## 5 Experimental Analysis
### Example for Evaluating Uncertainty of Diameter Measurement  
The height of the measured cylinder is 14 mm, the upper tolerance limit of diameter is +0.020 mm and the lower tolerance limit of the diameter is \(-0.015\) mm. The measured cylinder is made by a milling-tool, its roughness is 0.8 \(\upmu\)m, and the effect of surface roughness for diameter measurement is negligible. According to Formula (4), the uncertainty component caused by indication errors can be calculated as follows:  
\[u_{\text{E}}=\frac{E_{\text{L, MPE}}}{\sqrt{3}}=\frac{1}{\sqrt{3}}\times(3+ \frac{62}{250})=1.875\ \upmu\text{m} \tag{26}\]  
Under the condition of repeatability, complete 10 times' continuous and rapid measurement of the circle to be measured. See Table 2 for data on the repeatability experiment. Calculate the standard deviation of the repeatability experiment according to the experimental data in Table 2:  
\[S_{\text{r}}=\sqrt{\frac{\sum\limits_{i=1}^{n}\left(d_{i}-\overline{d} \right)^{2}}{(n-1)}}=0.51\ \upmu\text{m} \tag{27}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**No.** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline measured value \(d_{i}\) (mm) & 62.0010 & 62.0011 & 62.0010 & 61.9998 & 62.0011 & 62.0002 & 62.0008 & 61.9998 & 62.0007 & 62.0008 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The experimental data of diameter measurement for repeatability detection. Figure 4: The experimental of measurement. Figure 3: The drawing marking of the measured part. If the average value of three times' measurements is taken as the best estimate of the diameter measurement, the standard uncertainty caused by measurement repeatability is as follows:  
\[u_{\mathrm{r}}=\frac{S_{\mathrm{r}}}{\sqrt{3}}=0.294\ \mathrm{\upmu m} \tag{28}\]  
Carry out three groups of repeatability measurements independently by three surveyors with professional measurement knowledge and operation level according to their habits and in accordance with the measurement requirements. Try to have the sampling strategy, measurement starting point and other factors in the experiment kept as different as possible. The probe configurations that can be selected by the surveyors include 2 BY20, 2 BY40, 3 BY20, 3 BY40, 4 BY20, and 4 BY40. The number and distribution of sampling points conform to BS 7172 [12]. See Table 3 for the data in the reproducibility experiment:  
Total mean of the measurement for calculating reproducibility:  
\[\overline{\overline{\overline{d}}}=\frac{\sum\limits_{j=1}^{9}\overline{ \overline{d}_{j}}}{9}=62.0007\ \mathrm{mm} \tag{29}\]  
According to Formula (7), the standard uncertainty caused by the reproducibility of the diameter measurement is as follows:  
\[u_{\mathrm{R}}=\sqrt{\frac{1}{(9-1)}\sum\limits_{j=1}^{9}\left(\overline{ \overline{d}_{j}}-\overline{\overline{d}}\right)^{2}}=0.850\ \mathrm{\upmu m} \tag{30}\]  
Then see Table 4 for the uncertainty component of the diameter measurement:  
According to Formula (3), the combined standard uncertainty of the diameter measurement task can be calculated as follows:  
\[u_{\mathrm{c}}=\sqrt{u_{\mathrm{E}}^{2}+u_{\mathrm{r}}^{2}+u_{\mathrm{R}}^{2}}= 2.1\ \mathrm{\upmu m} \tag{31}\]  
According to GUM, take \(p=95\%\)\(k=2\) and the expanded uncertainty is as follows:  
\[U=k\times u_{\mathrm{c}}=4.2\ \mathrm{\upmu m} \tag{32}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Standard Uncertainty** & **Source of Uncertainty** & **Evaluation Result** \\ \hline \(u_{\mathrm{E}}\) & Indication error & 1.875 \(\mathrm{\upmu m}\) \\ \(u_{\mathrm{r}}\) & Repeatability & 0.294 \(\mathrm{\upmu m}\) \\ \(u_{\mathrm{R}}\) & Reproducibility & 0.850 \(\mathrm{\upmu m}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The uncertainty budget for the diameter measurement.