In this work, we explored the role of a single electron in the energy of neutral and charged clusters of \({Na}_{39}\) using data visualization and statistical techniques as a new insight. Initially, we studied the effects of one electron, time, and temperature on energy using multiple linear regression analysis with dummy variables, and the results demonstrated that all three predictors significantly affected the energy. Time had a positive impact (direct ratio effect) on the energy of \({{Na}^{-}}_{39}\) , and \({Na}_{39},\) and a negative impact (inverse ratio effect) on the energy of \({{Na}^{+}}_{39},\) while temperature had a positive effect on the energy of all three sodium clusters. Then, to study the thermodynamic properties of each cluster, we employed the fuzzy clustering technique. The results verified that each sodium cluster is divided into three groups based on the different temperatures used to investigate the thermodynamic properties of each cluster. Finally, time series analysis was applied to investigate the behavior of the energy in each sodium cluster and each temperature. We used the statistical software R version 4.3.3 to perform all statistical computations.
