When the Start Simulation button is pressed, a bar can be seen under the button that continuously indicates the progress of random sampling. After completing the simulation with the indicated data, the application is automatically located in the results panel.

This panel shows a graphic selector at the top left, which allows to toggle between the results of the Monte Carlo simulation and the analysis according to the GUM approach, which is also estimated as well in all cases.

Next, we analyze the information provided by the MCM results view:

1. Name of the application.
2. Software version.
3. Time involved by the application in obtaining results (in mm.ss, minutes and seconds)
4. Technical data, with the text that we indicated in Step 1
5. Number of iterations. 500 000 in our case.
6. Statistical analysis of simulation
   1. Media = 100.04111
   2. Variance = 1.947357e-4
   3. Standard deviation = 1.39548e-2
   4. Skew = -1.01559e-3
   5. Kurtosis = 2.64392
   6. Maximum value = 100.11192
   7. Minimum value = 99,97472
   8. Median = 100.0396
   9. Range = 0.1372
7. Normality Test (Jarque – Bera) = 2641.5964 (does not fit)
8. Result = 100.04111. This indicates the result of the calibration without rounding figures.
9. Confidence interval (p = 95.45%): [100.01376, 100.06841] (Half-width of the interval = 2.733e-2)
10. Classic format: 100.041 ± 0.027 ml. This represents the test result and its associated expanded uncertainty for the coverage probability indicated in step 3. Note that this is not the result from GUM framework but the MCM results expressed as usual in GUM framework
11. List of contributions to the uncertainty for each parameter of the model.
12. Section with input data and mathematical model describing completely what was entered when creating the simulation project.

More help

• First project – Step 1
• First project – Step 2-A: Basic model
• First project – Step 2-B: Input uncertainties
• First project – Step 2-C: Final model
• First project – Step 3.
• First project – Results

Simulation data

The only thing left for us to do in our first project is to assign a probability distribution function to each random variable that represents our project. As seen in the program, the grid is composed of several rows, some of them with a white background, and others with a gray background. The rows that have a gray background correspond to magnitudes for which a distribution function can not be assigned, because they are intermediate results or the final result, that is, magnitudes that will be obtained in the simulation as a result of secondary equations.

The lower part of the work panel has a command box that will give us a dropdown list of probability distribution functions so that we can select the one that fits the input magnitude. After we have selected a probability distribution, the lower box will ask us to type the necessary parameters to perform the simulation. Therefore the process of this step would be as follows:

1. It is recommended to leave the number of iterations in 500 000 since it is obtained in excellent performance in the application and the results are absolutely reliable with this number of iterations (can be consulted the paper: Computational Aspects in the Estimation of Test Uncertainties by the Monte Carlo Method (Spanish only)
2. We select the probability of coverage for the results. It is recommended to use 95.45% in order to obtain results for K = 2.
3. We click on the first row with white background (or we reach it with the cursor keys of our keyboard)
4. We select a probability distribution function in the drop-down list.
5. We fill the simulation parameters in the lower panel (please, take units into account).
6. We click on Apply. If everything is correct, the row will have a green background, indicating that the magnitude has its simulation data correctly assigned.

Going back to our volumetric flask we will have so:

1. V20 : Disabled because it is Result, it is not possible to assign an PDF(Probability Distribution Function)
2. Ml : Disabled because it is Intermediate result, It is not possible to assign a PDF
3. Mv : Disabled because it is Intermediate result, It is not possible to assign an PDF
4. Dens_w : We will assign the PDF Constant with a Value = 0.99829 (g / ml)
5. Dens_a : Assign the PDF Constant with a Value = 1.2E-3 (g / ml)
6. Dens_b : We will assign the PDF Constant with a Value = 8,000 (g / ml)
7. CDT : Assign the PDF Constant with Value = 3.3E-6 (1 / ºC)
8. t : Disabled because it is Intermediate result, It is not possible to assign an FDP
9. Ml_cal : Due to an expanded uncertainty, we will assign the PDF Normal with the average value of our readings, Mean = 162,416, entering the certificate information in “Use certificate “, with uncertainty = 0.0047 (g) and k = 2
10. SM_res : We will assign a PDF Rectangular , with Mean = 0 (corresponding to all the variables that are entered only for the purpose of estimating uncertainties) and Half interval = 0.0005 , that is, half of the division of the digital weighing scale.
11. SM_rep : For repeatability, MCM Alchimia provides an experimental PDF where we can directly put the measured values ​​and the application will be responsible for making the statistical calculations necessary for us, to use for simulation. As it is only for uncertainty purposes we will have to select “Force Mean = 0”. We will use the “Direct” option and clicking on the Values ​​button we will enter the 5 readings of our essay: 162,384; 162,431; 162,409; 162.417; 162,439
12. Mv_cal : Due to an expanded uncertainty, we will assign the PDF Normal with a mean in the reading of the balance in order to weigh the empty flask, Mean = 62.651, entering the certificate information in “Use certificate”, with uncertainty = 0.0047 (g) and k = 2
13. SMv_res : We will assign a PDF Rectangular , with Media = 0 and Half interval = 0.0005.
14. ti : Disabled because it is Intermediate result, It is not possible to assign a PDF
15. tf : Disabled because it is Intermediate result, It is not possible to assign a PDF
16. corr_t : We will assign the PDF Constant with Value = -0.022 (ºC)
17. ti_cal : Because it comes from a calibration certificate, we will assign the PDF Normal with the value of our reading of the mean thermometer = 20.05. At the same time we will select “Use certificate” and we will put the expanded uncertainty = 0.021 (ºC) and k = 2.
18. Sti_res : For this measure we use a mercury thermometer in division glass: 0.1ºC, from which we can visually estimate 1/4 of the division. According to this, we will assign a Triangular distribution with Mean = 0 and half-interval = 0.125 (this is estimate / 2)
19. tf_cal : The same is the case as in ti_cal but now our average will be the final temperature: Mean = 20,075, select “Use certificate” and indicate the expanded uncertainty = 0.021 (ºC) and k = 2.
20. Stf_res : Same as Sti_res, that is, Triangular with Media = 0 and semi-interval = 0.125.

