4.12. Gross quantity: Variance interpolation for a mean

4.12.1. Definitions

According to section Mathematical background the following definitions of means and associated variances are applied:

\(\bar{x}_{g} = \frac{1}{n_{g}}\sum_{i=1}^{n_{g}}x_{gi}\)	\(\bar{x}_{b} = \frac{1}{n_{b}}\sum_{i=1}^{n_{b}}x_{bi}\)
\(s_{g}^{2} = \frac{1}{n_{g}-1}\sum_{i=1}^{n_{g}}\left(x_{gi} - \bar{x}_{g}\right)^{2}\)	\(s_{b}^{2} = \frac{1}{n_{b}-1}\sum_{i=1}^{n_{b}}\left(x_{bi} - \bar{x}_{b}\right)^{2}\)
\(f_{g} = \frac{n_{g}-1}{n_{g}-3}\frac{1}{n_{g}}\)	\(f_{b} = \frac{n_{b}-1}{n_{b}-3}\frac{1}{n_{b}}\)
\(u^{2}(\bar{x}_{g}) = f_{g}\,s_{g}^{2}\)	\(u^{2}(\bar{x}_{b}) = f_{b}\,s_{b}^{2}\)

For those input quantities, to which mean values are attributed, the t-distribution is taken as type of distribution. The possible values, which can be attributed to the three parameters (number of degrees of freedom \(\upsilon\), mean value \(\widehat{\mu}\), standard uncertainty \(\widehat{\sigma}\) (scaling)) of the t-distribution, are given in the following table for the example of the gross quantity (subscript g); the table for the background quantity (subscript b) would look similarly.

	Method A “not being counts“	Method B “counts, with influence“	Method C “classical“
	t-distribution	distribution unclear	normal distribution
\(\upsilon\)	\(n_{g} - 1\)	\(m - 1\)	\(n_{g} - 1\)
\(\widehat{\mu}\)	\(\frac{1}{m} \sum_{i=1}^{m} n_{i}\)	\(\frac{1}{n_{g}} \sum_{i=1}^{n_{g}} x_{gi}\)	\(\frac{1}{m} \sum_{i=1}^{m} n_{i}\)
\(\widehat{\sigma}\)	\(\sqrt{f_{g} s_{g}^{2}}\)	\(\sqrt{\left( \frac{\overline{n}}{m} + f_{g} \left( \overline{n} + s_{n}^{2} \right) \right)}\)	\(\sqrt{f_{g} s_{g}^{2}}\)
\(f_{g}\)	\(\frac{n_{g} - 1}{n_{g} - 3} \frac{1}{n_{g}}\)	\(\frac{m - 1}{m - 3} \frac{1}{m}\)	\(\frac{1}{n_{g}}\)

(\(\overline{n}\ \) and \(s_{n}^{2}\) are estimated in the correspondent manner like \(\overline{x}\ \) and \(s_{x}^{2}\) .)

Notes:

The distribution type of the type of mean “counts, with influence“ is a superposition of a shifted t-distribution (mean \({(\overline{n} + s}_{n}^{2}\)) and a normal distribution (mean 0); see section Principle of the MC simulation. The case in the third column is program-internally treated as “normal distributed“, even if in UR the t-distribution has been chosen as distribution type.

The variance of the sum of a t-distributed and a normal distributed quantity is given by the sum of their variances only if more than 5 individual values of the t-distributed quantity are used.

If the background-related quantity \(x_{b}\) is not treated as a mean, this means \(f_{b} = 1\) is applied.

4.12.2. Principle of the MC simulation

The theoretical treatment of Method B from Table 2 was described by Weise et al. (2013), in its Appendix C, especially section C.2 It is shown there, how the expression for the variance \(u^{2}(\overline{n})\) of the mean \(\overline{n}\) of the numbers of counts

(4.12.1)\[u^{2}\left( \overline{n} \right) = \frac{1}{m}\left( \overline{n} + \frac{(m - 1)}{(m - 3)}{(\overline{n} + s_{n}^{2})} \right) = \frac{\overline{n}}{m} + \frac{1}{m}\frac{(m - 1)}{(m - 3)}{(\overline{n} + s_{n}^{2})}\]

was derived. The first term therein, \(\frac{\overline{n}}{m},\) is interpreted as a counting uncertainty contribution. The second term is a t-distributed contribution of additional random influences to the variance.

