## 7 Numerical Illustration: Measurement of a Step Gauge
More insight into the uncertainty behaviour can be found by examining Tables 4 and 5, which give the standard uncertainties (column two) in micrometres associated with the distances between the first and second, first and 51 st and first and 52 nd (last) faces of thestep gauge for scenarios I and I and statistical parameters MPE1, and MPE2. We note the following from Table 4, scenario I, involving only one probe offset. * The distance measurements are not sensitive to probe qualification effects. * Measurements involving left-facing faces only are not sensitive to probing effects as they both involve the same probing direction. * The uncertainties associated with scale effects are proportional to the distances, as would be expected. * For the short distances, the spatially correlated translation and rotational effects make a very small uncertainty contribution. We note the following from Table 5, scenario II, involving two probe offsets. * Measurements involving left-facing faces only are not sensitive to probe qualification effects as they only involve one probe offset. * Measurements involving left-facing faces only are not sensitive to probing effects as they both involve the same probing direction (as in scenario I). * The rotational effects are significant for the measurement of the distance between the first and second face. The measurement of the first face involves one probe offset while the measurement of the second involves the second probe offset, which means that the CMM position differs by 40 mm as measured along the \(y\)-axis, so that yaw along the \(y\)-axis has an effect. Because the distance along the \(y\)-axis is small, 40 mm, relative to the spatial correlation length, these yaw errors are significantly correlated but because the two probe offsets are in different directions along the \(y\)-axis, these yaw errors do not cancel but instead have approximately double the effect. Only by including the relationship between rotational errors and probe offsets in the model, as described in Section 5.4, is it possible to capture this uncertainty behaviour. Figure 6: Estimated expanded uncertainty \(Ku(d)\) with \(K=2\) associated with distances derived from measurements of a step gauge under scenario I and statistical parameters given by the third column of Table 3, MPE1. The label ’LL’ relates to distances between left-facing faces, ’RR’ to distances between right-facing faces and ’FF’ to the distances between the step faces for each step. Additionally plotted is the MPE function \(A+d/B\) (upper straight line). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \(d\)/mm & \(u\) & \(u_{R}\) & \(u_{PQ}\) & \(u_{S}\) & \(u_{ET}\) & \(u_{ER}\) & \(u_{P}\) \\ \hline LR & 10.00 & 0.37 & 0.28 & 0.00 & 0.01 & 0.01 & 0.00 & 0.24 \\ \hline LL & 20.00 & 0.28 & 0.28 & 0.00 & 0.02 & 0.03 & 0.01 & 0.00 \\ \hline LL & 500.00 & 0.64 & 0.28 & 0.00 & 0.49 & 0.28 & 0.10 & 0.00 \\ \hline LR & 510.00 & 0.70 & 0.28 & 0.00 & 0.50 & 0.28 & 0.10 & 0.24 \\ \hline LR & 10.00 & 1.13 & 0.85 & 0.00 & 0.03 & 0.08 & 0.02 & 0.73 \\ \hline LL & 20.00 & 0.87 & 0.85 & 0.00 & 0.06 & 0.17 & 0.03 & 0.00 \\ \hline LL & 500.00 & 1.94 & 0.85 & 0.00 & 1.48 & 0.85 & 0.34 & 0.00 \\ \hline LR & 510.00 & 2.10 & 0.85 & 0.00 & 1.51 & 0.85 & 0.34 & 0.73 \\ \hline \end{tabular}
\end{table} 
Table 4: Standard uncertainties (column three) in micrometres associated with the distances between the first and second, 51 st and 52 nd faces for the step gauge for scenario I and statistical parameters labelled MPE1, first four rows, and MPE2, second four rows, given in columns 4 and 5 of Table 3. Additionally shown (columns three and above) are the uncertainty contributions from the different influence factors, also in micrometres.