1. Introduction

The quality of analytical results is crucial because future decisions will be based on them. Uncertainty [1] is a good indicator of this quality. For example, two measurements made with the same ruler on different days by different people would be equivalent depending on their individual uncertainties.

Quality assurance measurements are a formal requirement in most of the analytical laboratories. As a consequence, to ensure that laboratories provide quality data, they are under continuous pressure to demonstrate their fitness for purpose, i.e., by giving confidence levels on the results. Measurement uncertainty will show the degree of agreement among results.

> © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

This concept of measurement uncertainty will be applicable to many cases, besides of quality control and quality assurance in production, such as complying with and enforcing laws and regulations, conducting a basic research, calibrating standards and instruments or developing, maintaining, and comparing international and national physical reference standards.

The ISO Guide to the expression of uncertainty in measurement, also known as the ISO/GUM or "bottom-up" [1], is one of the best approaches to estimate the uncertainty of analytical procedures. This procedure, originally conceived for use in physical measurements, has been suitably adapted to chemical ones in the EURACHEM/CITAC (Cooperation on International Traceability in Analytical Chemistry) guide [2] "Traceability in Chemical Measurement." However, this approach is tedious, time-consuming and unrealistic from the analytical viewpoint because their principles are significantly different from current procedures applied in analytical chemistry dealing with matrix effects, sampling operations and interferences [3, 4]. A strategy for reconciling the information requirements of ISO/GUM approach and the information coming from in-house method validation has been described by Ellison and Barwick [5]. The use of "cause" and "effect" analysis is the key for estimating the uncertainty of an analytical assay. In practice, this approach is performed by using a cause and effect diagram called Ishikawa or fishbone plot [6], consisting of a hierarchical structure that culminates in the "analytical result." In order to carry out the cause and effect analysis, the specification equation for the result is of utmost importance. The factors appearing in the equation (that contribute to the uncertainty of the result) are the main branches of the fishbone plot. For each branch, secondary factors can be considered, and so on, until their contribution to the result uncertainty is negligible. Two additional main branches (Recovery and Precision) come from the method validation. Nevertheless, these approaches exhibit some risks. The blind consideration of uncertainties coming from different sources of variation may lead to "double counting" in some instances. The analysts have to clearly identify the relationships among the sources of uncertainty in order to avoid duplications. Also, some sources of uncertainty that can be evaluated in a unique set of experiments must be suitably combined.

The combined uncertainty of the analytical measurand is the heritance of the uncertainties of all contributing variables (xi) involved in the specification relationship where the value of measurand (Z) is defined as

$$Z = F(\mathbf{x}\_1, \mathbf{x}\_2, \dots \mathbf{x}\_n) \tag{1}$$

But this uncertainty does not consider the uncertainty contributions due to the intermediate precision of the assay and the trueness evaluated from recovery experiments. Nevertheless, it is possible to include these ones into the specification relationship either directly or by

A Practical Way to ISO/GUM Measurement Uncertainty for Analytical Assays Including In-House Validation Data

do contribute to its uncertainty [7, 8]. Accordingly, the modified specification relationship

The new involved parameters are the recovery, R, and the intermediate precision of the assay, f prec. These contributions are issued from the data of method validation study. Accordingly, the

relð Þþ xi <sup>u</sup><sup>2</sup>

At this step, the considerations regarding to the sources of uncertainties have to be taken into

The specification relationship involves a given set of parameters depending on the analytical procedure applied. Common factors are: mass determinations (obviously for sample weight and used standards), volumetric measurements (glassware and other devices delivering volume), analyte concentration coming from indirect calibration, and the precision and recovery

<sup>Z</sup> <sup>¼</sup> F xð Þ <sup>1</sup>; <sup>x</sup>2;…xn

� � � � which do not contribute to the measurand value, but

relð Þþ <sup>R</sup> RSD<sup>2</sup>

<sup>R</sup> <sup>f</sup> prec (4)

http://dx.doi.org/10.5772/intechopen.72048

111

prec (5)

(6)

using unit-value factors f <sup>i</sup> ¼ 1 � u f <sup>i</sup>

uncertainty of measurand can be written as:

u2

of the analytical assay established in the validation study.

measurements (without considering buoyancy) is given by [9]

S2 <sup>r</sup> <sup>þ</sup> <sup>S</sup><sup>2</sup> env þ 2 3 a2 <sup>L</sup> þ

u mð Þ¼

1.1. Uncertainty of sample mass

where S<sup>2</sup>

<sup>r</sup> <sup>þ</sup> <sup>S</sup><sup>2</sup>

balance calibration.

relð Þ¼ <sup>Z</sup> <sup>X</sup><sup>n</sup>

i¼1 u2

account in order to avoid either under- or over-estimations of the result uncertainty.

