New Concepts of Quality Assurance in Analytical Chemistry: Will They Influence the Way We Conduct Science in General? Journal: Chemical Engineering… [600468]

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New Concepts of Quality Assurance in Analytical Chemistry:
Will They Influence the Way We Conduct Science in
General?

Journal: Chemical Engineering Communications
Manuscript ID GCEC-2015-0497.R4
Manuscript Type: 19th Romanian International Conference on Chemistry & Chemical
Engineering (RICCCE 19)
Date Submitted by the Author: 29-Mar-2016
Complete List of Authors: Andersen, Jens; Botswana International University o f Science and
Technology, Chemistry and Forensic Sciences
Glasdam, Sidsel-Marie; Technical University of Denm ark, Chemistry
Larsen, Daniel; Technical University of Denmark, Ch emistry
Molenaar, Nicolaas; Molenaar Geoconsulting,
Keywords: Mathematical modeling, Process control, Quality ass urance, Optimization,
Calibration, Spectrometry

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New Concepts of Quality Assurance in Analytical Che mistry: Will They Influence the
Way We Conduct Science in General?
Jens E.T. Andersen, Sidsel6Marie Glasdam,* Daniel B o Larsen* and Nicolaas Molenaar +

Botswana International University of Science and Te chnology (BIUST), Department of
Chemistry and Forensic Sciences, Private Mail Bag 1 6, Palapye, Botswana
*Technical University of Denmark, Department of Che mistry, Kemitorvet Building 207, DK6
2800 Kgs. Lyngby, Denmark
+Molenaar Geoconsulting, Sophocleslaan 3, 3584 AS Ut recht, Netherlands

Abstract
According to the guide Vocabulary in Metrology (VIM 3) (JCGM 2008), the definition of the
concepts of trueness and accuracy have been revised , which has an important impact on
analytical chemistry. Additionally, Eurachem/CITAC has published a new edition of the
guide to Quantifying Uncertainty in Analytical Meas urement (QUAM) (CITAC & Eurachem
2012). These two documents together, form a new bas is for the evaluation of data. Results of
prominent technologies of inductively6coupled plasm a mass spectrometry (ICP6MS) for
determination of chloride6isotope ratios ( 35 Cl/ 37 Cl) and inductively6coupled plasma optical
emission spectrometry (ICP6OES) for determination o f sodium, were evaluated in terms of
the true level of uncertainty, and revealed a genui ne problem of science that was not treated
in VIM3 and QUAM. Comparison of theory with experim ents definitely requires statistics
tools, but in contemporary science, two approaches to the implementation of statistics in
decision making are used: 1. Short6term precision a nd 2. Long6term precision. Both
approaches are valid and both are described with th e same methods of statistics. However,
they lead to completely different conclusions and d ecisions. Despite good intentions and new
concepts, as well as practices and procedures for Q A, it is shown by the two examples, that
these efforts might be insufficient or mislead scie ntists to making major mistakes in the
decision making process . A set of equations is su pplied, which are based on propagation of
uncertainty and the implication of results and conc lusions for other fields of science are
discussed. Page 1 of 44
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Introduction
Over the past two decades, the Eurachem/CITAC guide (CITAC & Eurachem 2012) has been
revised three times, and together with numerous ISO guides on quality assurance (QA), are
facing scientists with a genuine problem on how to properly report results. It is the obligation
of scientists to prove that results of their invest igations are reproducible, and colleagues from
laboratories anywhere in the World must obtain the same result of measurements of the same
sample, irrespective of the method that was applied to the chemical analysis. Analytical
chemistry is in focus because it is the tool for co rrect quantitation and decision making.
Earlier confusion among scientists between results has been eliminated by realising that large
deviations between apparently similar results could be explained by correctly estimating the
uncertainty of measurement. The uncertainty of meas urement may be associated with either
short6term precision (Einax & Reichenbächer 2006) o r long6term precision (Jens E. T.
Andersen 2014), which occasionally are confused wit h the conventional concepts of precision
and accuracy, respectively. Recent results of QA se em to have introduced some issues of
general interest to science and research. New conce pts for precision and accuracy have been
introduced by Bureau International des Poids et Mes ures (BIPM) and the European network
of analytical chemistry (Eurachem) and the Cooperat ion of International Traceability in
Analytical Chemistry (CITAC), and expanded with ano ther concept denoted as trueness
(JCGM 2008). Revision of these classical concepts h as been undertaken, in order to increase
understanding of quality assurance and statistics i n contemporary analytical chemistry.
Revision of concepts is important but practices and procedures of data management are
equally important. It may seem unnecessary to ‘patc h up’ the old classical statistics but it has
some serious shortcomings that have been elucidated by results of quality assurance and
analytical chemistry (CITAC & Eurachem 2012). It is fairly easy to obtain consensus about
an analytical result, but it is much more difficult to agree about the level of uncertainty,
which effectively hampers the reproducibility that is a fundamental concept of scientific
methodology (Jens E.T. Andersen 2014). In addition to the difficulty of estimating actual
values of uncertainty, it is also difficult to agre e about how results should be interpreted. The
latter is a serious flaw of scientific methodology because everyone must agree about how
statistics should be understood. Still, starting fr om today, it is still a mystery to science how it
is possible to obtain significantly different resul ts for the same sample using the same
apparatus and the same operator. This should not be possible. Otherwise science is open to Page 2 of 44
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consolidation of theories that are, in fact, incorr ect. Determination of chemical species of the
environment in different places around the world mo st frequently exhibit large fluctuations,
which may be interpreted as local differences but, alternatively, there is also the option that
the large differences originate from a poor perform ance of the method (Jens E. T. Andersen
2014). Scientific results must be confronted with w orst6case scenarios of uncertainty that are
related to the concept of accuracy rather than focu sing on precision. All apparatuses for
analytical chemistry may be used to perform measure ments at very high precision but it
should be emphasised that science is not particular ly interested in precision; trueness and
accuracy are thus the concepts in the focus of inte rest. Precision is taken care of by
manufacturers who deliver apparatuses with an ever increasing precision but it may be
difficult to tell anything about trueness and accur acy, unless additional investigations are
performed: This is the duty of science.
A number of investigations point at the option that the level of uncertainty associated with
one particular technology can be predicted for all future measurements by performing a
thorough validation of the method (Konieczka 2007; Jens E.T. Andersen 2014). This option
is investigated in further detail in the present wo rk, where it was demonstrated by results of
ICP6MS and ICP6OES that uncertainty of the calibrat ions (u(cal)) corresponded well to
uncertainty of repetitions (u(rep)) (Andersen & Alf aloje 2013). Equations for calculation of
standard deviations (SD’s), relative SD’s and coeff icients of variations (CV’s) are provided,
and they may be used to the case of linear regressi on. The impact of results of ICP6MS and
ICP6OES on other fields of science is discussed.

Experimental

ICP6MS

Chemicals
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Standards of chloride at concentrations between 0 a nd 165 mg/L of total chloride were
prepared by dilution of stock solution (1000 mg/L) with MilliQ water. Concentrations of
isotopes were calculated by using the value of natu ral ratio of isotopes that was derived upon
the basis of natural abundances. Already at this st age we have some controversy because
several different values of abundancies are availab le. An isotope6ratio 35 Cl/ 37 Cl of 3.0866 was
obtained from the manufacturers’ information sheet, whereas IUPAC gives a ratio of 3.125
(de Laeter et al. 2003). The former value was used since the value could be specific to the
apparatus. Samples of rock material were ground by pestle and mortar, and chloride was
extracted into MilliQ water for 20 min. in quartz v essels of a high6pressure microwave oven
at a temperature of 200 °C and a pressure of 80 bars. Samples were not filte red so remaining
particles of rock materials had to precipitate and settle before liquid was introduced into the
nebulizer of the apparatus.

Apparatus

The measurements were performed by a quadrupole ICP 6MS (Perkin6Elmer Elan 6000)
equipped with a cross6flow nebulizer, and liquids w ere propelled at a flow rate of 0.5
mL/min. by a peristaltic pump (Gilson). The plasma power was 1400 W and the lens voltage
was 7.5 V. The dwell time was 100 ms and the argon6 flow rate to the nebulizer was 0.95
L/min.

ICP6OES

Chemicals
Standards of sodium ions with concentrations of 0 – 5 mg/L were prepared by dilution of a
stock solution of sodium ions at a concentration of 1000 mg/L (Merck). Samples were
prepared from certified reference materials NIST Bo ne Meal (NIST SRM 1486) and NIST
Bone Ash (NIST SRM 1400). Page 4 of 44
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Apparatus
The content of sodium was determined by ICP6OES (Pe rkin6Elmer Optima 3300 DV) using
the sodium6emission line at 589.592 nm. The plasma power was 1300 W, the plasma6gas
flow was 15 L/min., the auxiliary6gas flow was 0.5 L/min. and the nebulizer6gas flow was 0.8
L/min. Standards and sample solutions were introduc ed at a flow rate of 1.5 mL/min.

