# A Method for Estimating the Resultant Expanded Uncertainty Value Based on Interval Arithmetic
## 1 Introduction  
In both direct and indirect measurements, calculating the value of the measured quantity \(y(t)\) is always encumbered with a certain error \(e_{y}(t)\). This error is typically delineated as the variance between the actual value of the measured quantity \(\tilde{y}(t)\) and the ideal value of this quantity \(\dot{y}(t)\) (achieved when the measurement remains unaffected by any phenomenon, commonly referred to as an error source) [1]:  
\[e_{y}(t)=\tilde{y}(t)-\dot{y}(t). \tag{1}\]  
Through repeated measurements, a specific set of instances of the discussed error signal \(e_{y}(t)\) is acquired. Parameters like the expected value \(E[e_{y}(t)]\), variance \(\sigma_{e,y}^{2}\), standard deviation \(\sigma_{e,y}\), and the probability density function's shape \(\xi_{e,y}(x)\) of the signal's distribution can depict the signal's characteristics. It is crucial to mention that when multiple measurements are conducted for various values of the measured quantity, the set of error signal instances \(e_{y}(t)\) encompasses both A and B type components, detailed in the guide [1]. In measurement practice, besides stating the measured quantity's obtained value, it is essential to specify a quantity that describes the precision with which the value was determined. The two primary measures commonly utilized for this purpose are the standard uncertainty and the expanded uncertainty [1]. In the classical definition of the error signal, as referenced in Equation (1), under the assumption of a symmetric distribution of instances and a signal with a zero expected value, the standard uncertainty equals the signal's standard deviation. In this scenario, the expanded uncertainty can be calculated using the equation:  
\[P\big{(}|\xi_{y}(t)\big{|}\leq U_{y}\big{)}=\gamma, \tag{2}\]  
where \(|\xi_{y}(t)|\) represents the absolute value of any instance of the error signal \(e_{y}(t)\), \(U_{y}\) signifies the expanded uncertainty value, and \(\gamma=1-\alpha\) represents the specified confidencelevel [1], where \(\gamma\in(0;100)\%\). The concept behind the expanded uncertainty metric lies in assuming that if the error signal value is equal to or lower than the expanded uncertainty value, this applies to \(\gamma\) across all signal instances. This metric can also be established by correlating its value with the standard uncertainty value:  
\[U_{y}=c_{e,y}\sigma_{e,y}, \tag{3}\]  
where \(c_{e,y}\) represents the coverage factor derived from the distribution shape of the analyzed error signal implementations and the chosen confidence level [1]. The characteristics of specific distribution types and the connection between the expansion coefficient values and the provided confidence level are extensively discussed in the study by [PERSON] [2]. The determination of the expanded uncertainty value, as defined in Equation (2), can also be achieved through experimental means. To do so, successive realizations of the signal outlined in Equation (1) need to be acquired via measurement or simulation experiments. Given a discrete set comprising \(N\) instances of the studied error signal, and assuming \(h_{e,y}(x)\) as a function that specifies the count of error signal instances \(e_{y}(i)\) with a value of \(x\) for \(i\in[0;N-1]\), where the signal's mean value is zero, Equation (2) is expressed as:  
\[\frac{1}{N}\sum_{x=-U_{y}}^{U_{y}}h_{e,y}(x)=\gamma, \tag{4}\]  
for cases where the probability density function of the analyzed signal is positive and symmetrical with respect to the ordinate axes. A graphical representation of Equations (2) and (4) is shown in Figure 1.  
Figure 1: Graphical representation of the expanded uncertainty measure \(U_{y}\) for the \(\gamma=1-\alpha\) confidence level assuming a zero expected value and a symmetric distribution of subsequent error signal \(e_{y}(i)\) implementations. Considering the information above, it is evident that the expanded uncertainty metric effectively quantifies the accuracy of determining the measured quantities' values.