## 5 Calculating the Average Rank, the Consistency of Expert Estimates and the Criteria Weights
The calculation data for sub-criteria beginning from the most important (E) to the least important (D) ones are shown in Figure 4.  
The calculated average ranks \(\overline{R}_{j}\) (Figure 3) and global weights \(\omega_{j}\) and \(\omega^{\prime}_{j}\) (Figure 4) of sub-criteria, determining the choice by the respondents to travel by train rather than by bus, show that sub-criteria E, M and H were much more important than sub-criteria N, C and D. This implies that their priority order should be as follows:Moreover, there should be an inverse straight-line relationship between the average ranks \(\overline{R}_{j}\) and the global weights \(\omega_{j}\) calculated by the ARTIW-L method. The determination coefficient of the regression equation of 15 sub-criteria, R\({}^{2}\) = 1, and coefficient of correlation \(r\) = \(-1\), show that this is a functional linear relationship \(\omega_{j}=-0.0083\overline{R}_{j}+0.1333\). The correlation between \(\overline{R}_{j}\) and the weights \(\omega^{\prime}{}_{j}\) calculated using the ARTIW-N method is non-linear (Figure 5a). Ranks and weights are related by the quadratic regression equation \(\omega^{\prime}{}_{j}\) = 0.0009 \(\overline{R}_{j}^{2}-0.0234\)\(\overline{R}_{j}+0.1932\), coefficient of determination, which is R\({}^{2}\) = 0.9955. Figure 4: Relative weights of sub-criteria in descending order. Figure 3: The average ranks of sub-criteria \(\overline{R}_{j}\). The data obtained in the performed study show that the estimates of the significance (importance) of sub-criteria determining the passenger choice of traveling by train rather than by bus were consistent (in agreement) and reflect their general opinion, shown by the averages \(\overline{\omega_{j}}\) of criteria weights \(\omega_{j}\) and \(\omega^{\prime}{}_{j}\) calculated according to Equation (5) and their correlation with the rank averages \(\overline{R}_{j}\) (Figure 5b). With reference to the principle of determining the sample size, the absolute error \(\Delta_{j}\) for determining the average rank \(\overline{R}_{j}\) of each \(j\)-th sub-criterion was calculated from formula (10) (Table 3). For the calculation of \(\Delta_{j}\), the values of the standard deviation \(\sigma_{R_{j}}\) of each sub-criterion were taken from Table 2 when 52 passengers were interviewed and \(\alpha=0.05\) significance level. The results (Table 3) show that the mean ranks \(\overline{R}_{j}\) of the 15 sub-criteria were identified with an absolute error \(\Delta_{j}\), the sample range of which was \(\Delta_{j\max}\) - \(\Delta_{j\min}\) = \(\Delta_{jA}\) - \(\Delta_{jK}\) = 1.20 - 0.88 = 0.32, and the mean value \(\overline{\Delta}_{j}\) = 1.06. By taking \(\Delta_{j}\) for each sub-criterion or the mean value \(\overline{\Delta}_{j}\) of 15 sub-criteria, it was possible to calculate the confidence interval \(\overline{R}_{j}\)\(\pm\Delta_{j}\) of ranks average \(\overline{R}_{j}\), with the population mean rank \(\overline{R}_{jP}\) of 95% confidence. For example, the population mean rank \(\overline{R}_{jA}\) of the sub-criterion A with the highest rank variation was in the range 6.751 \(\pm\) 1.20, i.e., between 5.551 and 7.951. The range would decrease if more than 52 passengers were interviewed. We believe that \(\overline{\Delta}_{j}\) = 1.06 is close to one, so the number of passengers \(m=52\) who completed the survey was sufficient and allowed us to reliably assess the factors determining the choice of passengers to travel by train as an alternative to bus. Figure 5: Correlation between the means of the sub-criteria ranks and the normalized weights of these sub-criteria calculated using: (**a**)—ARTIW-L and ARTIW-N methods; (**b**)—average values of two methods.