In this paper is presented an original approach to the definition of the metrological characteristics of the measurement systems used in process index monitoring. The model proposed allows the evaluation of the probability that the process under statistical control shows a different behaviour from that inferred by the measured data, relating to the input conditions. The proposed model is developed in general hypothesis on the statistic characteristics of the process index monitoring and a typical application cases is presented.

Linear Regression Analysis (LRA) is a technique commonly applied in many different branches of science. The present study investigates the use of LRA in Metrology and aims to develop a mathematical approach to adequately take into account its contribution for the uncertainty budget in a measurement.
In a calibration involving many standards and measuring instruments, the LRA technique is an important tool for the estimation of conventional true values based on certificate results. This statistical treatment usually intends to reduce the errors measured in the calibration process in order to achieve lower residual errors. The operation, however, introduces statistical uncertainties, which can be of significance when compared with the uncertainty contributions from other input quantities.
This document also presents the results of a measurement uncertainty evaluation related to the calibration of a length measuring machine, including the LRA contribution based on the application of the mathematical expression proposed. The relative influence of this contribution is also investigated.

Co-ordinate measuring machines (CMMs) are often used for calibrating different kinds of squares. A procedure for calibrating squares on the CMM Zeiss UMC 850 was developed in our laboratory to cover primarily industrial needs. The uncertainty analysis that was performed for this procedure is introduced in the article. The most important uncertainty contributions were evaluated experimentally. Many experiments were also performed in order to find the most appropriate calibration position in the CMMs measurement space. About 2000 measurement points were taken in order to get reliable results for uncertainty components. The analysis results expressed as the best measurement capability were checked by participation in the Euromet 570 project, which has not been finished yet. The current value of the best measurement capability is 0,9 arc sec.

In practice, quantization of a measured quantity often significantly influences observation values. A typical example is found in measurements using digital instruments. In some cases, due to the quantization, no dispersion is observed among repeated measurements. The type A evaluation then gives zero standard uncertainty. In such a case, the most common practice is to assume, as an a priori distribution in type B evaluation, a uniform distribution, the width of which is given by the quantization interval, and take the width divided by square root of 12 as the standard uncertainty.
This practice, however, is justified only when the population standard deviation is exactly zero. But generally this condition does not hold true even if the sample standard deviation appears to be zero. In the present study, we use the Bayesian approach to evaluate the uncertainty of a measurement based on quantized data with due consideration to the difference between the standard deviation of the apparent sample and the population standard deviation.
We assume that the quantity before quantization obeys a normal distribution having the average µ and standard deviation σ. A measurement data corresponds to a value of the quantity after quantization. Based on a specific combination of n repeated measurements, we can construct the probability density p(µ, σ) using the Bayesian method. The standard deviation of the function, p~(µ) = ∫p(µ, σ)dσ , in terms of µ gives the uncertainty of the measurement result. We have shown that when all of the measurement data take the same value, the conventional type B evaluation described the above results in an underestimate of the uncertainty, if the number of data is less than five. Analysis is also conducted in cases in which not all of the data take the same value.

It is possible to compute uncertainty in the form of confidence intervals. In this article is exploited the confidence intervals determination through a simulation study that enables to evaluate uncertainty in the form of confidence intervals for real measurements of inter-laboratory comparison (ILC) even for small numbers of observations.
Here are listed estimation approaches (mathematical-statistical algorithms) for the determination of the consensus value (the true measured value); confidence interval for the true measured quantity in different laboratories; determination (estimate) of the inter-laboratory variance; confidence interval for the inter-laboratory variance; and within-laboratory variance in experiment of the inter- laboratory comparison with homoscedastic as well as heteroscedastic measurements. Different possibilities of evaluation of ILC when the model applied is a linear model with one random effect "laboratory" and estimation procedures are listed and discussed (also describing of the statistical features) in this contribution.
The merit of the simulation study is for a statistician (evaluator of ILC) in better approximation of needed confidence level to obtain the expected result precision in balanced and unbalanced experiment design for homoscedastic as well as heteroscedastic measurements when having small number of observations.