When entering data for the last magnitude, the “Run the simulation” button will light up so that we can run our simulation and obtain the results. Watch results after simulation .

More help

• First project – Step 1
• First project – Step 2-A: Basic model
• First project – Step 2-B: Input uncertainties
• First project – Step 2-C: Final model
• First project – Step 3.
• First project – Results.

Determination of the final mathematical model for the test.

In the previous article we saw that our flask calibration test could be represented by the following diagram;

Breaking down the input magnitudes in their contributions we would get:

• Ml = Ml_cal + SMl_res + SMl_rep, where the suffixes “cal”, “res” and “rep” will be by calibration, instrument resolution and repeatability respectively. The prefixes “S” indicate that this component will have zero value since it is added only for the purpose of evaluation of uncertainties.
• Mv = Mv_cal + SMv_res
• The temperature, as we said before will be obtained from an average, so we can break down the equation in this average of readings. On the other hand, this thermometer has corrections. We could add this correction value (corr_t) to the average as a constant value without uncertainty, since this will be associated to the original values ​​of temperature.
  t = (ti + tf) / 2 + corr_t
• but each one of these values ​​will be affected by the calibration and resolution of the thermometer. So:
  ti = ti_cal + Sti_res
  tf = tf_cal + Stf_res

Therefore to conclude this step 2:

1. We write the complete model in the text area for the Set of equations:
   V20 = ((Ml-Mv) / (Dens_w-Dens_a)) * (1- (Dens_a / Dens_b)) * (1-CDT * (t-20))
   Ml = Ml_cal + SMl_res + SMl_rep
   Mv = Mv_cal + SMv_res
   t = (ti + tf) / 2 + corr_t
   ti = ti_cal + Sti_res
   tf = tf_cal + Stf_res
2. We fill the parameters grid with the following data:

More help

• First project – Step 1
• First project – Step 2-A: Basic model
• First project – Step 2-B: Input uncertainties
• First project – Step 2-C: Final model
• First project – Step 3.
• First project – Results.

Analysis of input values and their uncertainties

At this stage it is advisable to analyze the magnitudes that make up the input model to determine the contributions of uncertainty that they contribute. The advisable thing in this case is to define a model that represents as faithfully as possible the development of our analysis or trial, so that the contributions of uncertainty are adjusted to those that will really represent the model. Although we will then propose a way to do it, it is understood that each expert will do it in their own way and, therefore, the different processes will involve different final mathematical models for the trial. It is not the purpose of this aid the background discussion on volume metrology but for a given test model, finding a way to represent it faithfully.

Thus, we are assuming for our analysis that the characteristics of the test will be:

• The flask is initially washed, dried in an oven, allowed to cool and weighed once at 20 ° C on a digital scale. This magnitude, then, will mean contributions of uncertainty for the calibration of the balance
• The mass of the flask filled with water will be determined by repeating 5 times the measurement, weighing on a calibrated scale. Therefore this magnitude will bring uncertainty due to the calibration of the scale, its resolution and the repeatability of the measurement.
• The temperature of the test water will be taken at the beginning and at the end of the test with a mercury glass thermometer, the average of these two measurements being used for the calculation. The contributions will then be for the calibration of the thermometer, the division thereof and for the drift found in the measurement throughout the test.
• The other magnitudes (densities and coefficient of expansion) will be taken from tables and will be assumed constant in this opportunity (without contribution of uncertainty)

Therefore, the Cause-Effect diagram (Ishikawa) for our essay will be:

To write the complete model, then you should break down the basic magnitudes in their components, adding new lines to the model. This will be discussed in the next stage by clicking this link .

