# Estimating the Uncertainty of Measurements for Various Methods and 3D Printed Parts
## 1 Introduction
In the lack of information on the number of samples tested, it can be concluded that the tests were carried out on a single sample, which affects the reliability of the results obtained and is undoubtedly an inappropriate action [9; 11]. The results of testing for a small number of samples in determining the type A standard uncertainty and for the expanded uncertainty are crucial and affect the final results [12]. The most common instruments for measuring geometric quantities such as shaft diameters are hand-held measuring instruments such as calipers and bore gages. These instruments usually have a measurement resolution of about 0.01 mm, and their accuracy is at a similar level. Measurements made with these devices are subject to numerous errors such as observer error and low-precision pressing force. In Ref. [13], the authors described the results of tests conducted using hand-held measuring instruments such as calipers and the errors associated with this type of measurement. The authors pointed out that upper limb motor skills (especially finger coordination) significantly affect the accuracy of measurement using a caliper. Coordinate measuring machines (CMMs) are another common method for measuring the geometry of 3D printed parts. Measurement using CMM for 3D printing involves the need for stable mounting of the test sample, proper selection of the diameter of the probe tip, and appropriate force of the probe tip on the test piece. In an article [9] on the measurement of 3D printed parts manufactured using fused deposition modeling/fused filament fabrication (FDM/FFF) technology on a CMM, the authors pointed out that the results obtained from the measurement of internal diameters were lower than the nominal diameters of these holes. On the other hand, the authors of Ref. [13], who conducted a study on the geometric accuracy of stationary 3D printers using a CMM, obtained similar results for internal diameters, where in each case the diameter had a value lower than the nominal one. However, when measuring external diameters for a nominal value of 20 mm, the diameter value was higher, and for a nominal value of 30 mm, the value was lower than expected. One of the popular methods of measuring the geometry and surface of a component is 3D scanning. It is distinguished by its remarkable measurement speed as well as technological intuitiveness. However, it has its limitations [14], such as the requirement for adequate illumination of the sample under examination; for example, if it has deep narrow holes, errors in the virtual model and inaccurate representation of the geometry are possible. Another important aspect of the scanning method is that it is a non-contact measurement,which prevents potential changes in the surface of the measured sample, although there may be larger deviations created in the intermediate stages and the performance of 3D model fitting [15; 16; 17; 18; 19]. Associated with the measurement of the geometry of parts manufactured by 3D printing technologies is the problem of estimating the uncertainty of the results of these measurements. Measurement uncertainty \(u\) is a parameter that allows one to determine the limits of the interval containing, with assumed probability, the unknown true value of the measured quantity. Measurement uncertainty is affected by a number of uncertainty components. Their determination can be made based on the evaluation of standard deviation estimators for the obtained scatter of the results of a series of measurements or, in the case of single measurements, by means of standard deviations determined on the basis of predicted probability distributions [20]. Ref. [21] proposed a general method for establishing a mathematical model of spatial uncertainty based on the measured geometry of microstructures created in the manufacturing of parts by 3D printing technology. Whereby, the spatial uncertainty was defined as the deviation between the planned and actual geometry of the model structure made by 3D printing.