For a normal distribution, the variable

(4.12.2)\[t = \frac{\mu - \overline{n}}{s_{n}/\sqrt{m}}\]

follows a Student t-distribution with \((m - 1)\) degrees of freedom, expectation value of zero and variance \((m - 1)/(m - 3)\); \(\mu\) is the expectation value of \(\overline{n}\). Solving this equation for \(\mu\) leads to the equation

(4.12.3)\[\mu = \overline{n} + t\sqrt{\frac{s_{n}^{2}}{m}}\]

This is taken as a recipe for generating t-distributed random numbers. With standard-t-distributed random numbers \(t_{rnd}\), the MC values for simulating the distribution of \(u^{2}\left( \overline{n} \right)\) according to Eq. (4.12.1) are derived as follows.

With

(4.12.4)\[\mu = \overline{n} + t_{rnd}\sqrt{(s_{n}^{2} + \overline{n})/m}\]

random values with mean \(\overline{n}\) and variance

(4.12.5)\[\frac{1}{m}\frac{(m - 1)}{(m - 3)}{(s_{n}^{2} + \overline{n})}\]

are obtained; in a second step normal-distributed random values

(4.12.6)\[z_{rnd}\sqrt{\overline{n}/m}\]

are added to this, where \(z_{rnd}\) are standard-normal distributed random values:

(4.12.7)\[\mu = \overline{n} + t_{rnd}\sqrt{(s_{n}^{2} + \overline{n})/m} + z_{rnd}\sqrt{\overline{n}/m}\]

This last step contributes to broadening the distribution.

For the less complicated case of Method A, only equation (2; \(t\) is replaced by \(t_{rnd}\) ) is applied for generating random values.

Notes:

By using t-distributed values the multiplicative factor \((m - 1)/(m - 3)\) is generated automatically; therefore, this factor must not be supplied in equations (4.12.3) and (4.12.4).

In the TAB “Values, Uncertainties” in UncertRadio those uncertainties \(u(x)\) are displayed, which correspond to the row for \(\widehat{\sigma}\) in table 2. Before generating MC values for an assumed value \(\widetilde{y}\) according to Eqs (4.12.3) or (4.12.4), the value \(s_{n}^{2}\) is calculated from the associated \(u(x)\) by reversing equations (4.12.3) or (4.12.4).

From the uncertainty \(u(.)\) one calculates:

Eq. (4.12.3):

(4.12.8)\[s_{n}^{2} = u^{2}(.)\ m\ \left( \frac{(m - 1)}{(m - 3)} \right)^{- 1}\]

Eq. (4.12.4):

(4.12.9)\[s_{n}^{2} = \left\lbrack \left( u^{2}(.)\ m - \overline{n} \right)\left( \frac{(m - 1)}{(m - 3)} \right)^{- 1} - \overline{n} \right\rbrack\]

Special feature of the MC simulation of decision threshold and detection limit:

In these cases, the factor of \(\sqrt{(m - 1)/(m - 3)\ }\) for the gross count rate is already contained in the expression of its uncertainty varied according to Eq. (11, see below). As already indicated in the notes above, this factor is implied by generating random values \(t_{rnd}\) : it is identical with the standard uncertainty of the standard \(t\) distribution. To prevent from applying this factor twice, \(t_{rnd}\) is simply replaced by \(t_{rnd}/\sqrt{(m - 1)/(m - 3)\ }\) in the equations (4.12.3) through (4.12.7).

4.12.3. Procedures with unknown random influences

It is assumed that repeated measurements underly unknown random influences, which are not small and lead to increased fluctuations. This requires running some measurement series for estimating the gross and background count rate (or gross and background quantities).