In the following, these factors will be outlined and their uncertainties will be discussed.

In a typical mass determination, the analytical balance is zeroed with the empty container on the pan, and the container is the filled and weighed. In this case, the uncertainty of mass

expressed as an weighting intermediate precision, aL is the linearity specification of the balance, aT is the sensitivity temperature coefficient, ΔT is the difference between the room temperature and the calibration temperature (20�C) and ucal is the standard uncertainty for

Because the intermediate precision study is carried out for the entire analytical assay at the validation stage, individual contributions to the intermediate precision (here, weighting intermediate precision) cannot be taken into account for avoiding redundant counting of uncertainty. Thus, the uncertainty of mass will include the uncertainty contribution of lack

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

CAL <sup>s</sup>

m<sup>2</sup>a<sup>2</sup>

env is the variance of replication (repeatability and environmental variances)

<sup>T</sup>ð Þ <sup>Δ</sup><sup>T</sup> <sup>2</sup> <sup>9</sup> <sup>þ</sup> <sup>u</sup><sup>2</sup>

turns to:

Thus, the general expression for the combined uncertainty of measurand according to the law of propagation of uncertainty is given by

$$u^2(Z) = \sum\_{i=1}^{n} \left(\frac{\partial F}{\partial \mathbf{x}\_i}\right)^2 u^2(\mathbf{x}\_i) + 2\sum\_{i=1}^{n-1} \sum\_{j=i+1}^{n} \left(\frac{\partial F}{\partial \mathbf{x}\_i}\right) \left(\frac{\partial F}{\partial \mathbf{x}\_j}\right) \text{cov}\left(\mathbf{x}\_i, \mathbf{x}\_j\right) \tag{2}$$

When the specification function consists of products or ratios only, and the factors are considered to be independent, then

$$\begin{aligned} \left(\frac{\mu(Z)}{Z}\right)^2 &= \sum\_{i=1}^n \left(\frac{\mu(\mathbf{x}\_i)}{\mathbf{x}\_i}\right)^2\\ \mu\_{rel}^2(Z) &= \sum\_{i=1}^n \mu\_{rel}^2(\mathbf{x}\_i) \end{aligned} \tag{3}$$

But this uncertainty does not consider the uncertainty contributions due to the intermediate precision of the assay and the trueness evaluated from recovery experiments. Nevertheless, it is possible to include these ones into the specification relationship either directly or by using unit-value factors f <sup>i</sup> ¼ 1 � u f <sup>i</sup> � � � � which do not contribute to the measurand value, but do contribute to its uncertainty [7, 8]. Accordingly, the modified specification relationship turns to:

$$Z = \frac{F(\mathbf{x}\_1, \mathbf{x}\_2, \dots \mathbf{x}\_n)}{R} f\_{\text{prec}} \tag{4}$$

The new involved parameters are the recovery, R, and the intermediate precision of the assay, f prec. These contributions are issued from the data of method validation study. Accordingly, the uncertainty of measurand can be written as:

$$\mu\_{rel}^2(Z) = \sum\_{i=1}^n \mu\_{rel}^2(\mathbf{x}\_i) + \mu\_{rel}^2(\mathbf{R}) + RSD\_{prec}^2 \tag{5}$$

At this step, the considerations regarding to the sources of uncertainties have to be taken into account in order to avoid either under- or over-estimations of the result uncertainty.

The specification relationship involves a given set of parameters depending on the analytical procedure applied. Common factors are: mass determinations (obviously for sample weight and used standards), volumetric measurements (glassware and other devices delivering volume), analyte concentration coming from indirect calibration, and the precision and recovery of the analytical assay established in the validation study.

In the following, these factors will be outlined and their uncertainties will be discussed.

