Confidence interval expanded

This confidence interval arises from the variance in the calibration line and instrument response to the test sample only. Any other uncertainties must be combined to give the overall confidence interval (expanded uncertainty). The form of equation 2.43 is instructive. The standard deviation of... 

For a measurement result to be metrologically traceable, the measurement uncertainty at each level of the calibration hierarchy must be known. Therefore, a calibration standard must have a known uncertainty concerning the quantity value. For a CRM this is included in the certificate. The uncertainty is usually in the form of a confidence interval (expanded uncertainty see chapter 6), which is a range about the certified value that contains the value of the measurand witha particular degree of confidence (usually 95%). There should be sufficient information to convert this confidence interval to a standard uncertainty. Usually the coverage factor ( see chapter 6) is 2, corresponding to infinite degrees of freedom in the calculation of measurement uncertainty, and so the confidence interval can be divided by 2 to obtain uc, the combined standard uncertainty. Suppose this CRM is used to calibrate... 

In Figure 3.13A the data of 95% confidence limits are presented curve-fitted to a Daphnia toxicity test. Test concentrations are from very high to very low with no replication. The confidence interval at the EC, and EC20 are relatively low in each instance. In contrast, Figure 3.13B has fewer test concentrations for replicates. The test concentrations do not extend to levels corresponding to EC20. Note that the confidence interval is very narrow within the area of the graph represented by data. However, as extrapolation is required at lower concentrations, the confidence interval expands. 

The process of providing an answer to a particular analytical problem is presented in Figure 2. The analytical system—which is a defined method protocol, applicable to a specified type of test material and to a defined concentration rate of the analyte —must be fit for a particular analytical purpose [4]. This analytical purpose reflects the achievement of analytical results with an acceptable standard of accuracy. Without a statement of uncertainty, a result cannot be interpreted and, as such, has no value [8]. A result must be expressed with its expanded uncertainty, which in general represents a 95% confidence interval around the result. The probability that the mean measurement value is included in the expanded uncertainty is 95%, provided that it is an unbiased value which is made traceable to an internationally recognized reference or standard. In this way, the establishment of trace-ability and the calculation of MU are linked to each other. Before MU is estimated, it must be demonstrated that the result is traceable to a reference or standard which is assumed to represent the truth [9,10]. 

However, the GUM [Guide to the Expression of Uncertainty of Measurement approach (ISO 1993a), which leads to the verbose statement concerning expanded uncertainty quoted above, might not have been followed, and all the analyst wants to to do is say something about the standard deviation of replicates. The best that can be done is to say what fraction of the confidence intervals of repeated experiments will contain the population mean. The confidence interval in terms of the population parameters is calculated as... 

When n becomes large the t value tends toward the standardized normal value of 1.96 (z = 1.96), which was approximated to 2 above. The 95% confidence interval, calculated by equation 2.13, is sometimes explained much like the expanded uncertainty, as a range in which the true value lies with 95% confidence. In fact, the situation is more complicated. The correct statistical statement is if the experiment of n measurements were repeated under identical conditions a large number of times, 95% of the 95% confidence intervals would contain the population mean. ... 

The major components of uncertainty are combined according to the rules of propagation of uncertainty, often with the assumption of independence of effects, to give the combined uncertainty. If the measurement uncertainty is to be quoted as a confidence interval, for example, a 95% confidence interval, an appropriate coverage factor is chosen by which to multiply the combined uncertainty and thus yield the expanded uncertainty. The coverage factor should be justified, and any assumptions about degrees of freedom stated. 

As described already, the expanded uncertainty of a consensus value is often calculated as the 95% confidence interval, which entails dividing the standard deviation of the laboratory means by the square root of n, the number of laboratories. Whilst this is an approach suggested in ISO Guide 35 when individual laboratory uncertainties are not available, if the number of participant laboratories is large, the uncertainty estimate could perhaps become unrealistically small. In such circumstances it may be necessary to limit n to some upper value, regardless of the actual number of data points, although currently there appear to be no recommended procedures for this. This is an issue that could usefully be considered in future interlaboratory certification activities. 

Statistical evaluation of results (detection and rejection of outliers calculation of the means, standard deviations, confidence intervals, combined expanded uncertainties, etc.)... 