4.12.3.1. Using the gross count rate for interpolation

If the value of a gross count rate Rg or a gross quantity xg is estimated by a mean of a measurement series, its uncertainty can no longer be estimated by, e.g., u(Rg)=sqrt(Rg/t). Instead, this requires an interpolation between two known values of the variance. According to ISO 11929, this is solved for an assumed value \(\widetilde{y}\) of the output quantity by interpolating between the variance \(u^{2}(y)\) of the primary result and the variance \(u^{2}(\widetilde{y} = 0)\):

(4.12.10)\[u^{2}\left( \widetilde{y} \right) = u^{2}(0)\left( 1 - \frac{\widetilde{y}}{y} \right) + u^{2}(y)\frac{\widetilde{y}}{y}\]

In UncertRadio, however, such an interpolation refers to corresponding two variance values of the gross quantity \({\widetilde{x}}_{g}\). This case can be deduced from the one in Eq. (4.12.10). A measurement model with quantities \({\widetilde{\mathbf{x}}}_{\mathbf{g}}\mathbf{,}\mathbf{\ \ }\mathbf{x}_{\mathbf{b}}\mathbf{\ ,}\mathbf{x}_{\mathbf{int}}\) (gross, background, interference) is assumed, in which both, \({\widetilde{x}}_{g}\) and \(x_{b}\) are treated as mean values:

(4.12.11)\[\widetilde{y} = w\left( {\widetilde{x}}_{g} - x_{0} - x_{int} \right)\]

This means

(4.12.12)\[u^{2}\left( \widetilde{y} \right) = w^{2}\left( u^{2}\left( {\widetilde{x}}_{g} \right) + u^{2}\left( x_{b} \right) + u^{2}\left( x_{int} \right) \right) + \left( {\widetilde{x}}_{g} - x_{b} - x_{int} \right)^{2}u^{2}(w)\]

Equating the right-hand sides of (4.12.1) und (3) yields

(4.12.13)\[w^{2}u^{2}\left( {\widetilde{x}}_{g} \right) = u^{2}(0)\left( 1 - \frac{\widetilde{y}}{y} \right) + u^{2}(y)\frac{\widetilde{y}}{y} - w^{2}u^{2}\left( x_{b} \right) - w^{2}u^{2}\left( x_{int} \right) - \left( {\widetilde{x}}_{g} - x_{b} - x_{int} \right)^{2}u^{2}(w)\]

With setting

(4.12.14)\[\frac{\widetilde{y}}{y} = \widetilde{q} = \frac{\left( {\widetilde{x}}_{g} - x_{0} - x_{int} \right)}{\left( x_{g} - x_{0} - x_{int} \right)} = \frac{{\widetilde{R}}_{n}}{R_{n}}, \quad \widetilde{q} \geq 0\]

It follows:

(4.12.15)\[w^{2}u^{2}\left( {\widetilde{x}}_{g} \right) = u^{2}(0)\left( 1 - \widetilde{q} \right) + u^{2}(y)\widetilde{q} - w^{2}u^{2}\left( x_{0} \right) - w^{2}u^{2}\left( x_{int} \right) - {\widetilde{R}}_{n}^{2}u^{2}(w)\]

Now, with an expression for \(u^{2}(0)\):

(4.12.16)\[u^{2}(0) = u^{2}\left( \widetilde{y} = 0 \right) = w^{2}\left( u^{2}\left( x_{0} \right) + u^{2}\left( x_{int} \right) + u^{2}\left( x_{0} \right) + u^{2}\left( x_{int} \right) \right)\]

(4.12.17)\[u^{2}(0) = w^{2}2\left( u^{2}\left( x_{0} \right) + u^{2}(x_{int}) \right)\]

the expression for the variance \(u^{2}({\widetilde{x}}_{g})\) becomes:

\(w^{2}u^{2}\left( {\widetilde{x}}_{g} \right) = w^{2}2\left( u^{2}\left( x_{0} \right) + u^{2}(x_{int}) \right)\left( 1 - \widetilde{q} \right) + u^{2}(y)\widetilde{q} - w^{2}u^{2}\left( x_{0} \right) - w^{2}u^{2}\left( x_{int} \right) - {\widetilde{R}}_{n}^{2}u^{2}(w)\)

\(u^{2}\left( {\widetilde{x}}_{g} \right) = 2\left( u^{2}\left( x_{0} \right) + u^{2}(x_{int}) \right)\left( 1 - \widetilde{q} \right) + \frac{u^{2}(y)}{w^{2}}\widetilde{q} - u^{2}\left( x_{0} \right) - u^{2}\left( x_{int} \right) - {\widetilde{R}}_{n}^{2}u_{rel}^{2}(w)\)