#### 1.1. Uncertainty of sample mass

This concept of measurement uncertainty will be applicable to many cases, besides of quality control and quality assurance in production, such as complying with and enforcing laws and regulations, conducting a basic research, calibrating standards and instruments or developing,

The ISO Guide to the expression of uncertainty in measurement, also known as the ISO/GUM or "bottom-up" [1], is one of the best approaches to estimate the uncertainty of analytical procedures. This procedure, originally conceived for use in physical measurements, has been suitably adapted to chemical ones in the EURACHEM/CITAC (Cooperation on International Traceability in Analytical Chemistry) guide [2] "Traceability in Chemical Measurement." However, this approach is tedious, time-consuming and unrealistic from the analytical viewpoint because their principles are significantly different from current procedures applied in analytical chemistry dealing with matrix effects, sampling operations and interferences [3, 4]. A strategy for reconciling the information requirements of ISO/GUM approach and the information coming from in-house method validation has been described by Ellison and Barwick [5]. The use of "cause" and "effect" analysis is the key for estimating the uncertainty of an analytical assay. In practice, this approach is performed by using a cause and effect diagram called Ishikawa or fishbone plot [6], consisting of a hierarchical structure that culminates in the "analytical result." In order to carry out the cause and effect analysis, the specification equation for the result is of utmost importance. The factors appearing in the equation (that contribute to the uncertainty of the result) are the main branches of the fishbone plot. For each branch, secondary factors can be considered, and so on, until their contribution to the result uncertainty is negligible. Two additional main branches (Recovery and Precision) come from the method validation. Nevertheless, these approaches exhibit some risks. The blind consideration of uncertainties coming from different sources of variation may lead to "double counting" in some instances. The analysts have to clearly identify the relationships among the sources of uncertainty in order to avoid duplications. Also, some sources of uncertainty

maintaining, and comparing international and national physical reference standards.

that can be evaluated in a unique set of experiments must be suitably combined.

measurand (Z) is defined as

110 Quality Control in Laboratory

of propagation of uncertainty is given by

ð Þ¼ <sup>Z</sup> <sup>X</sup><sup>n</sup>

i¼1

∂F ∂xi � �<sup>2</sup>

u2

u Zð Þ Z � �<sup>2</sup>

u2

ð Þþ xi 2

u2

ered to be independent, then

The combined uncertainty of the analytical measurand is the heritance of the uncertainties of all contributing variables (xi) involved in the specification relationship where the value of

Thus, the general expression for the combined uncertainty of measurand according to the law

Xn�1 i¼1

When the specification function consists of products or ratios only, and the factors are consid-

<sup>¼</sup> <sup>X</sup><sup>n</sup> i¼1

> i¼1 u2 relð Þ xi

relð Þ¼ <sup>Z</sup> <sup>X</sup><sup>n</sup>

Xn j¼iþ1

u xð Þ<sup>i</sup> xi � �<sup>2</sup>

∂F ∂xi � � ∂F

Z ¼ F xð Þ <sup>1</sup>; x2;…xn (1)

∂xj � �

cov xi; xj

� � (2)

(3)

In a typical mass determination, the analytical balance is zeroed with the empty container on the pan, and the container is the filled and weighed. In this case, the uncertainty of mass measurements (without considering buoyancy) is given by [9]

$$
\mu(m) = \sqrt{S\_r^2 + S\_{env}^2 + \frac{2}{3}a\_L^2 + \frac{m^2 a\_T^2 (\Delta T)^2}{9} + \mathfrak{u}\_{\text{CAL}}^2} \tag{6}
$$

where S<sup>2</sup> <sup>r</sup> <sup>þ</sup> <sup>S</sup><sup>2</sup> env is the variance of replication (repeatability and environmental variances) expressed as an weighting intermediate precision, aL is the linearity specification of the balance, aT is the sensitivity temperature coefficient, ΔT is the difference between the room temperature and the calibration temperature (20�C) and ucal is the standard uncertainty for balance calibration.

Because the intermediate precision study is carried out for the entire analytical assay at the validation stage, individual contributions to the intermediate precision (here, weighting intermediate precision) cannot be taken into account for avoiding redundant counting of uncertainty. Thus, the uncertainty of mass will include the uncertainty contribution of lack

of linearity of balance, the uncertainty due to temperature effect and the calibration uncertainty

$$
\mu(m) = \sqrt{\frac{2}{3}a\_L^2 + \frac{m^2 a\_T^2 (\Delta T)^2}{9} + u\_{\text{CAL}}^2} \tag{7}
$$

• The independent variable x, is free from error ð Þ εð Þ¼ x 0 or at least, εð Þx << εð Þ Y .

