## 6 Results and Discussion  
The case presented is the calibration of a flow meter against a standard tank to obtain the meter factor (\(MF\)). The results are gathered in two tables: Table 2 shows the results for the meter factor, its standard uncertainty, and its lower and upper limits for AMCM and GUF; and Table 3 shows the results for the expanded uncertainty, the coverage factor, the relative expanded uncertainty (\(W\)), and the numerical tolerance \(\delta\) for both methods. As can be seen, AMCM \(U(MF)\) is lower than GUF. This means a 7.1% decrease in relative expanded uncertainty \(W(MF)\). \(MF\) from AMCM is identical to \(MF\) from GUF: which implies that the model function is linear within the range of the input quantities. The GUF value for \(u(MF)\) with two significant figures is identical to the value for AMCM: the law of propagation of uncertainty of the GUM method could be applied in this case to estimate the standard uncertainty. The AMCM value for de \(U(MF)\) is 6% lower than the GUF value due to the different coverage factors of both methods, so the \(MF\) distribution is not Gaussian. The tests applied reject the normality, and the kurtosis values (\(-0.55\)) are lower than in a normal distribution. The normality deviation index shows a deviation of \(-18.3\%\) in respect to a Gaussian. The classical GUM method would never result in the correct \(PDF\) as the Student's t-distribution has a positive excess kurtosis. In Figure 4, the \(MF\) histogram as obtained from the AMCM is fitted to a normal distribution. As can be observed, the resulting distribution is clearly deviated from the Gaussian with a negative excess kurtosis. Though the \(MF\) model function can be linearized, the deviation from normality indicates that the conditions for the central limit theorem are not fulfilled. According to GUM-S1 in NOTE 4 of the 5.7.2 subsection, the lack of normality is due to the fact that the rectangular distribution assigned to the \(MF\) repeatability is dominant. This can be proved by simulating the \(PDF\) and removing the repeatability contribution. The function model result of the sum of a Gaussian and a dominant rectangular distribution is a Flatten-Gaussian \(PDF\)[22] as it is the one proposed for the \(MF\). A specific simulation (non-adaptive) of \(10^{5}\) samples has been carried out to fit the empirical distribution to a Flatten-Gaussian \(PDF\) whose equation is [22] as follows:  
\[g_{\text{MF}}(\xi)=A\exp\Bigl{[}-a(\xi-c)^{2}-b(\xi-c)^{4}\Bigr{]} \tag{11}\]  
where \(A\) is a normalization constant, \(a\) and \(b\) are parameters dependent on the distribution momenta and, \(c\), is the best estimate of the \(MF\). The fitting plot is shown in Figure 5 and the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **AMCM** & \multicolumn{4}{c}{**GUF**} \\ \hline \(MF\) & 1.02451 & dimensionless & \(MF\) & 1.02451 & dimensionless \\ \(u(MF)\) & 0.00036 & dimensionless & \(u(MF)\) & 0.0003565 & dimensionless \\ \(\text{MF}_{\text{low}}\) & 1.02384 & dimensionless & \(MF_{\text{low}}\) & N/A & - \\ \(\text{MF}_{\text{high}}\) & 1.02518 & dimensionless & \(MF_{\text{high}}\) & N/A & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Meter factor and standard meter factor uncertainty for both methods: AMCM and GUF. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **AMCM** & \multicolumn{4}{c}{**GUF**} \\ \hline \(U(MF)\) & 0.00067 & dimensionless & \(U(MF)\) & 0.000713 & dimensionless \\ \(k\) & 1.86 & dimensionless & \(k\) & 2 & dimensionless \\ \(W(MF)\) & 0.065 & \% & \(W(MF)\) & 0.070 & \% \\ \(\delta\) & 0.000005 & dimensionless & \(\delta\) & N/A & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Meter factor expanded uncertainty, coverage factor, \(W\) and numerical tolerance \(\delta\) for both methods: AMCM and GUF. Finally, it has been checked that the estimated \(A\) is compatible with the \(A\) value (0.03868) calculated from \(a\) and \(b\) with the expression given in [22].