Results & Discussion

The relative uncertainty (u) of concentration (x) m ay be expressed by the relative standard
deviation (s); (u x/x) = (s x/x). However, uncertainty and standard deviation (S D) are not
similar, and most frequently, the designation u x is reserved for the expanded uncertainty
(CITAC & Eurachem 2012). Within the linear6response range of the apparatus, the SD of
responses is also proportional to concentration wit h a regression line of slope ‘a’ and
intercept ‘b’. The relative SD of a regression line of calibration may be calculated with the
IUPAC method (Danzer & Currie 1998) or with the law 6of6propagation of uncertainty
(CITAC & Eurachem 2012). In the present work, the l atter option was chosen with the
intention of creating an overview of uncertainties, for the sake of simplicity and also to derive
some tendencies of calibrations in general. With sl ope α and intercept β of a regression line,
the corresponding SD’s are denoted as s α and s β, and the RSD of calibration (s x(cal)/x) may
be expressed as (Andersen & Alfaloje 2013):

≠g>C>C≠g3αB>≠g4666≠g3α6α≠g3α5C≠g3α70 ≠g4667
≠g>C34=≠g>B6C
≠g>C6α∙≠g>C34∙≠g34C5≠g>C>C≠g3>α0≠g3ααB≠g>B7B≠g4666≠g>C>C≠g3>0C∙≠g>C34≠g4667≠g3ααB≠g>B7B≠g4666≠g>Cαα∙≠g>C34≠g>B7B≠g>Cα>≠g4667≠g3ααB
≠g>B70 (1)

The parameters ‘a’ and ‘b’ of eq. 1 represent slope and intercept, respectively, of a regression
line that is obtained by a depiction of the SD of c alibration points (s y) as a function of
concentration; hence the term ‘a .x+b’ in eq. 1. Equation 1 multiplied by 100 is the coefficient
of variation (CV) in percent, and it is used to ext ract information about the quality of Page 5 of 44
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measurements. It (eq. 1) may also be used to calcul ate the relative SD (RSD) of a single
unknown, and it shows that the RSD goes to infinity when x approaches zero. However, at
zero concentration, it has the standard deviation o f calibration ‘s x(cal)’ a finite value, which
may be used as a limit of detection (LOD) that take s into account the spread of data around
the calibration line at all concentrations: not exc lusively including signals of blanks in the
conventional formula of LOD.
Now, we consider a transition from a single calibra tion line to several calibration lines that
were determined in multiple experiments in several series of independent experiments. In that
case, eq. 1 remains valid for calculation of the RS D but a much larger value is expected in
comparison with the RSD of a single calibration lin e (Andersen & Alfaloje 2013). This
principle of pooled calibrations may be used to eva luate the results in a worst6case scenario
of uncertainty. It is interesting to consider the c ase when s x(cal)/x = ½ because the
corresponding concentration (x ½) may be used to estimate a lower limit of analysis (LLA)
above which, samples can be analysed with a CV valu e less than 50 %. The LLA is thus an
alternative to the conventional limit of quantitati on (LOQ) that is derived on the basis of
blank values and the slope of a single experiment.
With the condition of s x(cal)/x = ½, we have from eq. 1:

≠g4666≠g>C6α∙≠g>C34½≠g4667≠g3ααB
≠g>B70=s≠g>C6>≠g>B70+≠g4666s≠g>C6α∙x½≠g4667≠g>B70+≠g4666a∙x½+b≠g4667≠g>B70 (2)

Equation 2 is a quadratic equation of x½, as may be seen after expanding and collecting
terms:

≠g467>≠g>C6α≠g3ααB
≠g>B70−s≠g>C6α≠g>B70−a≠g>B70≠g4673∙x½≠g>B70−2∙a∙b∙x ½−s≠g>C6>≠g>B70−b≠g>B70= 0 (3)

The solution to eq. 3 provides LLA, as represented by x½, which may then be expressed as:
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LLA = x ½=≠g>B6C
≠g>C>6∙≠g>Cαα∙≠g>Cα>∙≠g34>B1±≠g34C51+p∙≠g3435s≠g>C6>≠g>B70+b≠g>B70≠g343C≠g343> p,a and b ≠ 0 (4)

Where the parameter ‘p’ is given by:

p=≠g3>0C≠g3ααB
≠g3ααB≠g>B7C≠g>C>C≠g3>0C≠g3ααB≠g>B7C≠g>Cαα≠g3ααB
≠g>Cαα≠g3ααB∙≠g>Cα>≠g3ααB α >≠g34C6s≠g>C6α≠g>B70+≠g>Cαα≠g3ααB
≠g>B6C≠g>B7B≠g467B≠g3α60
≠g3α77≠g3>α0≠g467C≠g3ααB (5)

A negative p6value (eq. 5) indicates that the CV6va lue never reaches 50 %, where eq. 4 then
becomes invalid. Both parameters ‘a’ and ‘b’ may al so become negative, in which case it
may be necessary to include the other solution to t he quadratic equation that was used to
derive eq. 4 (negative sign in bracket of eq. 4). I f these conditions of eq. 5 were not met, it
means that the CV would always reside above 50 %, a nd the CV6value of relevant
concentration may be calculated by the aid of eq. 1 . A simplified version of LLA (a = 0 and b
= 0 in eq. 4) may be found elsewhere (Andersen 2009 ).
The RSD of eq. 1 decreases as a function of concent ration and it has a corresponding limiting
value of CV, as found by the following eq. 6 which thus represents the best CV (BCV) of the
method:

BCV =≠g>C>C≠g3αB>≠g4666≠g3α6α≠g3α5C≠g3α70 ≠g4667
≠g>C34∙100 ≠g>C34→≠g>CCB≠g4657≠gα755≠g4654≠g>B6C≠g>B6B≠g>B6B
≠g>C6α∙≠g34C5≠g>C>C≠g3>0C≠g3ααB≠g>B7B≠g>Cαα≠g3ααB
≠g>B70 % (6)

The BCV of eq. 6 may be used as an estimate of CV6v alues at any concentration above a
certain concentration that is denoted as the start of best range (SBR). The SBR is determined
at the concentration where terms of the equation th at depend on concentration, become
greater than the constants. First, equation 1 may b e re6written as:
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≠g>C>C≠g3αB>≠g4666≠g3α6α≠g3α5C≠g3α70 ≠g4667
≠g>C34=≠g34C5≠g>C>C≠g3>α0≠g3ααB≠g>B7B≠g>Cα>≠g3ααB
≠g>B70∙≠g>C6α≠g3ααB∙≠g>C34≠g3ααB+≠g>Cαα∙≠g>Cα>
≠g>C6α≠g3ααB∙≠g>C34+≠g>C>C≠g3>0C≠g3ααB≠g>B7B≠g>Cαα≠g3ααB
≠g>B70∙≠g>C6α≠g3ααB (7)

Then, when the concentration approaches zero, the R SD thus becomes independent of the
third part under the square root of eq. 7, which le ads to the approximation:

≠g>C>C≠g3αB>≠g4666≠g3α6α≠g3α5C≠g3α70 ≠g4667
≠g>C34 ≠g>C34→≠g>B6B≠g4657≠gα755≠g4654 ≠g>B6C
≠g30B0∙≠g34C5≠g>C>C≠g3>α0≠g3ααB≠g>B7B≠g>Cα>≠g3ααB
≠g>B70∙≠g>C34≠g3ααB+≠g>Cαα∙≠g>Cα>
≠g>C34 (8)

Equation 8 shows that the SD approaches infinity wh en the concentration approaches zero, in
accordance with the Horwitz equation (Horwitz 1982) . Finally, in order to determine SBR,
we equate the SD (CV divided by 100) of eq. 6 with eq. 8, which yields:

SBR =x ≠g>C03≠g>BB6≠g>C0> =≠g>B6C
≠g>C>7∙≠g>Cαα∙≠g>Cα>∙≠g34>B1±≠g34C51+q∙≠g3435s≠g>C6>≠g>B70+b≠g>B70≠g343C≠g343> (9)

and the parameter ‘q’ is given by:

q=≠g>C>C≠g3>0C≠g3ααB≠g>B7B≠g>Cαα≠g3ααB
≠g>Cαα≠g3ααB∙≠g>Cα>≠g3ααB (10)