\(u^{2}\left( {\widetilde{x}}_{g} \right) = 2\left( u^{2}\left( x_{0} \right) + u^{2}(x_{int}) \right) - 2\left( u^{2}\left( x_{0} \right) + u^{2}(x_{int}) \right)\widetilde{q} + \frac{u^{2}(y)}{w^{2}}\widetilde{q} - u^{2}\left( x_{0} \right) - u^{2}\left( x_{int} \right)\) \(- {\widetilde{R}}_{n}^{2}u_{rel}^{2}(w)\)

\(u^{2}\left( {\widetilde{x}}_{g} \right) = \left( u^{2}\left( x_{0} \right) + u^{2}\left( x_{int} \right) \right) - 2\left( u^{2}\left( x_{0} \right) + u^{2}\left( x_{int} \right) \right)\widetilde{q} + \frac{u^{2}(y)}{w^{2}}\widetilde{q}\ - {\widetilde{R}}_{n}^{2}u_{rel}^{2}(w)\)

Setting now \({\widetilde{R}}_{n}^{2} = {\widetilde{q}}^{2}R_{n}^{2}\):

(4.12.18)\[\begin{split}\begin{align} u^{2}\left( {\widetilde{x}}_{g} \right) &= \left( u^{2}\left( x_{0} \right) + u^{2}\left( x_{int} \right) \right)\left( 1 - 2\widetilde{q} \right) + \frac{u^{2}(y)}{w^{2}}\widetilde{q} - {\widetilde{q}}^{2}R_{n}^{2}\ u_{rel}^{2}(w) \\ u^{2}\left( {\widetilde{x}}_{g} \right) &= \left( u^{2}\left( x_{0} \right) + u^{2}\left( x_{int} \right) \right)\left( 1 - 2\widetilde{q} \right) + \widetilde{q}\left( \frac{u^{2}(y)}{w^{2}} - \widetilde{q}R_{n}^{2}\ u_{rel}^{2}(w) \right) \end{align}\end{split}\]

For the program-internal application, \(y\) und \(u^{2}(y)\) are also replaced:

\(\frac{u^{2}(y)}{w^{2}} = u^{2}\left( x_{g} \right) + u^{2}\left( x_{0} \right) + u^{2}\left( x_{int} \right){+ \left( x_{g} - x_{0} - x_{int} \right)}^{2}u_{rel}^{2}(w)\)

\(\frac{u^{2}(y)}{w^{2}} = u^{2}\left( R_{n} \right) + R_{n}^{2}u_{rel}^{2}(w)\)

For the last round bracket in (4.12.18) one obtains:

(4.12.19)\[\begin{split}\begin{align} \left( \frac{u^{2}(y)}{w^{2}} - \widetilde{q}R_{n}^{2} u_{rel}^{2}(w) \right) &= u^{2}\left( R_{n} \right) + R_{n}^{2}u_{rel}^{2}(w) - R_{n}^{2}\widetilde{q}\ u_{rel}^{2}(w) \\ \left( \frac{u^{2}(y)}{w^{2}} - \widetilde{q}R_{n}^{2} u_{rel}^{2}(w) \right) &= u^{2}\left( R_{n} \right) + (1 - \widetilde{q})R_{n}^{2}u_{rel}^{2}(w) \end{align}\end{split}\]

which by inserting it into in (4.12.18) yields:

(4.12.20)\[u^{2}\left( {\widetilde{x}}_{g} \right) = \left( u^{2}\left( x_{0} \right) + u^{2}\left( x_{int} \right) \right)\left( 1 - 2\widetilde{q} \right) + \widetilde{q}\left( u^{2}\left( R_{n} \right) + (1 - \widetilde{q})R_{n}^{2} u_{rel}^{2}(w) \right)\]

In principle, equations (4.12.18) or (4.12.20) represent that equation or formula, which would have to be entered by the user into the “green cell“ in the table “values, uncertainties“ in UncertRadio. This would also imply to add several auxiliary quantities to the symbol list in UncertRadio. However, the already existing tool for treating mean values according to chapter 6.9 (see also section 6.12.), offers the opportunity to gather these auxiliary quantity values internally.

In equation (4.12.20), assumed values of the variable \({\widetilde{x}}_{g}\) are set by the program within the iterations for calculating the decision threshold and the detection limit. The fixed values \(n_{g},n_{b},x_{g},x_{b},s_{g},f_{g},s_{b},f_{b}\) are taken from the two tables in 6.12., or from the program-internal data arrays associated with the treatment of means. These also fixed values for \({w,x_{int},u}^{2}(x_{int})\) are read from the UR table „Values, uncertainties“.

