Spectrochimica Acta Part B 65 (2010) 509–523

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Spectrochimica Acta Part B

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Review

Calibration in atomic spectrometry: A tutorial review dealing with quality criteria,weighting procedures and possible curvatures

Jean-Michel MermetSpectroscopy Forever, 624, Chemin du Colombier, 01390 Tramoyes, France

E-mail address: jeanmichel.mermet@wanadoo.fr.

0584-8547/$ – see front matter © 2010 Elsevier B.V. Aldoi:10.1016/j.sab.2010.05.007

a b s t r a c t

a r t i c l e i n f o

Article history:Received 29 March 2010Accepted 17 May 2010Available online 31 May 2010

Keywords:Atomic spectrometryCalibrationRegressionWeightingCalibration uncertainty

Calibration is required to obtain analyte concentrations in atomic spectrometry. To take full benefit of it, theadequacy of the coefficient of determination r2 is discussed, and its use is compared with the uncertainty dueto the prediction bands of the regression. Also discussed from a tutorial point of view are the influence of theweighting procedure and of different weighting factors, and the comparison between linear and quadraticregression to cope with curvatures. They are illustrated with examples based on the use of ICP-AES withnebulization and laser ablation, and of LIBS. Use of a calibration graph over several orders of magnitude maybe problematic as well as the use of a quadratic regression to cope with possible curvatures. Instrumentsoftwares that allow reprocessing of the calibration by selecting standards around the expected analyteconcentration are convenient for optimizing the calibration procedure.

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© 2010 Elsevier B.V. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5092. Criteria for calibration optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

2.1. Calibration parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5102.2. Coefficient of correlation and of determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5102.3. Uncertainty resulting from the regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

3. Weighting procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5123.1. Weighting factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5123.2. Weighting selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5133.3. Alternative to the standard deviation for the weighting factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

4. Quadratic regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5144.1. Regression parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

5. Comparison between linear and quadratic regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5156. Alternative curve fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5157. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

7.1. Si calibration in solution using ICP-AES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5167.2. Mn calibration in steel using LA-ICP-AES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5177.3. Mg calibration in Al using LIBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5187.4. Mg calibration in ICP-AES with self-absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5187.5. Na calibration in ICP-AES with axial viewing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5197.6. Optimisation of a calibration graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520Appendix A. Uncertainty on the slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521Appendix B. Quadratic regression using a spreadsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522Appendix C. Matrix approach for a quadratic regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

510

510 J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

1. Introduction

Except when a primary method is used such as isotope dilutionmass spectrometry, quantitative analysis in atomic spectrometryrequires a calibration procedure for providing analyte concentrationsin unknown samples.

Calibration implies two steps. The first step consists of measuringsignal intensities using samples (e.g. solutions) with known concentra-tions (standards) so as to construct a calibration graph. In a second step,the concentration is deduced from the calibration graph using themeasurementof the signal intensity fromanunknownsample. A so-called“counter or inverse calibration” is therefore performed. When more thantwostandardsareused, it isnecessary tocarryout a regression,whichmaybe of the linear or quadratic (curvilinear) type. Regressions are mostlybased on the least-squares method (LSM), which minimizes the sum ofthe squares of the differences between the experimental data and thecomputed values. Regression is based on statistics, which implies someconsequences because of the LSM selected model.

The best calibration graph for an analyst is that which wouldprovide the closest computed value compared to the experimentalone, within an expected concentration range. It is the role of anoptimization to obtain such a result. A IUPAC guideline [1] and threeISO norms have been published [2–4] along with books dedicatedspecifically to regression [5,6] or more generally to statistics [7–11]that might be helpful to set a calibration strategy.

Usually, the correlation coefficient, r, or the determination coefficient,r2, is the only criterion provided by the calibration software incommercially available instruments. Depending on the analyticaltechnique, e.g. inductively coupled plasma-emission spectrometry (ICP-AES) or LIBS (Laser induced breakdown spectrometry), a value of 0.999or 0.99 may be considered as an acceptable value. It may be firstquestioned whether this criterion is appropriate to quote the quality ofthe calibration and sufficient for an optimization. Actually, the uncer-tainty due to the regression is certainly amore relevant approach [12,13].

