## 4 Discussion  
The XRFS reported an LOQ = 0.5 ppm for Ge and an LOQ = 1.0 ppm for Bi. Five outcomes were reported to be below the LOQ for Ge, and six for Bi. A robust statistical analysis of the data for Ge yielded a median of 2.2 ppm and a standard deviation of 1.9 ppm, while for Bi, a median of 3.2 ppm was obtained with a standard deviation of 1.9 ppm.  
Applying the conservative model proposed in this study, the mean concentration of Ge was calculated as 2.9 ppm, which is higher than that provided by the robust statistical analysis at approximately 0.7 ppm, i.e., approximately 32% greater. For Bi, the conservative approach yielded a mean concentration value of 3.3 ppm, which is very close to the robust estimation value of 3.2 ppm, representing an increase of approximately 3%. The conservative approach revealed higher standard uncertainties in these cases--approximately 3.4 ppm for Ge and 3.9 ppm for Bi--that exceeded the concentration values themselves. In such cases, the data are deemed unreliable for decision making regarding the recovery of Ge and Bi.  
The data analysis model we proposed herein produced contradictory results for Y and Ce (Table 3), with their concentrations exhibiting a clustering tendency (Figure 5a,b). This outcome aligns with the limitations of our model, as it is only valid for uniformly distributed data, such as in the cases of Ge and Bi (Figure 6a,b).  
Classical statistical analysis overestimates the Ge and Bi concentration values by approximately 59% and 15%, respectively. Similarly, robust statistical analysis overestimates  
\begin{table}
\begin{tabular}{l c c c c} \hline Statistical parameter & Y & Ce & Ge & Bi \\ \hline \multicolumn{5}{c}{Classical statistical analysis of the data} \\ \hline Arithmetic mean [ppm] & 22.01 & 36.50 & 3.54 & 3.05 \\ SD [ppm] & 1.85 & 2.62 & 1.97 & 1.82 \\ RSD (\%) & 8 & 7 & 56 & 60 \\ \hline \multicolumn{5}{c}{Robust statistical analysis of the data} \\ \hline Median [ppm] & 21.41 & 34.50 & 3.31 & 2.88 \\ SD * [ppm] & 0.89 & 2.97 & 2.69 & 2.32 \\ RSD ** (\%) & 4 & 9 & 81 & 81 \\ \hline \multicolumn{5}{c}{Conservative data analysis} \\ \hline Mean [ppm] & 11.15 & 19.25 & 2.31 & 2.68 \\ LOQ [ppm] & 0.50 & 1.00 & 0.50 & 1.00 \\ Uncertainty [ppm] & 9.12 & 15.75 & 1.95 & 2.40 \\ Relative uncertainty (\%) & 82 & 82 & 84 & 90 \\ \hline \end{tabular} \({}^{\star}\)SD: standard deviation; ** RSD: relative SD.  
\end{table}
Table 3: The outcomes of statistical data analyses using classical, robust, and conservative approaches.  
mates the Ge and Bi concentration values by approximately 43% and 7%, respectively. Additionally, the SDs assigned to the concentration values using classical and robust data analysis are underestimated (Table 3). This evidence demonstrates that both classical and robust statistical analyses of concentration values close to the LOQ of a suite of uniformly distributed results yield higher values than for real (true) ones. Such data analyses may lead to false positive decisions regarding investment in a trace element recovery business. Therefore, when there is evidence that analytical outcomes are uniformly distributed, it is prudent to apply a statistical analysis based on the conservative approach outlined earlier in this paper.