The expanded uncertainty can then be estimated after the calculation of the effective number of degrees of freedom, df (Eq. 7). Therefore the coverage factor used was the Student t defined for that number and a 95% significance level (/(df, 95%). The estimated confidence interval is defined by... 

Fig. 2 Repeatability test. The confidence intervals are represented by the average value plus the estimated expanded uncertainty for a 95% confidence level...

The analysis of experimental data shows that the average value 111 g/m2 of all corrosivity data (improved by rejecting outliers) corresponds to the value 140 40 g/m2 indicated in the standard. For the evaluation of the expanded combined uncertainty U with factor k=2 the corrosivity measurement gives the value of 215 g/m2 (at 95% confidence). It means that our data uncertainty is five-times higher than that specified in the standard as the data scattering interval 40 g/m2 and seven times as wide compared the statistic confidence interval in our own experimental data corrosivity (Table 2a and 2b). The main components of the combined uncertainty are mass loss and surface area determination. 

Each particular value found outside the plausibility bounds was manually inspected in its measurement context in order to decide between alternative decisions to be taken discard the value as wrong, correct the value for obvious reasons, or expand the confidence interval. The percentage of discarded data was surprisingly high with regard to the fact that the data were already quality-checked before. Frequent typical cases were data on land, obviously invalid, or erratic data, temperatures significantly below the freezing point and extremely low salinities likely measured inside fjords or near the shore. This detailed and careful procedure took most of the efforts in the BALTIC project development from 2000 to 2007. 

For a determined concentration of 200 pg/ml, the expanded uncertainty would be f/(y) = 200 x 0.292 = 58. This result denotes that, for a result of 200 pg/ml, the concentration of ochratoxin A would be expressed as ochratoxin A (200 58) pg/ml, where the stated uncertainty is an expanded uncertainty calculated using a coverage factor of 2 that corresponds approximately to the 95% confidence interval. ... 

Figure 14.1 Compass plots for penetrants in propylene glycol mixtures in PSFT. The interactions noted in this plot at left for triazine and phenol were positive and outside of the upper confidence interval for significant interactions (p <.05). One cell Is expanded to illustrate upper and lower bounds.

The parameter Cj for the point j is unity when the deviation between the value, in this case pressure, calculated from the equation of state py calc, T (meas), p (meas), n and the experimental value p jmeas, T (meas), p (meas) is equal to the experimental uncertainty Opj. Typically, for a reference equation of state, the should be less than unity for almost all data (typically > 95 % of the points if a is considered to be equal to two times the standard deviation as appropriate for an expanded uncertainty at a confidence interval of 0.95). In eq 12.1 the calculated pressure depends on the parameter vector n, thus on the coefficients of the equation of state that are fitted. In practice, the dimensionless compression factor Z is commonly used instead of the pressure. Thus, the residual becomes... 

The comprehensive certificate that accompanies each SRM 2100 set lists the fracture toughness for the set as well as the uncertainty associated with the estimate. For each billet the mean fracture toughness and the scatter in results as measured by the three test methods were statistically indistinguishable. The data were therefore pooled for each billet. The certified average fracture toughness in Table 6 is the grand mean of the pooled NIST database. The uncertainty U (with the subscript 1) is a 95 % prediction uncertainty for a single future observation and is based on the results of the NIST observations from the same normally-distributed population. The uncertainty Um (where the subscript m denotes the mean) is a 95 % confidence interval for the mean of five future observations, also based on the results of the NIST independently and randomly selected observations. The expanded uncertain-... 

The expanded uncertainty U is necessary in order to give a measure of uncertainty that define an interval about the measurement result within which the value of the measurand is confidently believed to lie. U is obtained by applying to the combined standard uncertainty Uc y) a coverage factor k, i.e. U = Uc y). The selected coverage factor is fc = 2 which defines an interval with a level of confidence of approximately 95%. Tables 3 and 4 show the expanded uncertainties of the 3D coordinates of the points in the spherical models obtained with the TLS equipment considered in this study both in field and lab conditions. 

Quantity defining an interval in which the true value of the measurand may fall with a specific level of confidence. Numerical factor for multiplying the combined standard uncertainty in order to obtain the expanded uncertainty. 

---