Together with spreadsheets, equations 1610 provide the tools to create an overview of
uncertainties and the most favourable range of conc entrations for measurements. An
expression for SBR when b is zero is also available (Andersen & Alfaloje 2013).
Determination of the content of an unknown yields a number of values with a corresponding
relative SD of repetition ‘RSD(rep)’ that is determ ined by the conventional formula for
calculation of SD. The purpose of introducing eqs. 1610 is partly to promote simplification
but it may be required to consider three cases that complete the overview. The three cases A, Page 8 of 44
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B and C are based on the assumption that the CV’s a re reliable, which means that a large
number of calibration data and a large number of re petitions are available.
A. CV(cal) >> CV(rep). In this situation the calibrati on is required for each series of
experiments because a large CV(cal) indicates that the operational calibration
effectively eliminates large fluctuations of the ap paratus, e.g. day6to6day variations.
Pooled calibrations cannot be used to predict the u ncertainty of repetitions.
B. CV(cal) = CV(rep). All uncertainties are explained and an uncertainty budget is
unnecessary. The experiments are in statistical con trol.
C. CV(cal) << CV(rep). When CV(cal) are determined by means of pooled calibrations,
it is expected that it accounts for all contributio ns to the uncertainty, except for the
presence of interferences from the sample matrix. T he influence of interferences may
be identified by an increased level of uncertainty of samples of unknowns.
None of the cases A6C indicate anything about the p resence of a potential method bias. The
bias can only be estimated by comparison with certi fied reference materials (CRM’s) (Jorhem
2004) of known contents. Any comparison and decisio n should be performed after a proper
method validation with at least 200 independent rep etitions, and a corresponding number of
data for calibration. The key issue is to make enou gh measurements as to reach a level of
uncertainty that does not change any further by the addition of more measurements; the most
reliable value of uncertainty is represented by lon g6term precision (Jens E. T. Andersen
2014). It may be suggested that the system is in st atistical control when uncertainty is
classified according to the cases that are represen ted by A, B and C (above).
In the following, the problem of classifying uncert ainties is demonstrated with two examples
of ICP6MS and ICP6OES. Figure 1 shows a series of e xperiments for determination of
chloride isotopes by ICP6MS. For a single experimen t, regression lines exhibit coefficients of
regression that are close to one (Figs. 1a and 1b), and low uncertainty of calibration
characterises the short6term precision of the metho d. Long6term precision may be estimated
by the aid of pooled calibrations (Jens E.T. Anders en 2014) (Figs. 1c and 1d) and by either
equations 1610 or the IUPAC formulae (Danzer & Curr ie 1998). Each point of Figs. 1a6b
represents an average of three repetitions of each standard. The regression line of 35 Cl
exhibited an intercept of nought, within limits of confidence ranges (Fig. 1a). The
corresponding regression line of 37 Cl in Fig. 1b has a large positive intercept, which
originates from interference of argon at amu = 37. Pooled calibrations of 35 Cl in Fig. 1c also Page 9 of 44
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have zero intercept, and a characteristic spread of data, which is proportional to
concentration. Similarly, pooled calibrations of 37 Cl in Fig. 1d have spread of data
proportional to concentration but with a higher deg ree of scatter, as compared to that of 35 Cl
in Fig. 1c. It is slightly unusual that SBR and LLA of 35 Cl are comparable (Table 1), and it
should also be noted that the BCV was found well be low 50 %, where eq. 4 is valid.
A lower value of p and q (Table 1) for 37 Cl seems thus to indicate a large scatter of
calibration data with pooled calibrations (Figs. 1c and 1d).
Both isotopes of chloride are influenced by interfe rences whereas 35 Cl is influenced only to a
minor degree by 16 O18 OH and 34 SH, and correction was therefore considered to be
unnecessary, owing to the relatively high level of uncertainty (Figs. 1a61d). Determination of
the concentration of chloride6isotope ion 37 Cl + is influenced by interference from 36 ArH + that
adds significantly to the signal at 37 amu. Correct ion is therefore needed for this argon6
hydride interference, according to the procedure of Fietzke et al. (Fietzke et al. 2008).
Correction for interference yielded a negative p6va lue (eq. 5), an SBR of 60 mg/L and a BCV
of 53 % that are less attractive figures of merits than those presented in table 1. Results of
determination of chloride by the aid of ICP6MS were not found in the literature but an LOQ
of 6 µg/g has however been reported, by applying a reacti on cell (Antes et al. 2010). It was
found that both signals of 35 Cl and 37 Cl correlated with the signal of 40 Ar 2+. Variations of the
40 Ar 2+6signal were interpreted as fluctuations imposed by the apparatus, and accordingly, all
signals should be divided by the signal of 40 Ar 2+ (Amais et al. 2011). By dividing all signals
with the signal of 40 Ar 2+, it was found that figures of merits improved mark edly thus reaching
BCV’s as low as 19 % and 25 % for 35 Cl and 37 Cl, respectively (Table 1, corrected signals).
The decrease in BCV upon correction for argon inter ference strongly indicates that the
procedure of correction was correct; if not correct , the uncertainty would increase owing to
propagation of uncertainty.
A series of samples were analysed in single experim ents (results not shown (Molenaar &
Andersen 2016)), in order to obtain isotope ratios and establish values of CV’s that were
based on repetitions, CV(rep). The average CV(rep) was determined as 21 %, of samples, that
were determined with calibrations of Figs. 1a61d, w hereas the CV(rep) was 12 % of samples
that were determined by calibrations of Figs. 1a an d 1b (N = 62). The latter CV(rep)
corresponds well to the CV6value of the empirical e quation of Horwitz (Horwitz 1982),
which was calculated as an average at 14 % for all samples. These values of CV(rep)’s were Page 10 of 44
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up to an order of magnitude lower than the correspo nding values that were predicted from
pooled calibrations. Results thus show that the exp eriments most likely belong to the type ‘A’
of results (see above) with CV(cal) >> CV(rep). Acc ordingly, it was found that the
uncertainty of pooled calibrations was too large to explain the observed CV(rep)’s. , it is
therefore important to perform calibrations in ever y single experiment because there are
indeed fluctuations of the apparatus that remain un explained. Thus, the true uncertainty is
represented by the uncertainty of repetition, CV(re p). From a scientific point of view, it might
be interesting to study the origin of fluctuations of the apparatus, but the knowledge thus
obtained, would mostly be of value to manufacturers who aim at increasing stability of
apparatuses.
Evaluation of uncertainty by equations 1610 is vali d for a large number of cases but there are
also examples where some of the equations are not v alid, e.g. where LLA is not defined, i.e.,
if CV(cal) were always above 50 %. Such an example is presented in Fig. 3 where the
method of determination of sodium with ICP6OES was investigated. Outliers in vast numbers
were observed in the concentration interval 0 – 2 m g/L but none were seen at higher
concentrations. Since no errors were identified, du ring the series of measurements, all outliers
were retained, thus characterising the true perform ance of the method. Maintaining of outliers
as part of the calibration, yields a negative a6val ue and a negative p6value (eq. 5). Therefore,
the CV6value never becomes lower than 50 % and SBR, when calculated by eq. 9, should be
used with the minus sign. Upper and lower confidenc e ranges are shown by broken lines. The
BCV was determined as 58 % (Fig. 3).
Determination of concentration of samples (not show n) with concentrations above 2 mg/L
(Fig. 3) provided results with CV(rep)’s of 5610 % that is less than the CV(cal) (> 58 %)
predicted by pooled calibrations (Fig. 3). The corr esponding CV’s of the Horwitz formula
(Horwitz 1982) were calculated as 5 %, which showed that CV(rep) most likely is the correct
expression for uncertainty of this particular exper iment.
From these investigations of the methods of ICP6OES and ICP6MS, it may seem that type A
experiments are common but this is definitely not t he case; these two examples are
exceptions to the rule. Most frequently, the predic ted uncertainty corresponds (CV(cal)) of
pooled calibrations well to CV(rep) (Jens E.T. Ande rsen 2014). Many statistical tools are
available to identify and remove outliers from data sets (Glasser 2007; Barbato et al. 2011)
but this practice should not prevail because it des troys the picture of the true uncertainty of Page 11 of 44
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the method, and it becomes impossible to classify t he method into types A, B or C. The
analysis of the method presented above ought to be an integrated part of method validations
because the analysis offers an opportunity to ident ify sources of uncertainty and it facilitates
decision making. Manufacturers rarely show results like those of Figs. 1a61d and Figs. 2a62b
in sales material because they focus on short6term precision which are important tools in
convincing customers of the high performance of the apparatus. Some customers may believe
that high precision underpins trueness and accuracy , which actually is not the case, as the
present investigation and also other investigations have shown (Jens E.T. Andersen 2014).
The LOD’s are approx. 0.001 mg/L for determination of chloride with ICP6MS and for
determination of sodium with ICP6OES, as given in t he information supplied by the
manufacturers. However, scientists frequently repor t LOD’s with orders of magnitude higher
than these values (Sapkota et al. 2005). Reports wi th determination of sodium by ICP6OES
are available with LOD’s within the range from 0.8 µg/L to 597 µg/L (Harrington et al. 2014;
Krachler et al. 2012), depending on the type of mat rix. The LOD of the best possible
regression line of experiments of Fig. 3 was determ ined as 3.4 µg/L, whereas the LLA was
determined as 2.1 mg/L, which is comparable to the LOD of Na in serum (Harrington et al.
2014) but much higher than LOD’s of manufacturers’ specifications.
It might be useful to apply confidence ranges (CR’s ) to the evaluation of the performance of
the method but there is a drawback to that procedur e, which influences decision making. CR
will narrow as a function of the number (N) of repe ated experiments, and CR’s approaching
zero when N is a very large number. It is thus anti cipated that the average value of numerous
experiments approach the one and genuinely true val ue but this is not reality; it is merely
theory. Reality shows that a very high number of re petitions provides a value that is
characteristic of the apparatus and the measurand ( CITAC & Eurachem 2012) in
combination, whereas the theory postulates that cal ibration eliminates the apparatus’
influence on the result. The apparatus may have a l arge influence on the result, as proven by
the present two cases, but they seem to be the exce ption to the rule.
Suppose that two independent laboratories Lab1 and Lab2, report results as average values
with corresponding SD’s x ≠g3364≠g>B6C±s≠g>B6C and x ≠g3364≠g>B70± s≠g>B70 of measurements of a certified reference
material with a certified value of x CRM . The theoretical difference between results is µ1 – µ2,
which should be zero if the two results were simila r. However, the similarity between results
depends on the number of measurements (N), and at a certain number of repetitions, as given Page 12 of 44
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by ‘N D’, the results start to disagree (Fig. 4a), accordi ng to Student’s t6test (t: Student’s t6
value) (Konieczka 2007):

