## 3 Results and Discussion
### Case Study 2: Interlaboratory Study of the Final Boiling Point of Gasoline  
Table 2 presents the data and their corresponding uncertainties obtained from 12 laboratories that participated in the ASTM interlaboratory study on gasoline's final boiling point (FBP). Mathematical model: \(m_{1}^{c}=m_{2}^{c}=\cdots=m_{12}^{c}\). Reconciled value: From Equations (17) and (19), respectively, the reconciled value is \(m^{c}=179.2\)\({}^{\circ}\)C and the minimized uncertainty is \(U_{m_{c}}=2.6\)\({}^{\circ}\)C.  
\(m_{c}=\frac{\frac{173.5}{6.0^{2}}+\frac{180.9}{11.0^{2}}+\frac{181.6}{6.0^{2} }+\frac{182.1}{81.2}+\frac{174.2}{6.0^{2}}+\frac{181.6}{6.2^{2}}+\frac{180.0}{ 6.7^{2}}}{\frac{1}{11.0^{2}}+\frac{183.0}{13.0^{2}}+\frac{183.6}{7.2^{2}}+\frac {1}{7.0^{2}}}{\frac{1}{6.0^{2}}+\frac{1}{12}(\frac{1}{1}+\frac{1}{7.4^{2}}+ \frac{1}{81.1}+\frac{1}{6.0^{2}}+\frac{1}{8.5^{2}}+\frac{1}{6.7^{2}}}}\)  
\(m_{c}=179.2\)\({}^{\circ}\)C  
\(U_{m^{c}}=\sqrt{\frac{1}{\frac{1}{6.0^{2}}+\frac{1}{12.0^{2}}+\frac{1}{7.0^{2 }}+\frac{1}{8.1^{2}}+\frac{1}{6.0^{2}}+\frac{1}{8.5^{2}}+\frac{1}{6.7^{2}}}{ \frac{1}{11.0^{2}}+\frac{1}{10.0^{2}}+\frac{1}{13.0^{2}}+\frac{1}{7.2^{2}}+ \frac{1}{7.0^{2}}}}}\)= \(2.6\)\({}^{\circ}\)C  
By observing Figure 3, from the FBPs and their respective expanded uncertainties, average value, and reconciled value, it became apparent that the reconciled value fell within the uncertainty range of all the laboratories. This behavior was not noticed when \begin{table}
\begin{tabular}{c c c} \hline \hline
**Laboratory** & **FBP** (\({}^{\circ}\)**C**) & **Expanded Uncertainty** (\({}^{\circ}\)**C**) \\ \hline
1 & 173.5 & 6.0 \\
2 & 180.9 & 12.0 \\
3 & 181.6 & 7.0 \\
4 & 182.1 & 8.1 \\
5 & 174.2 & 6.0 \\
6 & 181.6 & 8.5 \\
7 & 180.0 & 6.7 \\
8 & 188.0 & 11.0 \\
9 & 183.0 & 10.0 \\
10 & 183.6 & 13.0 \\
11 & 173.0 & 7.2 \\
12 & 185.5 & 7.0 \\ Average value & 180.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: FBP and respective uncertainties. Figure 2: Reconciled value and average value. the arithmetic mean of all the laboratories was used to represent the best estimate of the true value. In this case, 3 (laboratories 1, 5, and 11) of the 12 were outside the mean value and their respective uncertainty ranges. Using the reconciled values was more realistic than relying only on the mean value, as according to Grubbs' test, none of the values were considered outliers. Case Study 3: Metrological Evaluation of Three Distinct Test Methods to Determine Sulfur in Diesel Oil  
Table 3 compares the data derived from three distinct test methods for determining the sulfur mass fraction in the same sample of diesel oil. \begin{table}
\begin{tabular}{c c c} \hline \hline
** ASTM D453**[34] & **ASTM D4294**[33] & **ASTM D7039**[35] \\ \hline
6.2 & 5.4 & 6.3 \\
6.9 & 4.9 & 5.2 \\
6.1 & 5.3 & 6.3 \\
6.1 & 4.6 & 5.8 \\
7.5 & 4.9 & 6.3 \\
7.2 & 5.8 & 6.9 \\
7.0 & 4.9 & 5.6 \\
6.9 & 5.9 & 6.3 \\
6.5 & 4.9 & 6.2 \\
6.8 & 4.7 & 7.2 \\
6.6 & 3.8 & 6.2 \\
6.5 & 3.9 & 7.7 \\
6.7 & 4.4 & 5.9 \\
6.1 & 4.6 & 5.9 \\
6.0 & & 6.3 \\
7.0 & & \\ Arithmetic average & 6.6 & 4.9 & 6.3 \\ Standard deviation & 0.46 & 0.60 & 0.62 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Measurement data of sulfur mass fraction, mg kg\({}^{-1}\). Figure 3: Uncertainty bars, reconciled value, and average value. From the mathematical model \(m_{1}^{c}=m_{2}^{c}=m_{3}^{c}\), and from Equations (17) and (19), respectively, the following is obtained:  
\[ \begin{array}{c}m^{c}=\frac{\frac{6.66}{0.46^{2}}+\frac{4.49}{0.60^{2}}+\frac {6.3}{0.62^{2}}}{\frac{0.66^{2}}{0.62^{2}}+\frac{6.62^{2}}{0.62^{2}}}=6.1\ \text{mg kg}^{-1}\\ u_{m_{c}}=\sqrt{\frac{1}{\frac{1}{0.46^{2}}+\frac{1}{0.60^{2}}+\frac{1}{0.62^{ 2}}}}=0.30\ \text{mg kg}^{-1}\end{array}