Finally, it is no longer necessary to enter a formula into the „green cell“ for the standard deviation of the gross quantity, if the value of this quantity is given by a mean. This requires only to define the t-distribution type for the quantity symbol \(n_{g}\).

Example projects: ISO-Example-2a_EN.txp (with the old UR-treatment)

ISO-Example-2a_V2_EN.txp (with the new UR-treatment)

Equivalence of the linear interpolation alternatives

The interpolation of output quantity variances to be applied according to ISO 11929 shall be linear as in Eq. (4.12.1). As in this section the interpolation instead refers to gross count rate variances, it needs to be tested, whether the interpolated values according to these two interpolation variants would agree. This has been tested with small R program, separately for procedures A and B.

4.12.3.2. Application to Procedure A („not being counts“)

If the variances \(u^{2}(x_{g})\) und \(u^{2}(x_{b})\) are taken as products \(f_{g}s_{g}^{2}\) and \(f_{b}s_{b}^{2}\) according to 6.12.1, for the model case \(y = w(x_{g} - x_{b} - xint)\) the result for the interpolated variance of the gross quantity \({\widetilde{x}}_{g}\) is:

(4.12.21)\[u^{2}\left( {\widetilde{x}}_{g} \right) = f_{g}s_{b}^{2} + u^{2}\left( x_{int} \right) + \widetilde{q}\left\lbrack f_{g}\left( s_{g}^{2} - s_{b}^{2} \right) - u^{2}\left( x_{int} \right) + \left( 1 - \widetilde{q} \right)x_{net}^{2}u_{rel}^{2}(w) \right\rbrack\]

Testing the variance interpolation:

(Program Var_intpol_Ex13.R, for example 13 of ISO 11929-4)

The following formulae were applied for \(u^{2}\left( {\widetilde{x}}_{g} \right)\) which, after inserting it into Eq. (4.12.12), allows the comparison with variance values calculated according to Eq. (4.12.10) (\(x_{int}\) has been set zero):

var_Rg_tilde_a = fg*(sb^2) + uxint^2 +
q_tilde \* (fg*(sg^2-sb^2)- uxint^2 + xn^2*(uw/w)^2*(1- q_tilde) )
var_Rg_tilde_b = ((fg+fb)*sb^2 + 2*uxint^2)*(1. - q_tilde) +
q_tilde*( (uym/w)^2) - fb*sb^2 - uxint^2 - q_tilde^2*(ym/w)^2*(uw/w)^2
q_tilde = (xg_tilde – xb - xint) / (xg – xb - xint)

With using the following values:

sg= 71.71839 sb= 5.895336 fg= 0.03857143 fb= 0.04012346
xgtilde= 75.704 q_tilde= 2.542306e-06
xg= 192.25 uxg= 14.08521 xb= 75.7037 uxb= 1.180885
ym= 116.5463 uym= 37.71288 uy0= 1.653796

and Eq. (6) for calculation of xg_tilde from y_tilde, for 11 values of ytilde, between 0 and 116.54, the following variances were derived:

Index	y_tilde	xg_tilde	var_xg_tilde_a	var_xg_tilde_b	vary_tilde_lin	varytilde2
[1,]	0.00000	75.70370	1.340549	1.340549	2.73504	2.73504
[2,]	11.65463	87.35833	131.068433	131.068433	144.68766	144.68766
[3,]	23.30926	99.01296	236.346847	236.346847	286.64028	286.64028
[4,]	34.96389	110.66759	317.175789	317.175789	428.59290	428.59290
[5,]	46.61852	122.32222	373.555262	373.555262	570.54552	570.54552
[6,]	58.27315	133.97685	405.485263	405.485263	712.49814	712.49814
[7,]	69.92778	145.63148	412.965795	412.965795	854.45075	854.45075
[8,]	81.58241	157.28611	395.996855	395.996855	996.40337	996.40337
[9,]	93.23704	168.94074	354.578446	354.578446	1138.35599	1138.35599
[10,]	104.89167	180.59537	288.710565	288.710565	1280.30861	1280.30861
[11,]	116.54630	192.25000	198.393214	198.393214	1422.26123	1422.26123

There is no difference observed between the two compared variance values of the gross quantity (columns 3, 4). The same observation applies to the output quantity variance (columns 6, 6) calculated according to equations (4.12.10) and (4.12.12).