• The variance of the Y variable, σ2, remains uniform in the whole range of x (homoscedasticity).

A Practical Way to ISO/GUM Measurement Uncertainty for Analytical Assays Including In-House Validation Data

In our case, the independent variable is the concentration of standard ð Þ Ci and the Y variable is the analytical signal. In a typical case of multipoint calibration (external or internal), the three requirements mentioned above applies, and the ordinary least-squares procedure gives the calibration straight line Yb<sup>i</sup> ¼ b<sup>0</sup> þ b1Ci. The unknown analyte content is predicted from inter-

> Ccal <sup>¼</sup> <sup>Y</sup><sup>0</sup> � <sup>b</sup><sup>0</sup> b1

Eq. (10) can be rearranged to give the well-known formula recommended by EURACHEM [2]:

Here, sy=<sup>x</sup> is the residual standard deviation of the regression line, m is the number of replica-

Aside from the calibration uncertainty, an additional uncertainty contribution can be considered from the preparation of standards as indicated in Eq. (8) and may be accounted separately

Thus, the uncertainty of concentration is given by the uncertainty on sample analyte concentration coming from calibration, and the uncertainty due to the preparation of standards.

<sup>u</sup><sup>2</sup>ð Þ Vi V2 i þ

1 m þ 1 N þ

vuuuut

∂Ccal ∂b<sup>1</sup> � �<sup>2</sup>

> ð Þ Y<sup>0</sup> � b<sup>0</sup> b3 1

> ð Þ Y<sup>0</sup> � b<sup>0</sup> b3 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P N i¼1

Ccal � <sup>C</sup> � �<sup>2</sup>

Ci � <sup>C</sup> � �<sup>2</sup>

<sup>u</sup><sup>2</sup>ð Þ Vs V2 s þ u<sup>2</sup> Vf � � V2 f

u2

ð Þþ <sup>b</sup><sup>1</sup> <sup>2</sup> <sup>∂</sup>Ccal

cov ð Þ b0; b<sup>1</sup>

Cu<sup>2</sup> ð Þ b<sup>1</sup>

∂b<sup>0</sup> � � ∂Ccal

∂b<sup>1</sup>

http://dx.doi.org/10.5772/intechopen.72048

� � cov ð Þ <sup>b</sup>0; <sup>b</sup><sup>1</sup>

(11)

113

(12)

(13)

(14)

• The error associated to Y variable, is normally distributed, N 0; σ<sup>2</sup> � �.

whose uncertainty can be estimated from the variance propagation law:

u2 ð Þþ b<sup>0</sup>

2

2

Ci, N being the number of calibration points.

sy=<sup>x</sup> b1

tions measuring the sample signal and N the number of calibration points [13].

u2 ð Þþ b<sup>1</sup>

u2 ð Þ� b<sup>1</sup>

∂Ccal ∂b<sup>0</sup> � �<sup>2</sup>

<sup>þ</sup> ð Þ <sup>Y</sup><sup>0</sup> � <sup>b</sup><sup>0</sup>

<sup>þ</sup> ð Þ <sup>Y</sup><sup>0</sup> � <sup>b</sup><sup>0</sup>

u Cð Þ¼ cal

<sup>¼</sup> <sup>u</sup><sup>2</sup>ð Þ mstd m<sup>2</sup> std þ <sup>u</sup><sup>2</sup>ð Þ <sup>P</sup> <sup>P</sup><sup>2</sup> <sup>þ</sup>

b4 1

b4 1

polation of the sample response signal Y<sup>0</sup> according to

u2

ð Þ¼ Ccal

where <sup>C</sup> <sup>¼</sup> <sup>1</sup>

∂Ccal ∂Y � �<sup>2</sup>

<sup>¼</sup> <sup>u</sup><sup>2</sup>ð Þ <sup>Y</sup><sup>0</sup> b2 1 þ

<sup>¼</sup> <sup>u</sup><sup>2</sup>ð Þ <sup>Y</sup><sup>0</sup> b2 1 þ

> N P N i

in the uncertainty budget:

