# Estimating the Uncertainty of Measurements for Various Methods and 3D Printed Parts
## 2 Materials and Methods
### Statistical Analysis  
Statistical analysis is a key factor in the reliable presentation of measurement results, and also allows comparison of the results of different studies, so the calculation methodology used is presented [30]. The basic statistical parameter that allows further in-depth analysis is the standard deviation, which, in the presented test results, was calculated according to Equation (1) shown below.  
\[s=\sqrt{\frac{1}{(n-1)}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}} \tag{1}\]  
where:  
\(n\)--number of samples,  
\(x_{i}\)--results of sample measurement,  
\(\overline{x}\)--average value for series of measurements.  
Another parameter that makes it possible to assess the value of results in a much more precise way is the so-called standard uncertainty. The way of estimating the uncertainty is the criterion for its division into two types:  
- Type A uncertainties--determined by statistical methods,  
- Type B uncertainties--determined by other methods.  
Measurement results are subject to both type A and type B uncertainties, and they can have comparable values or significantly dominate each other. When the dominant uncertainty is:  
- Type A standard uncertainty specified as type A overall uncertainty,  
- Type B standard uncertainty shall be referred to as type B overall uncertainty.  
When the two uncertainties have comparable values, the total uncertainty will be the type AB uncertainty.  
The type A uncertainty for the study was calculated according to Formula (2), shown below.  
\[u_{A}=\sqrt{\frac{1}{n(n-1)}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}} \tag{2}\]  
The final statistical parameter that, for experimental studies, allows one to determine the result of a measurement that depends on the number of samples tested, is the Extended Uncertainty using Student's t, or according to Gauss normal distribution. In the case of the presented research, Formula (3) was used, which takes into account an additional expansion factor--k\({}_{\text{p}}\), the value of which in the following calculations is shown in Tables 3 and 4. The expanded uncertainty was calculated using the Student's t-distribution because the size of the largest series of samples did not exceed the number 30.  
\[U_{CA}=k_{p}u_{A} \tag{3}\]  
where:  
\(k_{p}\)--expansion factor for a series of measurements selected for a confidence level of 95%.  
There may be a situation where the measurement uncertainty calculated by the type A method is exceedingly small, which may be questionable. Therefore, it is necessary to refer to the limiting error of a single measurement.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Measurement** & **Series** & **Coefficient** & **Standard** & **Standard** & **Expanded** & **Result** \\
**Tool** & **n** & **k\({}_{p}\)** & **Ventilation** & **Uncertainty** & **Uncertainty** & **U\({}_{CA}\)** & **\(\overline{x}\) + \(U_{CA}\)** \\  & & & **s, mm** & **H\({}_{\text{\tiny{A}}}\), mm** & **mm** & **mm** \\ \hline Dial Caliper & 3 & 4.3 & 0.012 & 0.007 & 0.029 & 15.11 + 0.029 \\  & 5 & 2.78 & 0.023 & 0.010 & 0.029 & 15.10 + 0.029 \\  & 10 & 2.26 & 0.023 & 0.007 & 0.016 & 15.09 + 0.016 \\  & 20 & 2.09 & 0.023 & 0.005 & 0.011 & 15.08 + 0.011 \\  & 30 & 2.05 & 0.022 & 0.004 & 0.008 & 15.08 + 0.008 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Statistical analysis included statistical population size (number of samples) with seam.  
The next step was to divide the obtained results into a series of samples of 3-5-10-20-300 in a random manner. The arithmetic diameter was calculated for each series. Figures 11 and 12 show the results of the arithmetic means, and indicate the trend lines, where all decrease in this value can be observed as the number of samples increases.  
Figure 11: Calculated average value for a series of 3-5-10-20-30 samples with seam.  
Figure 12: Calculated average value for a series of 3-5-10-20-30 samples measurement excluding the seam site.  
Figure 10: Average diameter values—measurement excluding the seam point.