## 7 Numerical Illustration: Measurement of a Step Gauge  
The simulations involve characterisations of CMM behaviour based on the two MPE statements and statistical parameters given in Table 3. The calculations involve simulations of measurements of a step gauge with 26 steps each of nominal length 10 mm situated at 20 mm intervals along a measuring line. The simulations reported on here involve two scenarios. In each scenario, the step gauge is aligned with the \(x\)-axis so that measurements are not sensitive to squareness effects nor scale effects associated with the \(y\)- and \(z\)-axes. **Scenario I. **: This (realistic) scenario assumes (i) the 52 step faces are measured with the same probe with probe offset \(\mathbf{p}=(0.0,0.0,-20.0)^{\top}\) mm. In this scenario, the measured distances are not sensitive to probe qualification effects and the only rotation effect of importance is rotation about the \(y\)-axis, i.e., pitch along the \(x\)-axis. **Scenario II. **: This scenario (used for illustration) assumes (ii) the 26 left-facing faces are measured with a probe with offset \(\mathbf{p}_{L}=(0.0,20.0,0.0)^{\top}\) mm and the 26 right-facing faces are measured with a probe with offset \(\mathbf{p}_{R}=(0.0,-20.0,0.0)^{\top}\) mm. In this scenario, the measured distances are sensitive to probe qualification effects. The measurements are also sensitive to rotation about the \(z\)-axis, i.e., yaw along both the \(x\)- and \(y\)-axes. The associated features derived from the measurements are as follows:  
\(d_{ij}\): The distances between all faces, with \(d_{ij}=\|\mathbf{x}_{i}-\mathbf{x}_{j}\|\). \(d_{LL,ij}\): The distances between left-facing faces. \(d_{RR,ij}\): The distances between right-facing faces. \(d_{FF,k}\): The estimated length of each step. The uncertainties associated with these distances have been calculated by evaluating the variance matrix \(V_{X}\) associated with the point cloud \(\mathbf{x}_{i}\), \(i=1,\ldots,m=52\), representing points on the centre of the 52 step gauge faces. The variance matrix \(V_{X}\) is constructed as in Equation (28), involving the various influence factors. The uncertainties associated with distances are calculated using Equation (29), decomposed as in Equation (30). These calculations do not depend on any of the simplifying assumptions used in Section 6 to derive compact formulae for uncertainties associated with distances. Figures 6 and 7 plot the estimated expanded uncertainty \(Ku(d)\) with \(K=2\) associated with distances derived from measurements of a step gauge under scenario I and scenario II, respectively, and statistical parameters given in column 4 of Table 3, MPE1, for the different associated features discussed above. (The graphs corresponding to MPE2 show a similar behaviour.) 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. Also plotted is the MPE function \(A+d/B\) (upper straight line). The two figures show that the uncertainties associated with left-facing faces and right-facing faces are the same for both scenarios since these distances are not sensitive to probe qualification effects and the rotation effects are essentially the same. We have assumed that the rotation errors are isotropic, i.e., the same for each axis so that a pitch about the \(x\)-axis, scenario I, has the same effect as yaw along the \(x\)-axis, scenario II, bearing in mind that each probe offset is of length 20 mm. In both cases, the uncertainties associated with distances involving both a left- and right-facing face are significantly larger because they are also are sensitive to probing effects. In the case of scenario II, these distances are also sensitive to yaw errors along the \(y\)-axes.