## 2 Evaluation Method
### Uncertainty Evaluation Model  
The quantitative statistical analysis method can carry out direct statistics and analysis of the measurement results; the output and input quantities have identical units and the measurement results can be obtained without relying on any other quantity subject to a functional relationship with the quantity being measured; therefore, compared with the error traceability method, it features a simple transfer relation for uncertainty, it has convenient model use and it has richer adaptability to the requirements of simplicity, rapidness and practicability for task-oriented measurement uncertainty. However, disadvantageously, the quantitative statistical analysis method cannot clearly grasp the law of uncertainty transmission and may cause the consequence of \"excessive estimation\" of uncertainty components. Moreover, due to the complexity of error sources, it is difficult for general surveyors to ensure that no uncertainty source is repeated or missed in the analysis of error sources. The most obvious problem is that currently most of the relevant researches based on error traceability use indication errors to quantize individual points in the CMM space, and the indication errors essentially limit the measurement results, so the quantification of uncertainty in single point measurement is in itself characterized by the problem of \"over estimation\". The measurement system analysis method widely used in product quality management has summarized the characteristic indexes of the measurement of the whole measurement system as resolving power, bias, linearity, stability, repeatability, and reproducibility, which can comprehensively reflect the impact of uncertain system and random errors of the measurement system on the measurement results; the uncertainty modeling method based on statistics and analysis of the measurement characteristics has given the guiding ideology for evaluating the CMM task-oriented measurement uncertainty from the top design. The measurement task of dimensional and geometrical errors mainly includes two essential problems, namely, how to extract the actual elements and how to evaluate the ideal elements. The former determines the detection method while the latter depends on the evaluation method. The new generation of uncertainty evaluation theory has summarized the uncertainty caused by the evaluation method into the category of \"normative uncertainty\". Therefore, the influence of the method for ideal element evaluation is not considered in the evaluation of \"measurement uncertainty\", which is particularly important for evaluating the measurement uncertainty of geometrical errors [31]. Thus, the sources of the CMM task-oriented uncertainty can be analyzed as follows:  
(1) Uncertainty components caused by bias and linearity  
The influence of bias and linearity on the CMM measurement results is reflected in the uncertainty component \(u_{\text{E}}\) caused by the indication or detection errors of the measuring instrument. For the purpose of safety, the \"overestimation\" should be adopted, with the CMM \(E_{\text{L, MPE}}\) and \(\text{MPE}_{p}\) used to quantize \(u_{\text{E}}\). In calibration, the CMM indication or detection errors have considered the influence of such factors as probe configurations of the measuring instrument, method for coordinate system establishment, measuring object clamping, space position and environment. Therefore, when \(E_{\text{L, MPE}}\) and \(\text{MPE}_{p}\) are used to quantize \(u_{\text{E}}\), the influence of the above uncertainty sources is also included. (2) Uncertainty components caused by resolution and repeatability  
There is a certain correlation between the resolution and repeatability of the instrument, so CMM only needs to consider the uncertainty component \(u_{\text{r}}\) caused by repeatability.