Once a quality criterion has been selected, two major questionsmay be asked: (i) is weighting necessary or even compulsory, and if sowhich type ofweighting; and (ii) how to dealwith possible curvatures.The aim of this tutorial paper is to suggest answers to these questionsand to illustrate them with examples taken from ICP-AES and LIBS.

2. Criteria for calibration optimization

2.1. Calibration parameters

The major parameters involved in calibration are:

xi value of the concentration for a standard.n number of concentrations, xi, used for calibration, i.e. the

number of standards, with 1≤ i≤n.p number of replicates, j, per standard, assuming the same

number of replicates for each concentration with 1≤ j≤p.yi,j signal intensity corresponding to a concentration, xi, and a

replicate, j. The mean of the replicates is yi.yc intensity obtained from the regression for the concentra-

tion, xi.xm mean or centroid of the concentrations:

xm =∑xin

ð1Þ

ym mean of the intensities:

ym =∑yin

ð2Þ

In the case of a linear relationship between y and x:

y = b0 + b1⋅x ð3Þ

where the slope b1 is equal to:

b1 =∑ xi−xmð Þ yi−ymð Þ

∑ xi−xmð Þ2 ð4Þ

and the intercept b0:

b0 = ym−b1⋅xm ð5Þ

2.2. Coefficient of correlation and of determination

It may be written than:

yi−yc = yi−ym− yc−ymð Þ ð6Þ

or:

yi−ym = yc−ymð Þ + yi−ycð Þ ð7Þ

This relation is equivalent to:

∑ yi−ymð Þ2 = ∑ yc−ymð Þ2 + ∑ yi−ycð Þ2 ð8Þ

as the cross-product term 2×∑(yc−ym)(yi−yc) is equal to zero.The first term is the sum of squares of the experimental values

corrected from themean or the total sum of squares (SStot), the secondone is the sum of squares of deviations due to regression (SSreg), andthe last one is the sum of squares of the residuals, i.e. non explained bythe regression. The degrees of freedom are n−1, 1 and n−2,respectively. Using the SS nomenclature, relation (8) may be written:

SStot = SSreg + SSres ð9Þ

When using the LSM, SSres should tend to zero, and, therefore, theratio SSreg/SStot must tend to 1. The correlation coefficient, r, is then:

r =∑ yc−ymð Þ2∑ yi−ymð Þ2

!1=2

=SSregSStot

� �1=2ð10Þ

In the case of a linear relationship between the intensity and theconcentration, r is the Pearson coefficient:

r =∑ xi−xmð Þ yi−ymð Þ

∑ xi−xmð Þ2∑ yi−ymð Þ2� �1=2 ð11Þ

It may be also written that r is the correlation factor between twoseries of results:

r =cov y; xð Þsy⋅sx

ð12Þ

with:

cov y; xð Þ = 1n−1

∑n

i=1ðyi−ymÞ xi−xmð Þ ð13Þ

In this case, r is equivalent to θ. The s value is the standarddeviation of the x and y values. For a linear relationship, the threerelationships, 10, 11 and 12, lead to identical values of r.

The determination coefficient, r2, is more often used, as itemphasizes the departure from unity. The value of r allows theanalysts to verify whether the model, for instance a linear one, issuitable to describe the relation between y and x.

In ICP-AES (or MS), when solutions are used as standards, andassuming that there is no drift during the construction of the graph, itis relatively easy to obtain r2 value as good as 0.999 or even 0.9999.Scattering must be significant to observe values such as 0.98. An

Fig. 1. Schematic representation of the confidence and prediction bands.

511J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

example is given in Table 1, by using synthetic data. The first y columncorresponds to an ideal situation, with r2=1. The following columncorresponds to a 1% bias on each value, a positive one for x=1, 3 and5, and a negative one for x=2 and 4. It may be seen that r2 is stillhigher than 0.999. A bias has to be moved up to 5% to obtain an r2

value below 0.99. A 10% bias still leads to an r2 value near 0.96.Because the r2 value implies the use of absolute residuals (differencebetween the computed and the experimental values) and not relativeones (absolute residual divided by the experimental intensity), theeffect of bias will be more important when a % bias applies to highvalues rather than low values. In Table 1, it may be seen that a 10% biasfor x=1 leads to an r2N0.999, whereas the same %bias for x=5 leadsto r2=0.992. It will be seen below that this problemmay be solved byusing a weighting procedure.

