## 4 Conclusions  
This study aimed to statistically establish the \"Goodness of fit\" using adsorption isotherm models for MB adsorption on AG. Five different isotherm models were employed in this investigation to analyze the equilibrium adsorption data. A comprehensive set of isotherm parameters and residuals were optimized using five distinct error functions. The statistical optimization of the residuals, carried out using the SNE and CND, revealed that the R-P model provided the most accurate representation of the experimental data. Furthermore, the isotherm parameters were consistent with the magnitude of the error analysis. According to the data analysis, the variation margin in the residuals ranged from 0.0025 to 0.0048, and the lowest SNE value was generated by the SSE and HYBRID functions for the R-P isotherm. This suggests that the isotherm model prediction was accurate. The results of this research indicate that the prediction of the best-fitting isotherm model depends on the correlation between the errors and the theoretical basis of the isotherm parameters. This conclusion was drawn from the findings of the present investigation. It is important to note that applying various statistical approaches and isotherm models can enhance our understanding of the adsorption process and increase our confidence in the accuracy and reliability of model predictions. A deeper understanding of the interaction between the adsorbate and the adsorbent can be achieved by employing a range of statistical techniques and isotherm models. This is crucial for the development of efficient adsorption systems. Additionally, using multiple statistical methods and isotherm models can aid in the optimization of the operating parameters, such as the determination of the ideal adsorbent dosage and contact time. These factors can contribute to an increase in the effectiveness of the adsorption process, ultimately leading to sustainable and cost-effective solutions for water treatment and environmental remediation. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app13148106/s1](https://www.mdpi.com/article/10.3390/app13148106/s1)]([https://www.mdpi.com/article/10.3390/app13148106/s1](https://www.mdpi.com/article/10.3390/app13148106/s1)), Figure S1: Linear forms of Langmuir isotherm models for sorption of MB on AG at (a) 303, (b)313, (c) 323 and (d) 333 K; Table S1: Specification of chemicals used in the study; Table S2: Comparison of maximum adsorption capacity for MB sorption with other adsorbents [56; 57; 58; 59; 60; 61; 62; 63; 64; 65]; Table S3: Description of error functions; Table S4: Error analysis of Langmuir isotherm model at four different temperatures for MB adsorption on AG; Table S5: Error analysis of [PERSON] isotherm model at four different temperatures for MB adsorption on AG; Table S6: Error analysis of R-P isotherm model at four different temperatures for MB adsorption on AG; Table S7: Error analysis of [PERSON] isotherm model at four different temperatures for MB adsorption on AG; Table S8: Error analysis of Sips isotherm model at four different temperatures for MB adsorption on AG. **Author Contributions:** Conceptualization, H.D. and K.H. ; Methodology, H.D. ; Software, K.H. and M.Z.I. ; Validation, H.D., K.H. and M.Z.I. ; Formal Analysis, H.D. ; Investigation, H.D., K.H. and M.Z.I. ; Resources, L.K. and S.Y. ; Data Curation, H.D. ; Writing--Original Draft Preparation, H.D. and K.H. ; Writing--Review & Editing, H.D., K.H. and S.Y. ; Visualization, H.D. ; Supervision, J.Y. ; Project Administration, S.Y. ; Funding Acquisition, Y.Z. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the Key Research & Development Plan of Zhejiang Province of China (No. 2021C03160), the Natural Science Foundation of Zhejiang Province of China (No. LY19C160010), and the Science Foundation of Zhejiang Sci-Tech University (ZSTU) (No. 21202297-Y). **Institutional Review Board Statement:** Not applicable.