N≠g>BBB= t≠g>B70∙≠g467>≠g>C>C≠g3αα7≠g>B7B≠g>C>C≠g3ααB
≠g>C34≠g3364≠g3αα7≠g>B7C≠g>C34≠g3364≠g3ααB≠g4673≠g>B70 (11)

However, if the two laboratories were encouraged to share measurements and merge data
sets, disagreement would occur at an even lower num ber of repetitions, N D’:

N≠g>BBB≠g45C3= t≠g>B70∙≠g>C>C≠g3αα7≠g3ααB≠g>B7B≠g>C>C≠g3ααB≠g3ααB
≠g4666≠g>C34≠g3364≠g3αα7≠g>B7C≠g>C34≠g3364≠g3ααB≠g4667≠g3ααB (12)

Initial agreement (Fig. 4a) may be transformed into disagreement simply by increasing the
number of repetitions (N), and the disagreement is realised after a number of repetitions, as
given by N D (eq. 12). Application of expanded uncertainty of E urachem and CITAC may
improve the general level of agreement (Fig. 4b). H owever, in none of the cases Figs. 4a6b is
guaranteed compliance with CRM (dotted line).
It is a paradox that an increased number of measure ments automatically introduces
disagreement between results, according to theory. But theory does not take into account that
the combination of apparatus and sample provides re sults, which may deviate significantly
from the expected value of unknown, as mentioned ab ove. According to eqs. 11 and 12,
scientists would be prone to discuss differences th at appeared on an entirely accidental basis,
that is, if a laboratory were performing more exper iments than another laboratory did. In
principle, there is no need to struggle with calibr ation of all features of the apparatus because
improved precision is no guarantee that accuracy im proves (Figs. 4a64b). In recognition of
these problems with CR’s, Eurachem recommends creat ing a full overview of all
participating uncertainties that are involved in th e process of measurement, and from the
uncertainty budget calculate an expanded uncertaint y that may provide full agreement (Fig.
4b) (CITAC & Eurachem 2012). Despite these efforts there is still no guarantee of high
accuracy. Results may differ anyway. In addition to the problems of comparison, there is a Page 13 of 44
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problem of removing outliers from data sets of CRM’ s, which provides unrealistically low
uncertainties (Jens E.T. Andersen 2014) that makes it impossible for customers to compare
with own results (Jens E.T. Andersen 2014). Serious manipulations of data sets have been
identified for CRM’s of Institute of Reference Mate rials and Measurements (IRMM), where
poor results were converted into excellent results by elimination of outliers from data sets of
professional laboratories (Jens E.T. Andersen 2014) . Such CRM’s are unacceptable for
science and it is even more disagreeable that some manufactures of CRM’s do not even
provide any information about raw data and potentia l outliers that were used to produce the
respective CRM. It is very worrying that scientists cannot trust in the CRM’s since they are
supposed to be the fundamental standards of analyti cal chemistry. It would thus be more
feasible to accept the performance of the apparatus per se and report results with whichever
SD’s that may emerge. Elimination of day6to6day var iations of the apparatus by calibrations
was found true for two cases, one of chloride that was determined by ICP6MS and one with
determination of sodium by ICP6OES (see above). How ever, these examples may be
considered as rare and extreme cases of analytical chemistry (Jens E.T. Andersen 2014). In
recognition of this drawback in relation to compari son of results, Eurachem and CITAC have
introduced an expanded uncertainty that is equal to the combined uncertainty of the
uncertainty6budget multiplied by a coverage factor (CITAC & Eurachem 2012). There are
thus three major issues, number of measurements, ou tliers and CR’s, which need further
attention in order to facilitate comparisons (Magnu sson et al. 2008; Medeiros & Carla 2014;
Williams 2008). Although the two examples above ill ustrate some important issues in
analytical chemistry, the findings may also transfe r some implications to other fields of
science. Measurements within the health sector (Sch aller et al. 2002) must be under complete
statistical control. Otherwise, it might cause seri ous health issues of individual patients.
Diagnoses of patients rely on laboratory results (S zecsi & Ødum 2009; Bonini et al. 2002)
although not exclusively; diagnosis also depend on evaluation of symptoms and on
conversations. However, analytical chemistry focuse s on creating methods that have the final
say with respect to diagnosis and evaluation of the efficacy of treatment. Can we assume, that
all clinical measurements were properly validated w ith endless series of measurements that
have been performed in independent laboratories? Bu t in other areas of science, it may not be
so clear how many independent laboratories were inv olved in the measurements.
Fundamental constants of physics are known to be de termined experimentally at very high
accuracy; an accuracy that seems by far out of reac h of analytical chemistry. However, recent
discussions about revision of SI units (Leonard 201 4; de Bièvre 2012) focused on revising the Page 14 of 44
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mol unit, which has implications regarding Avogadro ’s number but in view of pooled
calibrations and classifications A, B and C above, some of the other fundamental constants
may also require renewed considerations. BIPM has p ublished a roadmap towards
redefinition of SI units, which is scheduled for 20 18 (BIPM 2014; Taylor 2011). Levels of
uncertainty of 2 parts in 10 8 have already at this stage of planning been mentio ned (BIPM
2014), which may also need further considerations i n terms of precision, trueness and
accuracy.

Conclusion

Two examples were presented where calibrations were important, in order to obtain the
correct and reliable values for concentrations in s amples. These may seem merely as
examples that illustrate common practice in analyti cal chemistry but, as previous results
indicate, these two results of the present work wer e exceptions to the rule in contradiction to
conventional practices and procedures of QA. It is important to provide an overview of
uncertainty of measurement during the method valida tion that accompanies any process
towards laboratory accreditations.
It is claimed that uncertainties may be characteris ed by three classes A, B and C that provide
information about the source of uncertainty. In cla ss A, uncertainty of pooled calibrations is
too large to explain observed uncertainty of repeti tion whilst uncertainties of class B provide
complete correspondence between predicted uncertain ty (u(cal)) and observed uncertainty
(u(rep)). Uncertainties of class C reveal additiona l uncertainty of sample interferences that
adds to the observed uncertainty (u(rep)). Classifi cation of uncertainty into classes A, B and
C is only possible, if a large number of independen t measurements are applied to the analysis.
Removing of outliers from the data set is not an op tion since such practice destroys the
general picture of uncertainty, particularly in exp eriments with a low number of repetitions.
A simplified procedure for evaluating uncertainties was presented in eqs. 1610, which was
derived by the law6of6propagation of uncertainty. C alculations were facilitated by using
spreadsheets with built6in functions for determinat ion of uncertainty of slope and intercept
from regression lines. Page 15 of 44
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Finally, it is claimed that the evaluation of uncer tainty that is applied to investigations within
the area of analytical chemistry has implications t o other fields of science. It is very
confusing to the general public and for science, if there is no clear distinction between the
concepts of precision, trueness and accuracy. CRM’s can only be trusted when manufacturers
provide complete information about all data involve d in the production process. Raw data are
needed, similar to those that are provided by CRM’s of IRMM. Application of fundamental
constants of physics, relies on very low levels of uncertainty but if the uncertainty of
measurement is related to precision rather than tru eness or accuracy, then it becomes an issue
regarding scientific importance. No genuinely true values of fundamental constants are
available and nobody knows the genuinely true conce ntration of chemicals in e.g, samples of
blood. Many independent series of measurements are required in order to obtain a consensus
value with the correct level of uncertainty that ma y be estimated after evaluation of data
according to eqs. 1610. Approximations made earlier (Andersen 2009) showed themselves
inadequate for description of measurements of large SD’s and a new set of simple equations
were suggested for calculation of LLA, SBR and BCV (Eqs. 1610).

Acknowledgements
Many thanks are due to the Organizers of the RICCCE 19 Conference in Sibiu for giving
corresponding author the opportunity to present res ults at a plenary lecture. The financial
support from Director Ib Henriksen’s Foundation and Brdr. Hartmann’s Foundation is
gratefully acknowledged. Many thanks are also due t o Engineers Flemming Hansen and
Claus Lilliecrona for their continued technical sup port.

References
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samples using inductively coupled plasma6optical em ission spectrometry. Analytica
Chimica Acta , 540(2), pp.247–256.
Schaller, K.H., Angerer, J. & Drexler, H., 2002. Qu ality assurance of biological monitoring
in occupational and environmental medicine. Journal of Chromatography B: Analytical
Technologies in the Biomedical and Life Sciences , 778, pp.403–417.
Szecsi, P.B. & Ødum, L., 2009. Error tracking in a clinical biochemistry laboratory. Clinical
Chemistry and Laboratory Medicine , 47(10), pp.1253–1257.
Taylor, B.N., 2011. The current SI seen from the pe rspective of the proposed new SI. Journal
of Research of the National Institute of Standards and Technology , 116(6), p.797.
Williams, A., 2008. Principles of the EURACHEM / CI TAC guide “Use of uncertainty
information in compliance assessment .” Accreditation and Quality Assurance , 13,
pp.633–638.