This verifies the equivalence of the two compared interpolation methods.

4.12.3.3. Application to Procedure B („counts, with influence“)

For the model \(y = w(x_{g} - x_{b} - x_{int})\), the variances \(u^{2}(x_{g})\) and \(u^{2}(x_{b})\), calculated according to Eq. (4.12.1), are given by:

(4.12.22)\[u^{2}\left( {\widetilde{x}}_{g} \right) = \frac{u^{2}\left( {\overline{n}}_{b} \right)}{t_{b}^{2}} + u^{2}(x_{int}) + \widetilde{q}\left\lbrack \frac{u^{2}\left( {\overline{n}}_{g} \right)}{t_{g}^{2}} - \frac{u^{2}\left( {\overline{n}}_{b} \right)}{t_{b}^{2}} - u^{2}(x_{int}) + \left( 1 - \widetilde{q} \right)x_{net}^{2}u_{rel}^{2}(w) \right\rbrack\]

with

\(\frac{u^{2}\left( {\overline{n}}_{g} \right)}{t_{g}^{2}} = \frac{1}{t_{g}^{2}}\left( \frac{{\overline{n}}_{g}}{m_{g}} + f_{g}{({\overline{n}}_{g} + s}_{g}^{2}) \right)\) ; \(\frac{u^{2}\left( {\overline{n}}_{b} \right)}{t_{b}^{2}} = \frac{1}{t_{b}^{2}}\left( \frac{{\overline{n}}_{b}}{m_{b}} + f_{b}{({\overline{n}}_{b} + s}_{b}^{2}) \right)\)

Testing the variance interpolation:

(Program Var_intpol_Ex14.R, for example 14 of ISO 11929-4)

The following variants of equations for \(u^{2}\left( {\widetilde{x}}_{g} \right)\) were applied for subsequent comparison with values from Eq. (5) (\(x_{int}\) has been set zero):

var_Rg_tilde_a = (un0_mean/t0)^2 + uxint^2 +
q_tilde \* ((ung_mean/tg)^2-(un0_mean/t0)^2 - uxint^2 +
Rn^2*(uw/w)^2*(1- q_tilde) )
var_Rg_tilde_b = ((un0_mean/t0)^2 + uxint^2) \* (1. - 2.*q_tilde) +
q_tilde*( (uym/w)^2 - q_tilde*(ym/w)^2*(uw/w)^2 )
var_Rg_green = urbt^2 + (uRg^2 - urbt^2)*(Rg_tilde-rbt)/(Rg-rbt) +
(uw/w)^2*(Rg_tilde-rbt)*(Rg-Rg_tilde)

with

q_tilde = (Rg_tilde - R0 - xint) / (Rg - R0 - xint)
Rg_tilde = ytilde/w + un0_mean/t0 + xint
rbt = R0 + xint; urbt = sqrt(uR0^2 + xint^2)

Note that „var_Rg_green“ denotes that formula that earlier had been manually inserted into the “green cell” within UncertRadio.

With using the following values:

R0= 0.02723333 u(R0)= 0.002929202 Rg= 0.06798667 u(Rg)= 0.006185528
Rn= 0.04075333 w= 34.39972 uw= 2.786688 ym= 1.401903 uym= 0.261393
uy0= 0.1425014

Table 4.12.1 Example List Table

Dataset	y_tilde	var_Rg_tilde_a	var_Rg_tilde_b	var_Rg_tilde_green
[1,]	0.0000000	8.580222e-06	8.580222e-06	8.580222e-06
[2,]	0.1401903	1.252920e-05	1.252920e-05	1.252920e-05
[3,]	0.2803807	1.626019e-05	1.626019e-05	1.626019e-05
[4,]	0.4205710	1.977321e-05	1.977321e-05	1.977321e-05
[5,]	0.5607614	2.306823e-05	2.306823e-05	2.306823e-05
[6,]	0.7009517	2.614528e-05	2.614528e-05	2.614528e-05
[7,]	0.8411421	2.900434e-05	2.900434e-05	2.900434e-05
[8,]	0.9813324	3.164542e-05	3.164542e-05	3.164542e-05
[9,]	1.1215228	3.406851e-05	3.406851e-05	3.406851e-05
[10,]	1.2617131	3.627363e-05	3.627363e-05	3.627363e-05
[11,]	1.4019035	3.826076e-05	3.826076e-05	3.826076e-05

There is no difference observed between the three compared variance values of the gross count rate.