<sup>u</sup><sup>2</sup>ð Þ Ci C2 i

u2 ð Þþ Y<sup>0</sup>

<sup>u</sup><sup>2</sup>ð Þ <sup>b</sup><sup>0</sup> b2 1

<sup>u</sup><sup>2</sup>ð Þ <sup>b</sup><sup>0</sup> b2 1

#### 1.2. Uncertainty of glassware volume

As R. Kadis pointed out [10], the evaluation of uncertainty of volumetric measurements consists of three kinds of contributions: specification limits for the glassware of a given class, repeatability of filling the glassware to the mark and temperature effects. Again, in order to avoid double counting and uncertainty redundancy, the precision of filling the flask is not considered here; thus, the uncertainty in the volume measurement is given by

$$
\mu(V) = \sqrt{\frac{a\_{\rm TOL}^2}{6} + \frac{\chi^2 V^2 \left(\Delta T\right)^2}{3}} \tag{8}
$$

where aTOL is the tolerance for a given class, χ is the dilatation coefficient for the filling liquid (2.1 � <sup>10</sup>�<sup>4</sup> <sup>K</sup>�<sup>1</sup> for water), and <sup>Δ</sup>T as indicated earlier.

#### 1.3. Uncertainty of concentration coming from calibration

Generally, in routine analysis, analytical determinations involve instrumental method where indirect calibration is applied. Common scenarios include external calibration, standard addition calibration (in case of matrix effects) and internal standard calibration (when intrinsic analytical signal variations appear or analyte losses may occur owing to sample preparation procedures [11]).

In case of linear calibration, the calibration straight line is established by preparing calibration standards. The primary stock standard solution is made by weighing the suitable mass of standard ð Þ mstd , of a given purity ð Þ P in the corresponding volume of solvent ð Þ Vs

$$\mathcal{C}\_{std} = \frac{m\_{std}P}{V\_s} \tag{9}$$

But this concentration has an uncertainty derived from the uncertainty in the weighting, in its purity and in the uncertainty of the glassware. The working standard solutions are prepared by diluting a volume ð Þ Vi of the stock standard solution to a final volume Vf . So, the concentration of any calibration standard is given by

$$\mathbf{C}\_{i} = \mathbf{C}\_{std} \frac{V\_{i}}{V\_{f}} = \frac{m\_{std} PV\_{i}}{V\_{s} V\_{f}} \tag{10}$$

and has an uncertainty that can be suitably calculated. However, when applying ordinary least-squares techniques (simple linear regression), three requisites have to be fulfilled [12]:


of linearity of balance, the uncertainty due to temperature effect and the calibration uncer-

m<sup>2</sup>a<sup>2</sup>

As R. Kadis pointed out [10], the evaluation of uncertainty of volumetric measurements consists of three kinds of contributions: specification limits for the glassware of a given class, repeatability of filling the glassware to the mark and temperature effects. Again, in order to avoid double counting and uncertainty redundancy, the precision of filling the flask is not

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ <sup>Δ</sup><sup>T</sup> <sup>2</sup> 3

<sup>T</sup>ð Þ <sup>Δ</sup><sup>T</sup> <sup>2</sup> <sup>9</sup> <sup>þ</sup> <sup>u</sup><sup>2</sup>

CAL

(7)

(8)

(9)

(10)

u mð Þ¼

1.2. Uncertainty of glassware volume

2 3 a2 <sup>L</sup> þ

s

considered here; thus, the uncertainty in the volume measurement is given by

a2 TOL <sup>6</sup> <sup>þ</sup> <sup>χ</sup><sup>2</sup>V<sup>2</sup>

where aTOL is the tolerance for a given class, χ is the dilatation coefficient for the filling liquid

Generally, in routine analysis, analytical determinations involve instrumental method where indirect calibration is applied. Common scenarios include external calibration, standard addition calibration (in case of matrix effects) and internal standard calibration (when intrinsic analytical signal variations appear or analyte losses may occur owing to sample preparation

In case of linear calibration, the calibration straight line is established by preparing calibration standards. The primary stock standard solution is made by weighing the suitable mass of

> Cstd <sup>¼</sup> mstdP Vs

But this concentration has an uncertainty derived from the uncertainty in the weighting, in its purity and in the uncertainty of the glassware. The working standard solutions are prepared by diluting a volume ð Þ Vi of the stock standard solution to a final volume Vf . So, the concen-

> Vi Vf

and has an uncertainty that can be suitably calculated. However, when applying ordinary least-squares techniques (simple linear regression), three requisites have to be fulfilled [12]:

<sup>¼</sup> mstdPVi VsVf

standard ð Þ mstd , of a given purity ð Þ P in the corresponding volume of solvent ð Þ Vs

Ci ¼ Cstd

s

u Vð Þ¼

(2.1 � <sup>10</sup>�<sup>4</sup> <sup>K</sup>�<sup>1</sup> for water), and <sup>Δ</sup>T as indicated earlier.

tration of any calibration standard is given by

1.3. Uncertainty of concentration coming from calibration

tainty

112 Quality Control in Laboratory

procedures [11]).