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For Peer Review OnlyParameter 35 Cl 37 Cl 35 Cl corrected 37Cl corrected
αα αα ( ( ( ( cps .L/mg) 25000 30000 0.0017 L/mg 0.0018 L/mg
sαα αα ( ( ( ( cps .L/mg) 3100 4700 0.00015 L/mg 0.00046 L/mg
ββ ββ (cps) 210000 2.7 .10 6 0.024 * 0.028 *
sββ ββ (cps) 210000 1.0 .10 5 0.010 * 0.010 *
a (( ((cps .L/mg) 13000 12000 4.5 .10 -4 L/mg 4.0 .10 -4 L/mg
b (cps) 15000 170000 7.6 .10 -3 * 3.1 .10 -2 *
p (1/cps 2) 3.9 .10 -9 7.2 .10 -11 110000 * 7700 *
q ((1/cps 2) 4.7 .10 -9 4.1 .10 -11 19000 * 2500 *
LLA (mg/L) 20 20 15 42
SBR (mg/L) 18 32 47 95
BCV (%) 37 30 19 25
*Unitless
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For Peer Review OnlyFigure captions

Figure 1. Operational calibration of ICP-MS for determinatio n of chloride isotopes 35 Cl and
37 Cl. a) 35 Cl, single experiment of ordinary-linear regression (OLR) b) 37Cl, single
experiment of OLR c) 35 Cl, pooled calibrations of four repetitive experime nts d) 37Cl,
pooled calibrations of four repetitive experiments.

Figure 2. After correction of signals (figs. 1c and 1d) for interference of 36 ArH + on 37 Cl +
(Fietzke et al. 2008) and normalisation with the si gnal of 40 Ar 2+ , scatter of data was reduced
mainly for a) 35 Cl and to some extent for results in b) 37 Cl, as compared with the scatter of
data in figs. 1c and 1d, respectively.

Figure 3. Pooled calibrations of determination of sodium (Na ) that were measured by ICP-
OES. Ten repetitive experiments.

Figure 4. Agreement and disagreement between tentative resul ts of laboratories illustrated by
depicting confidence ranges (CR’s) as a function of number of measurements (N) of two
independent laboratories Lab1 (Thick broken line) a nd Lab2 (Thin broken line) with
associated CR’s given by ( /circle6) and ( /rhombus6), respectively. a) Conventional t-testing b) Expa nded
uncertainty.
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New Concepts of Quality Assurance

in

Analytical Chemistry: Will They Influence the
Way We Conduct Science

in

General
?

Jens E.T. Andersen
, Sidsel
–
Marie Glasdam
,
*

Daniel Bo Larsen
*

and
Nicolaas Molenaar
+

Botswana International University of Science and
Technology (BIUST), Department of
Chemistry and Forensic Sciences, Private Mail Bag 16, Palapye, Botswana

*Technical University of Denmark, Department of Chemistry, Kemitorvet Building 207, DK
–
2800 Kgs. Lyngby, Denmark

+
Molenaar Geoconsulting,
Sophocleslaan 3, 3584 AS Utrecht, Netherlands

Abstract

A
ccording to the guide Vocabulary in Metrology (VIM3)
(JCGM 2008)
,
t
he
definition

of the
conce
pt
s

of trueness and accuracy
have

been revised
, which has an important impact on
analytical chemistry.
Additionally,
Eurachem/CITAC has published a new edition of the
guide to Quantifying Uncertainty in Analytical Measurement (QUAM)
(CITAC & Eurachem
2012)
. These two documents
together,
form a new basis for
the
ev
aluation of data.
Results of
prominent technologies of
inductively
–
coupled plasma mass spectrometry (ICP
–
MS)
for
determination of chloride
–
isotope ratios (
35
Cl/
37
Cl)
and inductively
–
coupled plasma optical
emission spectrometry
(ICP
–
OES)
for
determination
of sodium
,

were evaluated in terms of
the
true level of uncertainty
,
and
revealed a genuine problem of science

that was not treated
in VIM3 and QUAM
.
Comparison of theory with experiments

definitily
definitely

requires
statistics
tools
,

but in contemporary science
, t
wo approaches

to
the
implementation of
statistics
in

decision making

are used
: 1. Short
–
term precision and 2. Long
–
term precision.
Both approaches are valid and both
are
described
with

the same methods of statistics
.
However, they

lead to completely different conclusions and decisions. Despite good
intentions and new concepts
,

as well as

practices and procedures for QA, it is shown
by the
two

examples, that these efforts might be insufficient or mislead scientist
s

to making major
m
istakes in the
decision making

process .

A set of equations is supplied, which are based on
propagation of uncertainty

and the implication

of results and conclusions for other
fields of
science are

discussed.
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2

Introduction

Over the past two decades, the Eurachem/CITAC guide
(CITAC & Eurachem 2012)

has been
revised three times, and together with numerous ISO guides on quality assurance

(QA)
,
are
fac
ing

scientists with a genuine problem
on

how to properly report results.

.
The
days
are
over where perform
ing

a few measurements of preliminary or unproven data
was

the basis of
science.

It is the obligation of scientists to prove that results of their invest
igations are
reproducible, and colleagues
from

laboratories anywhere in the World must obtain the same
result of measurements of the same sample, irrespective of the method
that was
applied to the
chemical analysis. Analytical chemistry is in focus because it is the tool
for

correct
quantitation and decision making. Earlier confusion among scientist
s

between
results
,
results

has

been eliminated by realising that large deviations between appar
ently similar
results

could be explained by
correctly
estimating

the

uncertainty of measurement. The uncertainty
of measurement may be associated with either short
–
term precision
(Einax & Reichenbächer
2006)

or long
–
term precision

(Jens E. T. Andersen 2014)
, which occasionally are confused
with the
conventional
concepts of precision and accuracy, respectively.

Recent results of QA
seem to have introduced some issues of general interest to science and res
earch.
New
concepts
for

precision and accuracy have been introduced by
Bureau International de
s

Poids
et Mesures (
BIPM
)

and
the European network of analytical chemistry (
Eurachem
) and the
Cooperation of International Traceability in Analytical Chemistry (
CITAC
)
,

and expanded
with another concept denoted as trueness

(JCGM 2008
)
. Revision of these classical concepts
has been undertaken, in order to increase understanding of quality assurance and statistics in
contemporary
analytical chemistry.
Revision of concepts is important but practices and
procedures of data ma
nagement a
re equally

important.
It may seem unnecessary to ‘patch up’
the old classical statistics but it has some serious shortcomings that have been elucidated by
results of quality assurance and analytical chemistry

(CITAC & Eurachem 2012)
.
It is fairly
easy to obtain consensus about an analytical result
,

but it is much more

difficult to agree
about

the level of uncertainty, which effectively hampers
the
reproducibility that is a
fundamental concept of
scientific methodology

(Jens E.T. Andersen 2014)
. In addition to the
difficulty of
estimating

actual values of uncertainty, it is also d
ifficult to agree about

how
results should be interpreted. The latter is a serious flaw of scientific methodology because
e
veryone must agree about
how s
tatistics should be understood.
Still
,
starting from

today
, it is
still a mystery to science how it is possible to obtain significantly different results
for
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3

same sample using the same apparatus and the same operator
. T
his should not be possible.
Otherwise science
is
open to consolidation of theories that are, in fact,

in
correct.
Determination of chemical species of the environment
in
diffe
rent places around the world
most frequently

exhibit large fluctuations, which
may

be interpreted as local differences but,
alternatively, there is also the option that the large differences originate from
a
poor
performance of the method

(Jens E. T. Andersen 2014)
.
Scientific results must be confronted
with worst
–
case scenarios of uncertainty that are related to the concept of accuracy rather than
focusing on precision. All apparatuses
for

analytical chemistry may be used to perform
measurements at very high precisio
n but it should be emphasised that science is not
particularly interested in precision; trueness and accuracy are
thus
the concepts
in
the
focus of
interest
. Precision is taken care of by manufacturers who deliver apparatuses with
an
ever
increasing precis
ion but it may be difficult to tell anything about trueness and accuracy,
unless additional investigations are performed
: This

is the duty of science.

A number of investigations point
at

the option that the level of uncertainty associated with
one particular technology can be predicted for all future measurements by
performing

a
thorough validation

of the method

(Konieczka 2007; Jens E.T. Andersen 2014)
. This option
is investigated in further detail in the present work, where
it was demonstrated by results of
ICP
–
MS and ICP
–
OES

that uncertainty of
the
calibrations (u(cal)) correspond
ed

well

to
uncertainty of repetitions (u(rep))

(Andersen & Alfaloje 2013)
.

Equations for calculation of
standard deviations (SD’s), relative SD’s and

coefficients of variations (CV’s) are provided
,
and they may be
used
to the case of linear regression. The impact
of
results of ICP
–
MS and
ICP
–
OES
on other fields of science
is discussed.

Experimental

ICP
–
MS

Chemicals

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Standards of chloride
at concentrations between 0 and 165 mg/L of total chloride
were
prepared
by dilution of stock solution (1000 mg/L) with

MilliQ water. Concentrations of
isotopes were calculated by using the value of natural ratio

of isotopes

that was derived upon
the basis

of natural
abundances
.
Already at this stage we have some controvers
y because
several
different
values of abunda
ncies are available. An isotope
–
ratio
35
Cl/
37
Cl
of 3.0866 was
obtained from
the
manufacturers
’

information sheet
,

whereas IUPAC gives a ratio of 3.125
(de Laeter et al. 2003)
. The former value was used since the value
could
be specific
to
the
apparatus.
Samples of rock material were

ground by pestle and mortar
,

and chloride was
extracted into MilliQ water for
20 min. in
quartz vessels of
a high
–
pressure microwave oven
at
a temperature of
200

C and
a pressure of
80 bars.
Samples were not filtered so remaining
particles of rock materials had to precipitate and settle before liquid was introduced into the
nebulizer of the apparatus.

