# A Method for Estimating the Resultant Expanded Uncertainty Value Based on Interval Arithmetic
## 4 Verification of the Proposed Analysis Method  
To assess the efficacy of the proposed method, a plethora of simulation experiments were conducted. These endeavors aimed to elucidate the conditions warranting the method's application and to ascertain the typical discrepancies between the results derived from the proposed method and those derived from the Monte Carlo method. Typical discrepancies refer to the range encompassing 95% of the error values in estimating the resultant expanded uncertainty, calculated using the relationship:  
\[\delta_{U,*}=100\frac{U_{*}-U_{s}}{U_{s}}\ \%, \tag{17}\]  
where \(U_{*}\) denotes the estimated  
* A selected confidence level of \(\gamma\) was used, constant for each iteration of the experiment implementation;
* For a single iteration, the number of error signals was drawn from the interval \(N\in[N_{min};N_{max}]\) with the same probability of obtaining each possible value;
* For each error signal \(e_{j}(i)\), the expanded uncertainty value was drawn from the interval \(U_{j}\in[U_{min};U_{max}]\) with the same probability of obtaining each possible value associated with this signal;
* The shape of the distribution of the signal implementation value \(e_{j}(i)\) was each time drawn from a pool of distributions: normal, uniform, triangular, u-shape (sine function distribution), and each type of distribution was equally likely to be drawn. * The values \(N_{min}\), \(N_{max}\), \(U_{min}\), and \(U_{max}\) were constant for all iterations of the implemented variant of the experiment;  
Upon deriving the relative error values \(\delta_{U,*}\) for estimating the expanded uncertainty \(U_{*}\) in each experiment variant, the average, standard deviation, and expanded uncertainty values \(U_{-}\) and \(U_{+}\) were computed for a confidence level of 95%. Table 2 showcases specific results from the experiment variants, while Figure 5 displays histograms depicting the distribution of \(\delta_{U,*}\) values obtained. Analyzing the presented results, it can be noticed that the use of shape coefficients determined on the basis of Equation (11) causes the obtained uncertainty values to be smaller than those determined according to Equation (10). For selected cases, this allows to obtain more accurate results, while assessing the entire experimental results, it is concluded that it is more beneficial to use the coefficients determined in accordance with Equation (10). In measurement practice, it is a much better approach to slightly overestimate the uncertainty value than to underestimate it. Due to the above, the subsequent results presented  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\gamma\)} & \multirow{2}{*}{\(N\)} & \multirow{2}{*}{\(\upsilon_{I}\)} & \multicolumn{4}{c}{**(a) Coeficients \(s_{a,b}\) for case (10)**} & \multicolumn{4}{c}{**(b) Coeficients \(s_{a,b}\) for case (11)**} & \multicolumn{4}{c}{**(c) Assumming CLT is met [1]**} \\  & & & \(U_{1,-}\) & \(U_{1,+}\) & \(\delta_{U1,1}\) & \(\sigma_{\delta,1}\) & \(U_{2,-}\) & \(U_{2,+}\) & \(\delta_{U2,2}\) & \(\sigma_{\delta,2}\) & \(U_{3,-}\)  
will only concern the case of using Equation (10) and will be compared to those obtained by the Monte Carlo method and those obtained assuming the fulfillment of the conditions of the central limit theorem. Figure 6 shows selected experimental results for the indicated calculation methods. Figure 5: Obtained histograms for the implementation of the relative error in estimating the expanded uncertainty value, corresponding to the data listed in Table 2.  
Figure 6: Dependence of the average value of the relative error in estimating the expanded uncertainty value depending on the confidence level for the analyzed method **(a)** (left side) and assuming the fulfillment of the conditions of the central limit theorem **(c)** (right side), along with the expanded uncertainty for these quantities, determined for the confidence level \(\gamma=95\%\) (the estimated area is filled with color).