(\chi^{2}=141.5\) was \(4.8\) times the critical value \(\chi^{2}_{\alpha,\alpha}=29.1\), which allowed the researchers to assume that the respondents' estimates were consistent. The smallest value of the concordance coefficient \(W_{\min}\), with the significance level \(\alpha=0.01\) and the degree of freedom \(\upsilon=n-1=15-1=14\), allowing the authors to assume that the respondents' estimates were consistent, was calculated by Equation (9). The smallest value of the concordance coefficient \(W_{\min}=0.0400\) corresponded to only about one-fifth of the calculated concordance coefficient \(W=0.194\). The estimates of 52 passengers that took part in the survey were in agreement (or consistent) because the calculated concordance coefficient was equal to \(0.194\), while the value of Pearson's chi-squared statistic, equal to \(141.5\), was considerably larger than the critical value of \(29.14\), corresponding to degrees of freedom of \(14\) and a significance level of \(0.01\). The smallest concordance coefficient still allowing the estimates of all respondents to be considered consistent was equal to \(0.0400\), which was equivalent to only one-fifth of \(0.194\). It was hardly possible to expect very high consistency of the respondents' estimates because of their highly different experiences, wishes, habits and means. A bar diagram of the calculated average ranks \(\overline{R}_{j}\) of the 15 sub-criteria determining the passengers' choice of traveling by train rather than by bus is given in Figure 3.  
By applying the new ARTIW-L and ARTIW-N methods, the passengers' reasons for selecting a trip by train rather than a trip by bus, which were described by criteria (sub-criteria) and their weights \(\omega_{j}\), \(\omega^{\prime}_{j}\), and \(\overline{\omega}_{j}\), were determined.