Apparatus

The measurements were performed by a quadrupole ICP
–
MS (Perkin
–
Elmer Elan 6000)
equipped with a cross
–
flow nebulizer, and liquids were propelled at a flow rate of 0.5
mL/min. by a peristaltic pump (Gilson).
The plasm
a power was 1400

W and

the lens voltage
was 7.5 V
. T
he dwell time was 100 ms

and the argon
–
flow rate
to the nebulizer
was
0.95
L
/min.

ICP
–
OES

Chemicals

Standards of sodium ions with concentrations of 0
–

5 mg/L were prepared by dilution of a
stock solution of sodium ions at a concentration of 1000 mg/L (Merck).
Samples were
prepared from certified reference materials NIST Bone Meal

(NIST SRM 1486)

and NIS
T
Bone Ash

(NIST SRM 1400)
.
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5

Apparatus

The content of sodium was determined by ICP
–
OES
(Perkin
–
Elmer Optima 3300 DV)
using
the sodium
–
emission line at 589.592 nm. The plasma power was 1300 W, the plasma
–
gas
flow was 15 L/min., the auxiliary
–
gas flow was 0.
5 L/min. and the nebulizer
–
gas flow was 0.8
L/min. Standards and sample solutions were introduced at a flow rate of 1.5 mL/min.

Results

& Discussion

The relative uncertainty

(u)

of
concentration (x) may be expressed by the relative standard
deviation
(s); (
u
x
/x) =
(s
x
/x). However, uncertainty and standard deviation (SD) are not
similar, and mos
t frequently
,

the
designation
u
x

is reserved for the expanded uncertainty

(CITAC & Eurachem 2012)
. Within the linear
–
response range of the apparatus, the SD of
responses is also proportional to concentration with a regression line of slope
‘
a
’

and
intercept
‘
b
’
.

The relative SD
of a regression line of calibration
may be cal
culated with the
IUPAC metho
d
(Danzer & Currie 1998)

or wit
h the law
–
of
–
propagation of uncertainty

(CITAC & Eurachem 2012)
.

In the present work
,

the latter option

was

chosen
with the
intention of creating an overview of uncertainties, for the sake of simpli
city and also to derive
some tendencies of calibrations in general. With slope


and intercept


of a regression line,
the corresponding SD’s are denoted as s


and s

, and the RSD of calibration
(s
x
(cal
)/x)

may
be expressed as

(Andersen & Alfaloje 2013)
:

(

)

=


∙

∙





(


∙

)


(

∙



)

(1)

The parameters ‘a’ and ‘b’ of eq. 1 represent slope and intercept, respectively, of a regression
line that is obtained by
a
depicti
on

of
the SD of calibration points (s
y
) as a function of
concentration; hence the term ‘a
.
x+b’ in eq. 1.
Equation 1 multiplie
d by 100 is the coefficient
of variation (CV) in percent, and it is used to extract information about the quality of Page 30 of 44
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measurements. It (eq. 1) may also be used to calculate the relative SD (RSD)

of a single
unknown, and it shows that the RSD
goes to
infinit
y when x approaches zero.
However, a
t
zero concentration
,

it
has

the standard deviation of calibration ‘
s
x
(cal
)
’

a finite value
,

which

may be used as a limit of detection (LOD) that takes into account the spread of data around
the calibration line at all concentrations: not exclusively including signals of blanks in the
conventional formula of LOD.

Now, we consider a transition from a si
ngle calibration line
to several calibration lines
that
were

determined in multiple experiments
in

several series of independent
experiments.
In that
case, eq. 1
remains

valid for calculation of the RSD but a much larger value is expected in
comparison wit
h the RSD of a single calibration line

(Andersen & Alfaloje 2013)
. This
principle of pooled calibrations may be used to evaluate the results in a worst
–
case scenario
of uncertainty.
It is interesting to consider the case w
hen
s
x
(cal
)/x

= ½

because the
corresponding concentration (x
½
) may be used to estimate a lower limi
t of analysis (LLA)
above which, samples can be analysed
with
a CV value less than 50 %. The LLA is thus an
alternative to the conventional limit of quantitation (LOQ) that is derived on the basis of
blank values
and
the
slope
of a sin
gle experiment.

With the condition of s
x
(cal
)/x

= ½, we have from eq. 1:

(

∙

½
)


=
s


+
(
s

∙
x
½
)

+
(
a
∙
x
½
+
b
)

(2)

Equation 2 is a quadratic equation of
x
½
, as may be seen after expanding and collect
ing

terms:

−
s


−
a


∙
x
½

−
2
∙
a
∙
b
∙
x
½
−
s


−
b

=
0

(3)

The
solution to eq. 3 provides
LLA
, as represented by
x
½
, which

may then be expressed as:

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LLA
=
x
½
=


∙

∙

∙

1
±

1
+
p
∙

s


+
b

p
,
a

and

b

≠
0

(
4
)

Where the parameter ‘
p
’ is given by:

p
=












∙

α
>

s


+

(
5
)

A negative
p
–
value (eq.
5
) indicates
that the CV
–
value never reaches 50 %, w
here eq.
4
then
becomes
invalid
. Both
parameters ‘
a
’

and
‘
b
’

may also become negative, in which case it
may be necessary to include the other solution to the quadratic equation that was used to
derive eq.
4
(negative
sign in bracket of eq.
4
).

If
these conditions of eq.

5
were not met, it
means that the CV would always reside above 50 %, and the CV
–
value of relevant
concentration may be calculated by
the aid of
eq. 1. A simplified
version
of LLA
(a = 0 and b
= 0 in eq.
4
)
may be found elsewhere
(Andersen 2009)
.

T
he RSD of eq. 1 decreas
es

as a function of
concentration,
concentration

and it has a
corresponding limiting value of CV, as
found
by
the following
eq.
6
which thus represents
the best CV (BCV) of the method:

BCV
=


(

)

∙
100

→


⎯



∙

%

(
6
)

The BCV of eq.
6
may be used as an
estimate of CV
–
values at any concentration above a
certain concentration that is denoted as the start of best range (SBR)
. The SBR

is determined
at the
concentration
where terms
of the equation
that depend on concentration
,

become
greater

than the constant
s.

First, equation 1
may
be
re
–
written as:

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(

)

=








∙


∙


+

∙



∙

+







∙

(
7
)

Then, w
hen the concentration approaches zero, the RSD thus
becomes independent of the
third
part
under the square root of eq.
7
,
which leads to the
approximation
:

(

)

→


⎯

∙








∙


+

∙

(
8
)

Equation
8

shows that the SD approaches infinity when the concentration approaches zero, in
accordance with the Horwitz equation
(Horwitz 1982)
.
Finally, in order to determine

SBR
,

we equate the SD
(CV divided by 100)
of
eq.
6

with eq.
8
, which yields
:

SBR
=
x

=


∙

∙

∙

1
±

1
+
q
∙

s


+
b

(
9
)

and the parameter ‘q’ is given by:

q
=








∙

(
10
)

Together with spreadsheets
,

equations
1
–
10
provide
the tools to create an overview of
uncertainties and
the
most favourable range of concentrations
for
measurements. An
expression for SBR when
b

is zero is also available
(Andersen & Alfaloje 2013)
.

Determ
ination of the content of an unknown yields a number of values with a corresponding
relative SD of repetition ‘
RSD(rep)
’

that is determined by the conventional formula
for
calculation of
SD.
The purpose of introducing eqs. 1
–
10
is
partly
to promote simplif
ication
but it may be required to consider three cases that complete the overview. The three cases A, Page 33 of 44
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B and C are based on the assumption that the
CV
’s are reliable, which means that a large
number of calibration data and a large number of repetitions are
available.

A.

CV
(cal) >>
CV
(rep). In this situation the calibration is required for each series of
experiments because a large
CV
(cal) indicates that the operational calibration
effectively eliminates large fluctuations of the apparatus, e.g. day
–
to
–
day
variations.
Pooled calibrations cannot be used to predict the uncertainty of repetitions.

B.

CV
(cal) =
CV
(rep). All uncertainties are explained and an uncertainty budget is

unnecessary. The experiments are

in statistical control.

C.

CV
(cal) <<
CV
(rep).
When
CV
(
cal) are determined by means of pooled calibrations
,
it is

expected that it accounts for all contributions to the uncertainty, except for the
presence of interferences from the sample matrix. The influence of interferences may
be identified by an increased

level of uncertainty of samples of unknowns.

None of the cases A
–
C indicate anything about the presence of a potential method bias
. The
bias can only be estimated by comparison with certified reference materials
(CRM’s)

(Jorhem
2004)

of known contents.

Any comparison and decision should be performed after a proper
method validation with at least 200 independent repetitions, and a corresponding number of
data for calibration. The key issue is to make enough measurements as to reach a level of
uncertainty

that does
not
change
any
further by

the

addition of
more measurements; the most
reliable value of uncertainty is represented by long
–
term precision

(Jens E. T. Andersen
2014)
.

