**4. Examples**

#### *4.1. Univariate Real Quantity*

To illustrate the ability of DCCs to capture uncertainty information obtained from MCM, consider the example of gauge block calibration from clause 9.5 of GUMS1 [7]. In this example, the length of a gauge block of nominal length 50 mm is determined by comparing it with a known reference standard that has the same nominal length. The measurand is the deviation from the nominal length.

The measurand is expressed as an explicit function of nine input quantities. The probability distributions assigned to the input quantities comprise scaled and shifted t-distributions, a rectangular distribution, a normal distribution, an arc sine distribution and rectangular distributions with inexactly prescribed limits.

For this paper, the calculation stage has been implemented using a fixed number *M* = 106 of trials (c.f. the implementation in GUMS1 where an adaptive approach was used). Using the component structure of Table 3, summary information from the calculation stage can be encapsulated as follows:

```
<!-- MCM, 1e6 samples - Summary information -->
<si:real>
  <si:label>Deviation from nominal length</si:label>
  <si:value>838</si:value>
  <si:unit>\nano\metre</si:unit>
  <si:coverageInterval>
    <si:standardUnc>36</si:standardUnc>
    <si:intervalMin>745</si:intervalMin>
    <si:intervalMax>932</si:intervalMax>
    <si:coverageProbability>0.99</si:coverageProbability>
  </si:coverageInterval>
</si:real>
```
Using the component structure of Table 8, the full set of values *yk*, *k* = 1, ... , *M*, of the output quantity returned by the Monte Carlo calculation can be encapsulated as follows, showing only the first three (*k* = 1, 2, 3) and final three (*k* = *M* − 2, *M* − 1, *M*) values:

```
<!-- MCM, 1e6 samples - Output quantity values -->
<si:list>
  <si:listUnit>\nano\metre</si:listUnit>
  <si:real>
    <si:value>829.5221</si:value>
  </si:real>
  <si:real>
    <si:value>873.3864</si:value>
  </si:real>
  <si:real>
    <si:value>822.9225</si:value>
  </si:real>
    ...
  <si:real>
    <si:value>825.8857</si:value>
  </si:real>
  <si:real>
    <si:value>862.1964</si:value>
  </si:real>
  <si:real>
    <si:value>798.6789</si:value>
  </si:real>
</si:list>
```
As discussed in Section 1, it would be impractical to generate a paper certificate that contains such a large number of numerical values. When written to file (without any spaces or indentation), the full set of values in the format above takes up approximately 48.6 MB (and requires 3*M* + 3 lines). For comparison, were the representation that allows the same unit of measurement to be assigned to all quantities not available, using the component structure of Table 4 would lead to a file of approximate size 78.2 MB (4*M* + 2 lines).