At least when using standards in the form of solutions in ICP-AESor MS, a criterion based on r2N0.99 is very tolerant. Only when usingsolid standards as in LIBS or laser ablation ICP-spectrochemistry, r2

values below 0.98 may be observed.It is important to emphasize a note given in the IUPAC guideline [1]:

“Although those r and r2 coefficients are widely used, the guidelinediscourages their use in calibration as r is a measure of relationship oftwo random variables”, which is not the case for the concentration, x.

2.3. Uncertainty resulting from the regression

For the computation of the confidence and prediction bands seee.g. Ref. [6]. Assuming a linear relationship between the signal, y, andthe concentration, x, the concentration, xu, of an analyte in anunknown sample may be deduced from its signal intensity yu:

xu = − b0b1

+yub1

=yu−b0b1

ð14Þ

An alternative to the comparison of SSreg to SStot is the use of SSres.The standard deviation of the y values with respect to the straight line(also called the residual standard deviation), sy/x is related to SSres andto the degree of freedom n−2:

sy=x =∑ yi−ycð Þ2

n−2

" #1=2=

SSresn−2

� �1=2ð15Þ

Because of the uncertainty on b0 and b1, and for a givenconcentration, xo, the true mean value, yo, of repeated measurementsis within a confidence limit of ±t∙so.

y0 = b0 + b1x0Ft⋅s0 ð16Þ

where t is the Student's coefficient t (1−α/2) at a given risk α or at aconfidence percentage of [100(1−α)%] and n−2 degree of freedom.The value of s0 is given by:

s0 = sy =x1n

+x0−xmð Þ2

∑ xi−xmð Þ2" #1=2

ð17Þ

Table 1Synthetic data to illustrate the role of data scattering on the values of the coefficient of deterwhereas the next columns correspond to a given % bias on each value, a positive one for x

%bias 0 1 1.5 2 4 5

x y y y y y y1 10 10.1 10.15 10.2 10.4 10.52 20 19.8 19.7 19.6 19.2 193 30 30.3 30.45 30.6 31.2 31.54 40 39.6 39.4 39.2 38.4 385 50 50.5 50.75 51 52 52.5r2 1 0.9995 0.9989 0.9981 0.9925 0.9884

The equation describes two hyperbolas on each side of the straightline that correspond to two confidence bands. The confidence limit foryo is symmetrical and is minimum for the centroid.

s0 x0 = xmð Þ = sy=x1n

� �1=2ð18Þ

Once the confidence bands have been determined, it is necessaryto determine the confidence limits for a concentration, xu of anunknown solution, from p replicates of the yu intensity. In thisinstance, individual p measurements lead to the following predictionconfidence band:

su = sy =x1p

+1n

+xu−xmð Þ2

∑ xi−xmð Þ2" #1=2

ð19Þ

The term (1/p)1/2 has been added to take into account the preplicates. The uncertainty on yu decreases with (1/p)1/2. Predictionbands are slightly farther from the regression line than the confidencebands (Fig. 1).

Two values y1 and y2 of the two branches of the hyperbolaecorrespond to the intensity, yu (Fig. 2). It may be deduced two values,x1 and x2, which correspond to the uncertainty on xu due to theregression (Fig. 2). When the departure of the prediction bands fromthe regression line is significant, these two values are not strictlysymmetrical. The values of y1 and y2 are roots of the two equations:

yu = y2−t⋅sy= x1p

+1n

+y2−ymð Þ2

b21∑ xi−xmð Þ2" #1=2

ð20Þ

yu = y1 + t⋅sy=x1p

+1n

+y1−ymð Þ2

b21∑ xi−xmð Þ2" #1=2

ð21Þ

mination, r2, for several bias. The column with 0 bias corresponds to an ideal situation,= 1, 3 and 5, and a negative one for x = 2 and 4.