It may be suggested that the system
is
in st
atistical control when uncertainty is
classified according to the
cases
that are
represented by A, B and C (above).

In the following, the problem of classifying uncertainties is demonstrated
with

two examples
of ICP
–
MS and ICP
–
OES.
Figure 1

shows a series of experiments
for
determination of
chloride isotopes by ICP
–
MS
.
For a
single experiment
,

r
egre
ssion lines
exhibit coefficients of
regression that are close to one

(F
igs. 1a and 1
b), and low uncertainty of calibration
characterises the shor
t
–
term precision of the method. Long
–
term precision may be estimated
by the aid

of pooled calibrations

(Jens E.T. Andersen 2014)

(F
igs. 1c and 1
d)

and
by
either
eq
uation
s 1
–
10
or the IUPAC formulae
(Danzer & Currie 1998)
.

Each point
of Figs. 1a
–
b
represents an average of thre
e repetitions

of each standard. The regression line of

35
Cl
exhibited an intercept of

nought, within limits of confidence ranges

(Fig. 1a)
. The
corresponding regression line of
37
Cl in
Fig. 1b

has a large positive intercept, which
originates from interference of argon
at amu = 37. Pooled calibrations of
35
Cl in
Fig. 1c

also Page 34 of 44
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have zero intercept, and a characteristic spread of data, which is proportional to
concentration. Similarly, pooled calibrations of
37
Cl in
Fig. 1d

have spread of data
proportional to concentration but with a higher degree of scatter, as compared to that of
35
Cl
in Fig. 1c
.

It is slightly unusual that SBR
and
LLA
of
35
Cl are
comparable

(T
able 1
)
, and it
should also be noted that the BC
V
was found
wel
l below 50 %
,

where
eq.
4 is valid
.

A lower

value of p and q
(T
able 1
)
for
37
Cl
seems thus to indicate a large scatter of
calibration
data

with

pooled calibrations
(F
ig
s
. 1c and 1
d)
.

Both isotopes of chloride are influenced by interferences
whereas
35
Cl is influenced only to a
minor degree by
16
O
18
OH and
34
SH,
and
correction was th
erefore

considered
to be
unnecessary
, owing to the relatively high level of

uncertainty (F
igs. 1a
–
1
d)
.
Determination of
the
concentration of
chloride
–
isotope ion

37
Cl
+

is influenced by interference from
36
ArH
+

that
adds
significantly
to the signal at 37 amu.
C
orrection is
therefore
needed for this argon
–
hydride interference, according to the procedure of Fietzke et al.
(Fietzke et al. 2008)
.
Correction for interference
yielded
a negative p
–
value (eq.
5
), an SBR of 60 mg/L and a BC
V
of
53 % that are less attractive figures of merits t
han those presented

in table 1.

Result
s of
determination of chloride
by the aid of
ICP
–
MS
were not found

in the literature but an LOQ
of 6


g/g has

however

been reported, by applying a reaction cell
(Antes et al. 2010)
.
It w
as
found that both signals of
35
Cl and
37
Cl correlated with the signal of
40
Ar
2
+
. Variations of the
40
Ar
2
+
–
signal were interpreted as fluctuation
s imposed by the apparatus, and

accordingly, all
signals
should
be divided by the signal of
40
Ar
2
+

(Amais et al. 2011)
.
B
y
dividing

all signals
with the signal of
40
Ar
2
+
, it was found that figures of merits improved markedly
thus reaching
BCV’s as low
as 19

%

and 25 % for
35
Cl and
37
Cl, respectively
(T
able 1, corrected signals).
The decrease in BCV upon correction for argon interference strongly indicates that the
procedure of correction was correct; if not correct, the uncertainty would increase owing
to
propagation of uncertainty.

A series of samples were analysed in single experiments

(results not shown
(Molenaar &
Andersen 2016)
)
, in order to
obtain isotope ratios and
establish values of C
V
’s that were
based on repetitions
, CV(rep)
.
The average

CV
(rep)

was
determined as
21

%
,

of samples
,

that
were determined w
ith calibrations of F
igs. 1a
–
1
d
,

whe
reas the

C
V
(rep)

was 12 % of samples
that were determi
ned
by

calibrations of
F
igs. 1a and 1
b

(N = 62)
.
The latter CV(rep)
corresponds well to the CV
–
value of the
empirical
equation
of Horwitz
(Horwitz 1982)
,
which

was calculated as an average
at
14 % for all samples.
These values of C
V
(rep)
’s were Page 35 of 44
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up to an order of magnitude

lower than the corresponding values that were predicted f
rom
poo
led calibrations. R
esults
thus
show that the experiments
most likely belong to the type

‘
A
’

of results (see above) with
CV
(cal) >>
CV
(rep). Accordingly, it was
found
that the
uncertainty of pooled calibrations was too large to explain the observed
CV(rep)’
s
.
,

it is
therefore
important to perform calibrations in every single experiment because there are
indeed fluctuations of the apparatus that remain unexplained.
Thus, t
he true uncertainty is
represented b
y the uncertainty of repetition
, CV(rep)
.

From a scientific point of view, it might
be interesting to study the origin of fluctuations of the apparatus
,

but the
knowledge thus
obtained
,

would mostly be of value
to

m
anufacturers who aim at increasing

stability of
apparatuses.

Evaluation of uncerta
inty by eq
uations

1
–
10

is valid for a large number of cases but there are
also examples where some of the equations are not valid, e.g.
where
LLA is not defined
,

i
.e.,
if

CV
(cal) were

always above

50 %. Such a
n example is presented in
F
ig. 3

where the
method of determination of sodium
with
ICP
–
OES

was investigated
.
Outliers in

vast number
s

were observed in the concentration interval 0
–

2 mg/L but none were seen at higher
concentrations. Since no errors were identified
,

during the series of mea
surements
,

all outliers
were
retained
,
thus characterising the true performance of the method.

Maintaining
of
outliers
as part of the calibration
,

yields a negative a
–
value

and
a
negative p
–
value (eq.

5
). Therefore
,

the CV
–
value

ne
ve
r become
s

l
ower

than 50 % and SBR
,

when calculated
by
eq.
9
,
should

be
used with

the minus sign.

Upper and lower confidence ranges are shown by broken lines. The
BCV was determined as 58 %

(Fig. 3)
.

Determination of concentration of samples

(not shown)
with concentrations above 2 mg/L
(F
ig. 3
)
provided results with

CV
(rep)
’s of 5
–
10 % that is less than the CV
(cal)

(> 58 %)
predicted by pooled calibrations

(F
ig. 3
)
. The corresponding CV’s of the Horwitz
formula
(Horwitz 1982)

were

calculated as 5 %, which showed that CV
(rep)
most likel
y is the correct
expression
for
uncertainty

of this particular experiment
.

From these investigations of
the
methods of ICP
–
OES and ICP
–
MS, it may seem that type A
experiments are common but this is definitely not the case
;
these two examples are
exception
s to the rule. Most frequently
,

the predicted uncertainty

corresponds

(CV(cal)) of
pooled calibrations well to CV(rep)

(Jens E.T. Andersen 2014)
. Many statistical tools are
available to identify and remove outliers from data sets
(Glasser 2007; Barbato et al. 2011)

but this practice should not prevail because it destroys the picture of
the

true uncertainty of Page 36 of 44
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the method
,

and it becomes impossible to classify t
he method into types A, B or C.

The
analysis of the method presented above ought to be an integrated part of method validations
because
the analysis offers an opportunity to identify s
ources of uncertainty and it facilitates
decision making.
Manufactu
rers rarely
show
results
like those
of F
igs.
1
a
–
1d and F
igs. 2a
–
2
b
in sales material because they focus on short
–
term precision
which are
important tools
in
convincing
customer
s
of
the high performance of the apparatus. Some
customer
s may believe
that high precision underpins trueness and accuracy, which

actually
is not the case,
as
the
present investigation
and

also

other investigations
have

shown

(Jens E.T. Andersen 2014)
.

The LOD’s are
approx. 0.001 mg/L
for determination of chloride
with
ICP
–
MS and for
determination of
sodium
with
ICP
–
OES,
as
given in the
information

supplied
by the
manufacturers.

However,
scientists

frequently
report
LOD’s
with
orders of magnitude higher
than these values
(Sapkota

et al. 2005)
.

Reports with determination of sodium by ICP
–
OES
are available

with LOD’s
within the
rang
e

from 0.8

g/L to 597

g/L
(Harrington et al. 2014;
Krachler et al. 2012)
, depend
ing

on the type of matrix. The LOD of the best

possible
regression line of experiments of
Fig. 3 was determined as 3.4

g/L, whereas the LLA was
determined as 2.1 mg/L, which is comparable to the LOD of Na in serum
(Harrington et al.
2014)

but much higher than LOD’s of manufacturers’ specification
s
.

