Development of analytical method for calculation the expanded uncertainty in convolution of rectangular and Gaussian distribution

[Display omitted] ► The new analytical method for calculation the expanded uncertainty was presented. ► Very high precision (up to 15 decimal digits) was achieved, not available with other methods. ► We provide a practical MS Excel tool using the above algorithm. ► Results between the presented method and the MCM are compared in a practical example. The paper presents the new method for calculating a coverage factor. In this case the calculation is done in a linear combination of the both the Gaussian and the rectangular distribution. Its exact analytical form is difficult to find because the rectangular distribution is not in an analytical form at the borders, although a form can be built using multiplicative law of probabilities as the product of conditional probability. This approach was checked in MS Excel 2007, and achieved very high accuracy of calculations. Such accuracy is hard to be achieved using other known methods, e.g. Monte Carlo method. In the Appendices are a formal reasoning of the algorithm, an exemplary macro for MS Excel, as well as the R and MATLAB codes for the MC simulation used for comparing its performance with the new method. The paper presents an example of calculating the uncertainty for the case of investigating precise voltage measurement of a transducer using the method being discussed.