• The variance of the Y variable, σ2, remains uniform in the whole range of x (homoscedasticity).

In our case, the independent variable is the concentration of standard ð Þ Ci and the Y variable is the analytical signal. In a typical case of multipoint calibration (external or internal), the three requirements mentioned above applies, and the ordinary least-squares procedure gives the calibration straight line Yb<sup>i</sup> ¼ b<sup>0</sup> þ b1Ci. The unknown analyte content is predicted from interpolation of the sample response signal Y<sup>0</sup> according to

$$\mathbb{C}\_{cal} = \frac{Y\_0 - b\_0}{b\_1} \tag{11}$$

whose uncertainty can be estimated from the variance propagation law:

$$\begin{split} \mu^{2}(\mathsf{C}\_{\mathrm{cal}}) &= \left(\frac{\partial \mathsf{C}\_{\mathrm{cal}}}{\partial Y}\right)^{2} \mu^{2}(Y\_{0}) + \left(\frac{\partial \mathsf{C}\_{\mathrm{cal}}}{\partial b\_{0}}\right)^{2} \mu^{2}(b\_{0}) + \left(\frac{\partial \mathsf{C}\_{\mathrm{cal}}}{\partial b\_{1}}\right)^{2} \mu^{2}(b\_{1}) + 2\left(\frac{\partial \mathsf{C}\_{\mathrm{cal}}}{\partial b\_{0}}\right) \left(\frac{\partial \mathsf{C}\_{\mathrm{cal}}}{\partial b\_{1}}\right) \mathrm{cov}\left(b\_{0}, b\_{1}\right) \\ &= \frac{\mu^{2}(Y\_{0})}{b\_{1}^{2}} + \frac{\mu^{2}(b\_{0})}{b\_{1}^{2}} + \frac{\left(Y\_{0} - b\_{0}\right)^{2}}{b\_{1}^{4}} \mu^{2}(b\_{1}) + \frac{\left(Y\_{0} - b\_{0}\right)}{b\_{1}^{3}} \operatorname{cov}\left(b\_{0}, b\_{1}\right) \\ &= \frac{\mu^{2}(Y\_{0})}{b\_{1}^{2}} + \frac{\mu^{2}(b\_{0})}{b\_{1}^{2}} + \frac{\left(Y\_{0} - b\_{0}\right)^{2}}{b\_{1}^{4}} \mu^{2}(b\_{1}) - \frac{\left(Y\_{0} - b\_{0}\right)}{b\_{1}^{3}} \overline{\mathrm{C}}u^{2}(b\_{1}) \end{split} \tag{12}$$

where <sup>C</sup> <sup>¼</sup> <sup>1</sup> N P N i Ci, N being the number of calibration points.

Eq. (10) can be rearranged to give the well-known formula recommended by EURACHEM [2]:

$$\mu(\mathbb{C}\_{cal}) = \frac{s\_{y/x}}{b\_1} \sqrt{\frac{1}{m} + \frac{1}{N} + \frac{\left(\mathbb{C}\_{cal} - \overline{\mathbb{C}}\right)^2}{N}} \tag{13}$$

Here, sy=<sup>x</sup> is the residual standard deviation of the regression line, m is the number of replications measuring the sample signal and N the number of calibration points [13].

Aside from the calibration uncertainty, an additional uncertainty contribution can be considered from the preparation of standards as indicated in Eq. (8) and may be accounted separately in the uncertainty budget:

$$\frac{\mu^2(\mathbb{C}\_i)}{\mathbb{C}\_i^2} = \frac{\mu^2(m\_{std})}{m\_{std}^2} + \frac{\mu^2(P)}{P^2} + \frac{\mu^2(V\_i)}{V\_i^2} + \frac{\mu^2(V\_s)}{V\_s^2} + \frac{\mu^2(V\_f)}{V\_f^2} \tag{14}$$

Thus, the uncertainty of concentration is given by the uncertainty on sample analyte concentration coming from calibration, and the uncertainty due to the preparation of standards.