It might be
useful

to apply
confidence ranges (
CR’s
)

to the evaluation of the performance of
the method but there

is a drawback

to
that

procedure, which in
fluence
s

decision making.
CR

will
narrow as a function of the number (N) of repeated experiments
,

and CR’s approach
ing

zero when N is a very large number. It is thus anticipated that the average value of numerous
experiments approach th
e
one and
genuinely
true value
but this is not reality;

it is merely
theory. Reality shows that
a

very hig
h number of repetitions provide
s

a value that is
characteristic of the apparatus and the
measurand

(CITAC & Eurachem 2012)

in
combination
,

whereas the theory postulates that calibration eliminate
s

the
apparatus’
influence on the result
.

The apparatus may have a large influence on the result,
as proven by
the present two cases, but they seem to b
e the exception to the rule.

Suppose that two independent laboratories
Lab1 and Lab2
,

report results

as average value
s

with corresponding SD’s

x


±
s

and
x


±

s

of
measurements of
a certified reference
material
with a certified value of x
CRM
.

The theoretical difference between results is

1

–


2
,
which should be

zero if the two results were
similar
. However, the
similarity
between results
depends on the number of measurements (N), and at
a certain number of repetitions
, as

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by ‘
N
D
’
,

the results start to disagree

(Fig. 4
a)
, according to Student’
s t
–
test (t: Student’s t
–
value
)

(Konieczka 2007)
:

N

=
t

∙

(
11
)

However, if the two laboratories were
encouraged
to share measurements and
merge

data

sets
, disagreement would occur at a
n even

lower number of repetitions, N
D
’:

N


=
t

∙







(







)

(
12
)

Initial agreement
(Fig. 4a)
may be transformed into disagreement simply by increasing the
number of repetitions (N), and the disagreement is realised after a number of repetitions, as
given by N
D

(eq. 12).
Application of expanded uncertainty of Eurachem an
d CITAC may
improve the general level of agreement

(Fig. 4b)
. However, in none of the cases
Figs. 4a
–
b

is
guaranteed compliance with CRM (dotted line).

It is a paradox that an increased number of measurements automatically introduce
s

disagreement between results, according to theory. But theory does not take into account that
the
combination of apparatus and sample

provide
s

results
, which

may deviate si
gnificantly
from
the
expected value of
unknown
, as mentioned above.
According to eq
s.
11
and
12
,
scientists would be prone to discuss differences that appeared on an entirely accidental basis,
that is
,

if
a
laboratory
were performing

more experiments than
another

laboratory

did
. In
principle, there is no need to struggle with calibrati
on

of

all features of the apparatus because
improv
ed

precision is no guarantee that accuracy
improves

(Fig
s. 4a
–
4
b)
.
In recognition of
the
se

problems with CR’s
,

Eurachem
recommends
creat
ing

a full overview of all
participating uncertainties that are involved in the process of measurement, and from the
uncertainty budget calculate an expanded uncertainty that may provi
de
full

agreement (Fig.
4
b)

(CITAC & Eurachem 2012)
. Despite these efforts there is still no guarantee of high
accuracy. Results may differ anyway.
In addition

to the problems of comparison
,

the
re

is

a
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problem of removing outliers from data sets of CRM’s, which provides unrealistically low
uncertainties
(Jens E.T. Andersen 2014)

that makes it impossible for customers to compare
with own results

(Jens E.T. Andersen 2014)
. Serious manipulations of
data sets have been
identified for CRM’s of Institute of Reference Material
s and Measurements (IRMM), where
poor results were converted into excellent result
s

by elimination of outliers from data sets of
professional laboratories
(Jens E.T. Andersen 2014)
. Such CRM’s are unacceptable for
science and it is even more disagreeable that

some manufactures of CRM’s
do not
even
provide any information about raw data
and potential outliers
that w
ere

used to produce the
respective
CRM.
It is very worrying that scientist
s

cannot trust
in
the CRM’s since they are
supposed to be the fundamental standards of analytical chemistry.
It would thus be more
feasible to accept the performance of the apparatus per se and report results with
whichever
SD’s that may
emerge
.
Elimination of day
–
to
–
day va
riations of the apparatus by calibrations

was found true for
two

case
s, one

of chloride that was determined by ICP
–
MS
and one with
determination of sodium by ICP
–
OES (see above)
.
However, t
h
ese

example
s

may be
considered as rare and extreme case
s

of analyt
ical chemistry

(Jens E.T. Andersen 2014)
.
In
recognition of this drawback

in relation to comparison of results
, Eurachem and CITAC
have
introduced an expanded uncertainty that is
equal to
the combined uncertainty of the
uncertainty
–
budget multiplied by a
coverage factor

(CITAC & Eurachem 2012)
.
There are
thus three

major issues,
number of measurements,
outlier
s

and CR
’s
,
which

need
further
attention in order to facilitate comparisons

(Magnusson et al. 2008; Medeiros & Carla 2014;
Williams 2008)
.

Although the two examples above illustrate some

important issues

in
analytical chemistry, the findings may also
transfer
som
e

implications to other fields of
science. Measurements
within
the health sector
(Schaller et al. 2002)

must be under complete
statistical control. Otherwise, it might cause serious health issues of individual
patients.
Diagnoses of patients rely

on laboratory results

(Szecsi & Ødum 2009; Bonini et al. 2002)

although

not exclusively; diagnosis also depend on evaluation of symptoms and on
conversations. However, analytical chemistry
focu
ses

on
creating methods that have the final
say
with respect to
diagnosis and evaluation of
the
efficacy of treatment.
Can we assum
e
,
that
all clinical measurements
were
properly validated with endless series of measurements that
have been performed in ind
ependent laboratories
?

But in other areas of science, it may not be
so clear how many independent laboratories were involved in the measurements.
Fundamental constants of physics are known to be d
etermined experimentally at very high
accuracy; an accuracy
that seems
by far
out of reach of analytical chemistry. However, recent
discussions about revision of
SI units
(Leonard 2014; de Bièvre 2012)

focused on revising the Page 39 of 44
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mol

unit
, which has
implications regarding Avogadro’s number but in view of pooled
calibrations and classifications A, B and C above,
some of the other fundamenta
l constants
may also require renewed
consideration
s
.

BIPM has published a roadmap towards
redefinition of SI unit
s
, which is scheduled for 2018
(BIPM 2014; Taylor 2011)
.
L
evels of
uncertainty of 2 parts in 10
8

have
a
lready at this stage of planning
been
mentioned

(BIPM
2014)
, which may also need further considerations

in terms of precision, trueness and
accuracy
.

Conclusion

Two examples were presented where calibrations were important, in order to obtain the
most
correct and reliable values for concentrations in samples.
These
may seem
merely
as
examples that illustrate common practice in analytical chemistry but, as previous

results
indicate, these two results of the present work were ex
c
eptions to the rule

in contradiction to
conventional practices and procedures of QA
. It is important to provide an overview of
uncertainty of measurement during the
method validation that acc
ompanies

any process
towards laboratory accred
ita
tions.

It is claimed that uncertainties may be characterised by three classes A, B and C that provide
information about the source of uncertainty. In class A, uncertainty of pooled calibrations is
too large
to explain observed uncertainty of repetition whilst uncertainties of class B provide
complete correspondence between predicted uncertainty (u(cal)) and observed uncertainty
(u(rep)). Uncertainties of class C reveal additional uncertainty of sample interfe
rences that
adds to the observed uncertainty (u(rep)). Classification of uncertainty into classes A, B and
C is only possible, if a large number of independent measurements
are
applied to the analysis.
Removing
of
outliers from the data set is not an optio
n since such practice destroys th
e
general picture of uncertainty
, particularly in experiments
with
a low number of repetitions
.

A simplified procedur
e for evaluating uncertainties wa
s presented in eqs. 1
–
10
, which
was
derived by the law
–
of
–
propagation of
uncertainty. Calculations were facilitated by using Page 40 of 44
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spreadsheets with built
–
in functions
for
determin
ation of

uncertainty of slope and intercept
from
regression lines.

Finally, it is claimed that
the
evaluation of
uncertainty that is applied to investigati
ons within
the area of analytical chemistry

has implications to other fields of science. It is very
confusing to the general public and
for
science, if there
is
no clear distinction between
the
concepts of precision, trueness and accuracy.
CRM’s can only b
e trusted when
manufacturers

provide
complete information about all data involved in the production process. Raw data are
needed
, similar to those
that are
provided by CRM’s of IRMM.
Application of funda
mental
constants of physics
,

relies

on very low level
s of uncertainty but if
the uncertainty of
measurement is related to precision rather than trueness or accuracy
,

the
n it becomes an issue
regarding
scientific importance
. No genuinely true values of fundamental constants are
available and nobody knows the
genuinely true concentration of chemicals in e.g, samples of
blood. Many independent series of measurements are required in order to obtain a consensus
value with the correct level of uncertainty that may be estimated after evaluation of data
according to
eqs. 1
–
10
.

Approximations made earlier
(Andersen 2009)

showed
themselves
inadequate

for description of measurements of large SD’s and a new set of simple equations
were
suggested
for calculation of LLA, SBR and BCV (Eqs. 1
–
10
).

Acknowledgements

Many thanks are due to the Organizers of the RICCCE 19 Conference in Sibiu for giving
corresp
onding author
the opportunity to present results at a plenary lecture.
The financial
support from Director Ib Henriksen’s Foundation and Brdr. Hartmann’s Foundation is
gratefully acknowledged. Many thanks are
also
due to
Engineers
Flemming

Hansen and
Claus Lilliecrona for their continued technical support.

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