6 8 10 10% for x= 1 10% for x= 5

y10.6 10.8 11 11 1018.8 18.4 18 20 2031.8 32.4 33 30 3037.6 36.8 36 40 4053 54 55 50 550.9836 0.9719 0.9577 0.9996 0.9918

Fig. 2. Computation of an unknown xu concentration from the measured yu intensity,associated with its x1–x2 uncertainty range.

512 J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

It is more convenient to express the results as a function of x1 andx2. Assuming [5]:

g =t⋅sy=x� 2

b21∑ xi−xmð Þ2 ð22Þ

x2 = xu +g xu−xmð Þ + t⋅sy = x

b1xu−xmð Þ2

∑ xi−xmð Þ2 + 1p + 1

n

� 1−gð Þ

h i1=21−g

ð23Þ

x1 = xu +g xu−xmð Þ− t⋅sy = x

b1xu−xmð Þ2

∑ xi−xmð Þ2 + 1p + 1

n

� 1−gð Þ

h i1=21−g

ð24Þ

Eqs. (23) and (24) are exact mathematical solutions, but may lookcomplex to use.When g is negligible compared to unity, which is oftenthe case, the two equations may be simplified:

x2 = xu +t⋅sy =xb1

xu−xmð Þ2∑ xi−xmð Þ2 +

1p

+1n

� �" #1=2ð25Þ

x1 = xu−t⋅sy=xb1

xu−xmð Þ2∑ xi−xmð Þ2 +

1p

+1n

� �" #1=2ð26Þ

These two equations are those usually given in textbooks.It may be assumed near the centroid:

xu−xmð Þ2∑ xi−xmð Þ2 bb

1p

+1n

ð27Þ

And then:

x2 = xu +t⋅sy =xb1

1p

+1n

� �1=2ð28Þ

x1 = xu−t⋅sy=xb1

1p

+1n

� �1=2ð29Þ

However, the use of a computer avoids any simplification of therelationships. The % uncertainty on any xu is obtained by computing(x2−xu)/xu and (xu−x1)/xu. The advantage of such an approach isthe possibility of adapting the value of the uncertainty to theanalytical problem. There are currently no guidelines for thesethresholds, which must be selected by the analyst. When a hightrueness is required, a first threshold may be for instance, 2%, awarning between 2 and 5%, and rejection above. For environmentalsamples with high heterogeneity, the two limits can be 10 and 20%.

Besides, the two limits will also depend on the concentration level, i.e.close or far from the limit of detection, as a low uncertainty cannot beexpected near the limit of detection. The uncertainty resulting fromthe regression procedure is certainly a convenient way to evaluate thequality of calibration. A 1% uncertainty means that the regressionprocedure is certainly not the limiting step in the whole analyticalprocess, and may be probably neglected in the uncertainty budget. Incontrast, an uncertainty above 5% is no longer negligible.

3. Weighting procedure

A number of assumptions must be made in order to apply theleast-squares method:

(i) the yi−yc errors are normally distributed random variableswith mean 0;

(ii) there is no uncertainty on the value of x; in other words, theerror on the standard concentrationmust be small compared tothat on the signal, and

(iii) the standard deviation, s, of the deviation e is constant over theconcentration range used for the experiment.

This last point is a key issue. For instance, in ICP-AES, it is commonknowledge that for concentrations significantly above the limit ofdetection, the system is flicker-noise limited, which means that thestandard deviation is proportional to the signal. In LIBS, the signal maybe shot-noise limited, which means that in this case, the standarddeviation is proportional to the square root of the signal. Regardless ofthe origin of the noise, the standard deviation is usually not constantover a wide range of concentrations. This behavior is calledheteroscedasticity, in contrast to homoscedasticity for which homo-geneous standard deviations are obtained. It is, then, obvious, thatcondition (iii) does not apply, and it should be compulsory to use a so-called weighting procedure [12,14].

