2.4.3. TAB “Values, Uncertainties”¶

A major part of this TAB consists of a 10-column table for the input of measurement values and uncertainties of the independent measured quantities (input quantities). The columns are:

1:	Symbol	text field	formula symbol
2:	Type	text field	Parameter type: a for dependent, u for independent, m for a quantity mean, p for a parameter to be calculated, without uncertainty
3:	Unit	text field	input of unit
4:	Value	number field	value of measured quantity
5:	distribut.	select. field	selection of the quantity`s distribution (normal / rectangular / triangular / others) The Low-Level Applications, (N+x)-rule may be selected here, implying Gamma distributions of the associated counting rate variables
6:	StdDev formula	text field	formula of the standard deviation of the quantity; no formula if (N+x) rule has been selected; (the internal coverage factor is always 1); always “.” has to used for the decimal point
7:	StdDev value	number field	value of the uncertainty for normal distribution if the (N+x) rule has been selected, nothing shall be entered these cells!(the internal coverage factor is always 1)
8:	Half width	number field	Half width of rectangular/triangular distribution (the internal coverage factor is always 1)
9:	abs./rel.	select. field	select whether the uncertainty from col. Sp. 6, 7 or 8 are to be taken absolute or as relative value.
10:	abs. std. Unc.	number field	(combined) absolute standard uncertainty calculated by the program form the values of the columns 6, 7, 8 and 9; Note: a value entered by the user will always be over-written by the program!

The columns 1 to 3 have been taken from the preceding symbol list and are disabled here.

For the user’s support those cells in the table are colored with red background which are not filled in by the user: these are cells belonging to dependent quantities. Only cells with white background should be used for direct input by the user. That means, the rows belonging to dependent quantities are found in the uppermost part of the table, the first row (row 1) is the one for the output quantity, or the rows 1-3 in the case of three output quantities.

Special feature of non-normal distributed numbers of counts and count rates: :ref:`` .

For a data set-based quantity, mean and uncertainty are calculated internally from the associated data set and transferred to the “values, uncertainties” table with the button “Calculation of uncertainties”. The corresponding rows in that table therefore have also a red colored background.

The “white cells” in columns 4 to 9 must be filled in by the user. Numerical values must be input into the column “value” while this is not always necessary in columns 6 to 9 for each of the quantities. One can often abstain from giving uncertainties, for instance for counting times. The input of uncertainties is allowed in only one of the columns 6, 7 or 8. The selectable field “abs./rel.” however must be used, if in the columns 6 to 8 an entry exists.

For counting rates or numbers of counts the standard deviation can be input as a formula in column 6 (“StdDev formula”) - without preceding equal sign. For quantities declared as normal distributed - also possible for counting rates/numbers of counts - the numerical value of the uncertainty is input in column 7 (“StdDev value”). In the case of rectangular and triangular distributed quantities the uncertainty is characterized by the value of the half-width in column 8. The latter value is internally converted to a standard deviation (normal distribution) according to the GUM rules which then is transferred to the cell in column 10 (“abs. std.Unc.”).

One cell within the column “StdDev formula” is highlighted by green colour. Here, the standard deviation formula of the gross counting rate must be given. This formula is required later for the numerical-iterative calculation of the Decision threshold and the Detection limit. It is considered as “uncertainty function” of the gross counting rate (and implicitly of the (procedure dependent) net counting rate) which allows estimating its uncertainty for any varied (iterated) value of the gross counting rate. It has, however, no meaning for the calculation of the combined uncertainty of the output quantity.

Possible formulae of the standard deviation of the gross counting rate Rg (incomplete):

sqrt(Rg/tm) counting devices (counting time tm), single measurement

sqrt(Rg/tm/n) counting devices (counting time tm), n-times repeated measurement

sqrt(Rg/2/tau) ratemeter measurements (time constant tau)

If in addition to the gross count rate Rg the gross count number Ng is also used, e.g., by defining an equation Rg=Ng/tm, one should have in mind that the program also modifies Ng if Rg is modified. From this equation, Rg=Ng/tm, UncertRadio can identify the count number Ng belonging to Rg. This ensures, that the program, during evaluating the equations from the bottom upwards (Ng then is an independent quantity and Rg is dependent), does not overwrite the Rg value and its uncertainty primarily modified, by re-calculating Rg and its uncertainty from Ng and its uncertainty, if the latter would not have been modified also.