More help

• First project – Step 1
• First project – Step 2-A: Basic model
• First project – Step 2-B: Input uncertainties
• First project – Step 2-C: Final model
• First project – Step 3.
• First project – Results.

Determination of the mathematical model of the trial

In this step two we will define the mathematical model of the essay. The ISO 4787 guide details two calculation methods to obtain the volume contained in a volumetric flask. The first one indicates the calculation of the volume integrating all the magnitudes participating in the model into the equation. For a test, particularly for a defined temperature, a defined building material (glass), etc. Many of these values, such as water density and coefficient of expansion, can be considered constant, so this standard also proposes a tabulated factor to simplify the calculation. In this case we will use the complete equation to study the variables and their associated uncertainty. The equation (test model) will be:

The equation editor of MCM Alchimia is an ASCII editor that does not have special characters for which the Greek letters will have to be replaced by a common variable name. The main equation that will be put in Set equations will be V20 = ((Ml-Mv) / (Dens_w-Dens_a)) * (1- (Dens_a / Dens_b)) * (1-CDT * (t-20) ) where:

V20: Volume of the flask at 20ºC.
Ml: Mass (read on the scale) of the flask flush with detilated water (g).
Mv: Mass (read on the scale) of the empty flask (g).
Dens_w: Density of the water used in the calibration (g/ml).
Dens_a: Air density (g/ml).
Dens_b: Density of the masses with which the scale has been adjusted (g/ml).
CDT: Coefficient of cubic thermal expansion of the flask material (in 1/ºC).
t: Test water temperature (ºC).

It is advisable to make a deeper analysis of the sources of uncertainty of the input quantities of the model. In this way, if any magnitude has more than one source of uncertainty, it will be broken down into a sum of as many variables as there are sources of uncertainty, one of them will take the value of the magnitude and the rest will have zero value since they are added only for the purpose of evaluation of uncertainties.

More help

• First project – Step 1
• First project – Step 2-A: Basic model
• First project – Step 2-B: Input uncertainties
• First project – Step 2-C: Final model
• First project – Step 3.
• First project – Results.

Several estimations of uncertainty made with MCM Alchimia for advanced models of tests, calibrations or analytical techniques, can be consulted in several published scientific works, both by the software developer, as well as by other researchers in various branches of science. However, for a first contact with the application, in this document we will make a first simple calculation project to know the operation and the possibilities of the application. As an example, we will perform the calculations relevant to the calibration process of a calibration calibrated by the gravimetric method, consistent with the guide ISO 4787: 2010 Laboratory glassware – Volumetric glassware – Methods for use and testing of capacity

• Run the application from the desktop. Estra will already open in step 1.
• We write in the field Title : Calibration of a volumetric flask according to ISO 4787: 2010
• In the field Technical data we replace the pre-existing text with the technical description of our essay. There we can write the following text:
  This project consists of the evaluation of the calibration uncertainty of a 100 ml volumetric flask by the gravimetric method with ISO 4787: 2010. For this, a calibrated 1 mg division scale is available, with its calibration certificate, which tells us that if we assign the maximum correction of the reading in the working range to the uncertainty, it results in an expanded uncertainty of 0.0054 g for k (coverage factor) = 2.
  We also have a mercury glass thermometer of 0.01ºC, which also has a calibration certificate that indicates that at 20.00ºC it has an error of -0.022ºC and an expanded uncertainty of 0.012ºC for k = 2.
  The flask will be completely dried in an oven, cooled to 20°C, filled with distilled water and then weighed on the scale. This operation will be performed 5 times, taking the 5 mass values. To the average of these 5 mass values ​​and the temperature of the water filling, the equations of the standard will be applied to convert the mass value in Volume contained by the flask to 20ºC
• Having entered this data we can continue to the next step. By clicking on the link or the Step 2 button.

More help

• First project – Step 1
• First project – Step 2-A: Basic model
• First project – Step 2-B: Input uncertainties
• First project – Step 2-C: Final model
• First project – Step 3.
• First project – Results.

Although the MCM Alchimia interface has been designed in a modern way, it is also simple and intuitive enough so that the estimation of uncertainties of the test models, can be carry out by technicians who do not necessarily have computer or statistical skills. To facilitate access to the different software features, this interface was designed sequentially and with almost no nested menus. The interface consists of the following sections:

1 .- Main menu: This menu consists of only 6 buttons that manage the files and documents of the program as well as the language modules:

	Save: Allows you to update changes in a previously saved model. If you are working on a new model, it produces an effect identical to the Save as button.
	Save as: Create a new model file with the information added in the screen. The saved model will contain the model information and input quantities with information about its probability distribution function. This option does NOT allow to save curves connected to the model.
	Load model: loads the information of a previously saved model.
	Print results: Send the input data and results of the simulation project to the printer.
	Change language: This option allows you to change the active language of the application’s interface. The detailed procedure for changing the language of the application can be found by following this link .
	Info: Display property data of the application.