3.1. Weighting factors

With ICP-AES or MS instrument softwares, various weightingfactors are available, with different estimators. Usually, the weightingfactor is related to the standard deviation si:

wi =1s2i

ð30Þ

or

wi =1si

ð31Þ

The values previously defined become:

xm =∑wixi∑wi

ð32Þ

ym =∑wiyi∑wi

ð33Þ

r =∑wi xi−xmð Þ yi−ymð Þ

∑wi xi−xmð Þ2∑wi yi−ymð Þ2� �1=2 ð34Þ

sy=x =∑wi yi−ycð Þ2

n−2

" #1=2ð35Þ

When a non-weighted regression is used, the value of sy/x isexpressed in intensity units. In contrast, when using a weightedregression, the sy/x unit depends on the weighting factor, so it will not

Fig. 3. Based on data in Table 2, absolute residuals for the three concentrations. Blackbar: non-weighted regression; light gray bar: 1/sweighted regression; dark gray bar: 1/s2

weighted regression.

Table 3Absolute and relative residuals as a function of a non-weighted regression and 1/s and1/s2 based weighted regressions.

x yc Absoluteresiduals

Relative residuals(to 1/yc)

Non-weightedregression

1 1.333 0.333 0.3335 5.333 −0.666 −0.1119 9.333 0.333 0.037

Relative residuals(to 1/yc)

Weighted regression(1/s)

1 1.059 0.059 0.0595 5.294 −0.706 −0.1189 9.529 0.529 0.059

Relative residuals(to 1/yc)

Relative residuals(to 1/yc2)

Weighted regression(1/s2)

1 1.009 0.009 0.009 0.0095 5.363 −0.637 −0.106 −0.01779 9.717 0.717 0.079 0.009

513J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

be possible to compare its value to that obtained with a non-weightedregression. Note that the weighting result depends upon the relativevalues of the weighting factors and not upon their absolute ones. Ifeach weighting factor is multiplied by the same value, sy/x will vary,but obviously the same regression parameters will be obtained, e.g. b0and b1. A comparison is made possible by normalizing the weights tounity:

w′i =wi

∑wið36Þ

Another possibility will be described below, by normalizing thestandard deviation to the intensity.

The value of s0 becomes:

s0 = sy =x1

∑wi+

x0−xmð Þ2∑wi xi−xmð Þ2

" #1=2ð37Þ

and su:

su = sy =x1

p⋅wu+

1∑wi

+xu−xmð Þ2

∑wi xi−xmð Þ2" #1=2

ð38Þ

where 1/wu corresponds to xu and is equal to:

1wu

= s2u or1wu

= su ð39Þ

Then:

g =t⋅sy=x� 2

b21∑wi xi−xmð Þ2 ð40Þ

x2 = xu +g xu−xmð Þ + t⋅sy = x

b1xu−xmð Þ2

∑wi xi−xmð Þ2 + 1p⋅wu

+ 1∑wi

� 1−gð Þ

h i1=21−g

ð41Þ

x1 = xu +g xu−xmð Þ− t⋅sy = x

b1xu−xmð Þ2

∑wi xi−xmð Þ2 + 1p⋅wu

+ 1∑wi

� 1−gð Þ

h i1=21−g

ð42Þ

As usually xm does not correspond to one of the standardconcentrations, it is, then, necessary to interpolate the standarddeviation from the experimental values to be able to compute theweighting factor, which can be done by assuming that s isproportional to x or to x1/2.

3.2. Weighting selection

As seen above, the most commonly used weighting factors are:

1wu

= s2u or1wu

= su

Table 2Data to study the influence of different weighting procedures, based on a calibrationusing 3 equidistant concentrations and 3 replicates.

x y1 y2 y3 ym s

1 0.9 1 1.1 1 0.15 5.4 6 6.6 6 0.69 8.1 9 9.9 9 0.9

The consequences of these two different weighting factors may bequestioned. Assuming a very simple three-point calibration (Table 2)with three replicates, regressions without weighting and with either1/s or 1/s2 weighting were performed.

As may be seen in Table 3 and Figs. 3 and 4, the absolute residualsare similar for the 1 and 9 concentrations with no weighting. This

Fig. 4. Based on data in Table 2, relative residuals for the three concentrations. Blackbar: non-weighted regression; light gray bar: 1/sweighted regression; dark gray bar: 1/s2

weighted regression.