If the values of the gross count rate Rg is obtained as a mean of several individual values, a linear interpolation is necessary between its variance at measurement (u_Rgm^2) and the variance to be used for calculating the decision threshold (u_R0^2). The formula for the standard uncertainty may be expressed as:

(2.4.1)¶\[\sqrt{u(R_{0})^{2} + (u(R_g)^2 – u(R_{0})^{2})*(R_{g} - R_{0}) / (R_{gm} – R_{0})}\]

Herein, Rg und Rgm are the modified (varied) and the measured value of the gross count rate, respectively. If Rg takes the upper value, Rgm, the result is sqrt(u_Rbm^2), while the result is sqrt(u_R0^2) for the lower value Rg=R0. This requires appending the symbols Rgm, u_Rgm and u_R0 to the symbol list (TAB “Equations”) of independent quantities. They do not get an uncertainty and their values are to be given explicitly in the “Values, uncertainties” table.

Note

Since the program version 2.3.05 2020/01 it is no longer necessary that the user supplies the equations (2.4.1) or a similar formula to UncertRadio. The introduction of additional auxiliary quantities also can be omitted. The Section 4.9 generally informs how to supply datasets for calculating means. Based on that, the Section 4.12 describes the necessary calculations and manipulations within the program for deriving an extended version of equation (2.4.1).

The type of equation (2.4.1) for the uncertainty of thegross count rate-related mean depends on how the mean of single values suffers by additional random influences, which can be characterized as unknown or known. According to these two options, different versions of Eq. (2.4.1) are applied. This is also considered in Section 4.12:, where corresponding example projects are also indicated.

It may happen with special measurement problems that the length of the standard deviation formula of the gross counting rate is longer than the visible part of that cell. For better editing, one can then copy this formula by copy and paste (in the mouse context menu) from this cell to the longer text cell (“extra working cell for formulae editing”) located above the table and back again.

Below the uncertainty table is located a smaller table for the input of covariances. Its columns are as follows:

1:	Symbol A	symbol list box for quantity A
2:	Symbol A	symbol list box for quantity B
3:	Type	list box for choosing input between covariance and correlation coefficient
4:	Formula	text field for defining the covariance as formula being a function of already defined symbols
5:	(or) Value	number field for input of the value of the covariance / correlation coefficient

In the columns 1 and 2 the symbols of correlated measured quantities are selected. After having selected “covariance” in column 3 a formula for the covariance may be entered in the column “Formula”. Otherwise, a numerical value of the covariance / correlation coefficient can be entered directly into the column “(or) Value”.

Relation between correlation coefficient r and covariance cov:

\(r(SymbolA,\ SymbolB) = \frac{cov(SymbolA,\ SymbolB)}{\sqrt{var(SymbolA) \bullet var(SymbolB)}}\)

Having completed the input to the uncertainty and to the covariance tables a mouse click on the button “calculation of uncertainties” will initiate the following calculations:

Note: All calculations within this TAB and the following refer to the actually selected output quantity, if more than one output quantities have been defined for the project. Under the menu item “Edit – Select output quantity” another output quantity may be selected;
Uncertainty table: values of dependent quantities (red colored fields) are calculated and inserted into the corresponding cells; Formulae for standard deviations in the column “StdDev formula” are evaluated; standard uncertainties of independent quantities are evaluated into column “abs. std.Unc.“;
Covariance table: evaluation of covariance formulae as numerical values into column “(or) Value”;
Uncertainty table: now all variances/covariances are known for the complete uncertainty propagation; calculation of the standard uncertainties of the dependent quantities (a) below the output quantity and of the output quantity (row 1 in that table); the combined standard deviations of the dependent quantities (red colored) are exclusively calculated from the uncertainties/covariances of the independent quantities (white colored).

Only after finalisation of these calculations the TAB “Uncertainty budget” is enabled.

With more complex measurement problems and a more slowly PC it may be that the calculations take few seconds; it is indicated in the status bar segment at the lower right corner of the UncertRadio window with the entry “calculating…” that is still working. After termination of the calculations the entry within this field will be “Ready!”.

Note: Changes within the Options menu

For working with tables: see Menu Edit - Table