2 .- Application bar: : It fulfills the classic functions of the Windows title bar. From this area you can move the MCM Alchimia window through the screen.On the right contains Minimize, Close and Help icons.

3 .- Sequential menu: It determines the work flow recommended for the preparation of a project to estimate measurement uncertainties. While the application allows you to go from one step to another just by pressing the button, it is highly recommended to complete the fields, text areas and drop-down menus each step. In this way, you can not only obtain reliable results, but also reuse the project saved in the future, easily referencing the input data.

• Step 1 .- Project information. This panel allows to indicate a title and the technical description of the project. These fields are part of the final results report that can be sent to the printer, so it is advisable to complete these fields.
• Step 2 .- Model definitionThe upper field of this panel is an equation editor where you must indicate the mathematical model of the test. To break down the contributing parameters of the typed model just click on the Update Parameters link and the application will divide the mathematical model you typed, automatically filling the lower list of that panel, the one of step 3, and correlations matrix with null values.
• Step 3 .- Simulation data. In this panel, the operational characteristics of the simulation are determined. The number of iterations indicated here will determine the size of the vector of pseudo-random values that each variable will contain and, consequently, the amount of output values obtained for the measurand. At the top right, the probability of coverage for the uncertainty analysis may be chosen and in the lower part of the panel, the probability distributions for each contributor are indicated.
• Step 4 .- Correlations matrix. This panel allows the analysis of models with correlated variables, taking into account covariances between them. The parameters of the correlation matrix are automatically filled when the model is defined. The correlation values must be typed manually with values between -1 y 1.
• Step 5 .- Results. After executing the simulation, a list of results and statistical data of the vector is obtained. This panel opens automatically after finishing the simulation.
• Regression data. This version of MCM Alchimia includes support for regression and curves (calibration curve / linear regression), in order to use an interpolation of the curve (in both directions) in the math model of the test.

4 .- Work panel: This is the area of the screen where information is exchanged with the user.

5 .- Simulation button: This button remains inactive until all the necessary data for performing the simulation have been fulfilled. When the probability distribution functions have been indicated for all the parameters of the model, this button lights up allowing to run the simulation and obtain the result of the calculation with its associated uncertainty.

More help

• Installing MCM Alchimia
• Activate the application
• Set your default language
• How to understand the application interface
• Tips and tricks to become an expert in MCM Alchimia

MCM Alchimia can be downloaded in English and Spanish versions by clicking download buttons on the application’s website. Nevertheless, at any time, you can download a new language module (or even generate a new language online) and install it in the application:

• Check if there is a plugin for the language you are looking for, following this link.

1. If the language you are looking for exists, you can download it by clicking on the link with the name of the language.
2. If the language you are looking for does not exist, you can generate it by translating manually the terms from English to your language using the form below. The terms that are not translated will remain in English in the application.
3. By pressing the Submit button at the end of the form, a language file with the extension “.mcl” will be downloaded.

• Click on the button Install language file
• Select the downloaded language file and click Open
• Then MCM Alchimia offers you to use this language as default.If you wish so, click accept, otherwise the software language will be changed back to default language on the next run.

NOTE: Does the language file correspond to an earlier version?
As new software is updated, new terms may appear or change, on account of new specifications or possibilities of the software. In these cases it is possible that the language file has not been updated yet containing fewer terms than necessary.
What to do: In these cases the software automatically solves these problems by adding the missing terms in English. You only need to press Accept in that window and continue.

More help

• Installing MCM Alchimia
• Activate the application
• Set your default language
• How to understand the application interface
• Tips and tricks to become an expert in MCM Alchimia

The licenses of MCM Alchimia have a duration of 1 year and they are totally free.The first time you run the application after installation, you will see a screen with two fields called Short key and Long key. The same will happen after one year using the application with those keys. To activate MCM Alchimia follow these steps:
Run the application. A window with graphic indications to activate the software will be displayed

1. Click on the indicated link to get a free featured one-year licence.
2. If the previous action does not open the web browser on the MCM Alchimia page, follow this link.
3. Copy the short key text in the web page and paste it in the namesake field in the application. Next, do the same with Long key text.
4. Click on Continue button
5. If a message appears saying Successful !!!, that indicates that the program is activated correctly. Now you can press the button Close to start using the application or just wait 5 seconds.
6. If an error is raised, refresh the web page (keys will updated as well) and re-copy the keys in the application’s fields.

More help

• Installing MCM Alchimia
• Activate the application
• Set your default language
• How to understand the application interface
• Tips and tricks to become an expert in MCM Alchimia