514 J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

means that, considering the relative residuals, the regression line iscloser to the highest value than to the lowest one. The use of a 1/sweighting leads to a rotation of the regression line so that the relativeresiduals become equivalent for the highest and lowest concentra-tions. The effect of the 1/s2 weighting is even stronger, so that theregression line is closer to the lowest point, and therefore farther fromthe highest one. In this case, the relative residuals are equivalentwhen they are relative to 1/yc2. In terms of bias, in the case of a non-weighted regression, the deviations are 33% and 1% for the highestand lowest concentrations respectively, whereas they are 4% and 8%for the 1/s2 weighting. Note that, in this particular example based onthree equidistant points, the absolute and relative residuals remainpractically unchanged for the central concentration (Figs. 3 and 4).

The type of weighting will obviously have a consequence on theuseful working range of the calibration graph, in particular on theconfidence and prediction bands (Fig. 5). For a non-weightedregression, two hyperbolas are obtained, whereas V-shape bandscorrespond to a weighted regression. In other words, the uncertaintydue to regression will be significantly smaller at low concentrationswhen using a weighted regression.

3.3. Alternative to the standard deviation for the weighting factor

Usually, a limited number of replicates are used, which leads topoor estimations of the standard deviation [15]. At least 20–30replicates should be used to obtain a reliable estimation which isseldom the case for routine experiments. A weighting procedurebased on the use of the standard deviation may be, then, not fullyadequate. In the case of ICP spectrometry, when concentrations aresignificantly above the limit of detection, the standard deviation isproportional to the signal, and, therefore, to the concentration. Inother words, the relative standard deviation, RSD, is constant. This iswhy some softwares make use of either the intensity or theconcentration as an estimator of the weighting factor to replace thestandard deviation. As mentioned above, the weighting factordepends upon the relative values of the estimators, and not to theabsolute ones. However, the use of the concentration may bedangerous in the case of a blank, as the 1/w or 1/w2 factors becomemeaningless due to a division by zero. Usually a totally arbitrary factoris then used. It seems that the intensity is a more appropriateestimator than concentration, as it is seldom equal to zero, even for ablank. Besides, interpolation of intensity values is easy.

To keep the same unit for the value of sy/xwhen using the intensity yas the estimator, it is possible to compute standard deviations, scalc, thatare normalized to a median value of the RSD, so that scalc=RSDmed×yi.The value of scalc is then directly proportional to y.

Fig. 5. Schematic comparison of the prediction bands with a weighted regression (grayline) and a non-weighted regression (black line) for the lowest part of theconcentration range.

4. Quadratic regression

Although intensity in ICP-AES is supposed to be a linear function ofthe concentration, curvatures may be observed because of self-absorption phenomena. Calibration of alkali elementsmay also lead tocurvature, particularly with axial viewing. With LIBS, curvatures maybe also observed. This is why polynomial regressions are used, thesimplest one being the quadratic regression, whichmay be consideredas an extension of the linear regression.

y = b0 + b1⋅x + b2⋅x2 ð43Þ

It is often said that b1 represents the linear contribution, whereasb2 represents the curved contribution, which explains the alternativename of curvilinear regression.

4.1. Regression parameters

It is convenient to use intermediate Q terms:

Q1 = ∑x2i −∑xið Þ2n

ð44Þ

Q2 = ∑ðxi yiÞ− ∑xi∑yin

� �ð45Þ

Q3 = ∑x3i − ∑xi∑x2in

!ð46Þ

Q 4 = ∑x4i −∑x2i� 2

nð47Þ

Q5 = ∑ x2i yi�

− ∑yi∑x2in

!ð48Þ

Using data from Table 4, Q1=6730, Q2=3,793,243, Q3=672,000,Q4=71,288,750 and Q5=376,963,585.