In the following table, for every value of ytilde, the following calculated values are shown:

var_y_tilde_lin from Eq. (4.12.1);

Rg_tilde (from reversing Eq. (2),

var_Rg_tilde (the above mentioned var_rg_tilde_b) ,

varytilde2 (after inserting var_Rg_tilde into Eq. (3),

ratio (the ratio varytilde2 / vary_tilde_lin)

Dataset	y_tilde	vary_tilde_lin	Rg_tilde	var_Rg_tilde	varytilde2	ratio
[1,]	0.0000000	0.02030666	0.02723333	8.580222e-06	0.02030666	1
[2,]	0.1401903	0.02510862	0.03130867	1.252920e-05	0.02510862	1
[3,]	0.2803807	0.02991058	0.03538400	1.626019e-05	0.02991058	1
[4,]	0.4205710	0.03471254	0.03945933	1.977321e-05	0.03471254	1
[5,]	0.5607614	0.03951451	0.04353467	2.306823e-05	0.03951451	1
[6,]	0.7009517	0.04431647	0.04761000	2.614528e-05	0.04431647	1
[7,]	0.8411421	0.04911843	0.05168533	2.900434e-05	0.04911843	1
[8,]	0.9813324	0.05392039	0.05576067	3.164542e-05	0.05392039	1
[9,]	1.1215228	0.05872235	0.05983600	3.406851e-05	0.05872235	1
[10,]	1.2617131	0.06352432	0.06391133	3.627363e-05	0.06352432	1
[11,]	1.4019035	0.06832628	0.06798667	3.826076e-05	0.06832628	1

This verifies the equivalence of the two compared interpolation methods.

4.12.4. Procedures with known random influences

It is assumed that repeated measurements underly unknown random influences, which are small. It is furthermore assumed, that the gross and background and other counts are influenced in the same way, also in the case of different measurements but the same measurement conditions. With a reference analysis, i.e., with a larger number \(m\) of measurements of a sample (which gets a subscript r), the unknown influence can be quantified by parameter \(\vartheta\), which is applied also to the other involved measurement quantities like gross and background counts.

The parameter \(\vartheta\) has already been introduced in chapter 6.9.1. It is determined from the reference data set and applied to the uncertainty calculations of the other count numbers (subscript x):

\(u^{2}({\overline{n}}_{x}) = \left( {\overline{n}}_{x} + \vartheta^{2}{\overline{n}}_{x}^{2} \right)/m_{x}\)

or in the case of gross and background counts as well as for assumed gross counts \({\widetilde{n}}_{g}\) within the detection limit related iterations:

(4.12.23)\[u^{2}({\overline{n}}_{g}) = \frac{{\overline{n}}_{g} + \vartheta^{2}{\overline{n}}_{g}^{2}}{m_{g}}\]

(4.12.24)\[u^{2}({\overline{n}}_{b}) = \frac{{\overline{n}}_{b} + \vartheta^{2}{\overline{n}}_{b}^{2}}{m_{b}}\]

(4.12.25)\[u^{2}({\widetilde{n}}_{g}) = \frac{{\widetilde{n}}_{g} + \vartheta^{2}{\widetilde{n}}_{g}^{2}}{m_{g}}\]

With applying the tool for means (Using data sets for mean and variance; see also Definitions), the data necessary for calculating \(\vartheta\), but also those data referring to mean value-related datasets are available within the program. Therefore, the formulae corresponding to the equations (4.12.23), (4.12.24) and (4.12.25) are easily programmed and are part of the program. This means, in contrast to earlier UncertRadio versions, it is no longer necessary for the user to enter uncertainty formulae with the TAB “Values, Uncertainties” ; the introduction of further auxiliary quantities also is no longer necessary.

In addition, from the datasets supplied to UR, that one of them representing the reference measurements from which \(\vartheta\) has to be derived, has to be identified. This can be done with a combobox field within the dialog shown in chapter 6.9.2.

Example-projects: ISO-Example-2b_EN.txp, Mean-theta_EN.txp (with the old UR treatment)

ISO-Example-2b_V2_EN.txp (with the new UR treatment)