## 4 Results
### Clustering Validation Analysis  
The essence of cluster analysis assumes that little or nothing is known about the grouping structure that underlies the data set. The operational objective, in this case, is to discover a data grouping structure that is frequently stated as the problem of finding natural groups or clusters. The approach to this problem and the results achieved help the investigator choose to give operational and quantitative meaning to the phrases \"natural association\" and \"relatively distinct\". Cluster analysis remains the most relevant and robust tool to gain insight into previously unknown grouping structures and to disclose the intrinsic characteristics of data. The cluster analysis task may have more than one solution and, thus, is recognized as \"ill-posed\" [97; 98]. For instance, the result of the validation procedure depends on the choice of the data, clustering algorithms, dissimilarity measures, etc. [99]. Applying different clustering algorithms or the same clustering algorithm with varying input parameters on the same given data set can result in diverse partitions and, therefore, may greatly affect the goodness/quality of clustering results [100]. Such issues are related to clustering validation, which is one of the most challenging topics in data mining literature. The main objective of clustering validation is the effective evaluation of the goodness of clustering results. This involves finding not only the \"true\" number of clusters but also the optimal partition that best fits the underlying grouping structure of the data set. One common practice in clustering validation procedures is to apply clustering validation indices (CVIs). However, many authors agree that there is no universal CVI that always enables the correct decision to be made. In some research, multicriteria solutions used to reach the best and most adequate results have been commonly used [101; 102]. These solutions assume the adoption of several CVIs to achieve greater certainty and correctness in clustering results. For instance, when a group of CVIs takes part in the decision process, their votes, in many cases, disagree. Frequently, the CVIs have different sensitivity domains: some are heavily biased toward particular clustering algorithms and distance-based dissimilarity measures, such as the silhouette CVI; and some are sensitive to outliers, such as the Dunn validation index. Moreover, certain data characteristics, including high dimensionality, noise, types of attributes, and scales, may strongly affect clustering validation analysis [103]. In light of this fact, in the presented research, the authors used majority-based decision fusion clustering validation methods to reveal the \"true\" number of clusters and to justify the presence of significant differences in all estimated factors toward different educational modes. The authors applied the combination of two novel decision-making approaches: a non-invasive majority-based decision fusion method and an invasive clustering validation method. These clustering validation methods have been carefully described in previous works [81; 104], including the experimental setups, general input settings, and guidelines for benchmarking; the methods' execution with different data sets were also showcased. The combination itself should easily evaluate the correctness of obtaining the \"true\" number of clusters and the goodness of the clustering results. Moreover, the majority-based clustering validation methods will help us completely and utterly answer the research questions posed at the beginning of the manuscript. Figure 3 illustrates the simplified validation scheme of a non-invasive majority-based decision fusion method and invasive clustering validation methods. The result of the CVIs' voting process aligns with one of the most common scenarios in majority-based decision fusion validation methods. According to previous studies [81; 104], almost 99% of input benchmark data sets follow this pattern.