The regression parameters can be then deduced:

b2 =Q2⋅Q3ð Þ− Q5⋅Q1ð ÞQ2

3− Q1⋅Q 4ð Þ ð49Þ

b1 =Q5⋅Q3ð Þ− Q2⋅Q 4ð ÞQ2

3− Q1⋅Q 4ð Þ =Q2−b2⋅Q3

Q1ð50Þ

b0 =∑yi−b1∑xi−b2∑x2i

nð51Þ

With Table 4 data, b0=−83.4184, b1=606.4744 and b2=−0.42906.

Note that these parameters can also be obtained by using aspreadsheet such as Excel (see Appendix B).

Table 4Data used to illustrate the quadratic regression.

x y

5 293310 597450 29,03275 43,154100 56,214

Fig. 6. Based on data in Table 4, t∙su plot as a function of the concentration.

515J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

The residual standard deviation is:

sy=x =∑ yi−ycð Þ2

n−3

" #1=2ð52Þ

An alternative way is:

s2y = x =∑y2i −b0∑yi−b1∑xiyi−b2∑x2i yi�

n−3ð53Þ

Similarly to the linear regression, it is possible to determine theconfidence interval on the concentration:

t:sy=xðb1 + 2⋅b2⋅xuÞ

×1p

+1n

+xu−xmð Þ2⋅Q 4 + x2u−∑x2i

n

� 2⋅Q1−2⋅ðxu−xmÞ: x2u−∑x2i

n

� :Q3

Q 4⋅Q1−Q23

264

3751=2

ð54Þ

As in Section2.3, the uncertainty, t∙su,maybe computed toobtain theconfidence bands. For x0=50, t=4.3026 (95%, dof=2), sy/x=158.976,and t∙su=696.01 (677 for x0=5, and 786 for x0=100). Note that sy/x isalso called se.

Actually, the t∙su function is rather complex. Based on the data inTable 4, it can be seen that t∙su varies in the range 1–100 in Fig. 6.Confidence bands may be computed, but it is far more complex tocompute the uncertainty on the concentration as in Section 2.3. A fairestimation may be obtained by assuming that the bands are quasi

Fig. 7. Principle of the uncertainty computation on the concentration, assuming a linearbehavior over a small range of concentration in the case of a quadratic regression.

linear near the concentration of concern, which makes a linearinterpolation possible. An example is given in Fig. 7 where theuncertainty on the 50 concentration is in the range 48.6–51, i.e. about2%.

A more complete computation is based on the use of a matrixapproach, as described in detail in the chapter 10 of Ref. [6]. Based onthe same data (Table 4), the procedure is described in Appendix C. Thematrix approach is useful to compute the r value and the uncertainty,t∙su.

r =SSregSStot

� �1=2

ð55Þ

5. Comparison between linear and quadratic regressions

It may be questioned whether the use of a quadratic regression isjustified with respect to a linear regression. Some tests are available. Afirst possibility is to show that b2 from the quadratic regression is notstatistically different from zero, i.e. that the curved contribution isnegligible (section 13.6.4 in Ref. [6]). For that, the zero value must bewithin the uncertainty on b2, i.e. ±t.sb2. Still using the Table 4 data, wehave −0.09bb2b−0.76, i.e. zero is not contained, which means thatthe curvature cannot be negligible, and a quadratic regression shouldbe preferred.

Another approach is to compare the sy/x values, at least by usingnon-weighted regressions. Two F tests are possible [1,3]

F =s2y = x; lin

s2y = x; quadð56Þ

or

F =s2y = x; lin−s2y = x; quad

s2y = x; quad=

s2y = x; lin

s2y = x; quad−1 ð57Þ

F shouldbe compared to F(α,n−2,n−3).When FbF(α,n−2,n−3),the linear model is adequate. With table data, the two tests lead to 10.84and 9.84, respectively. These two values are smaller than F(α, n−2,n−3)=19.16, i.e. the linear model is adequate. It may be seen that thevarious tests are not necessarily coherent.

6. Alternative curve fitting

A different approach may be used, based on median-based robustregressions. The so-called single median method is the simplest one(section 12.1.5.1 in Ref. [6]). The slope, b1, is estimated as the medianof every combination of the two-point slopes, i.e. n(n−1)/2 slopes.Then, using b1, the intercept, yi−b1∙xi, is computed for each yi, xi pair,and b0 is the median of all the computed intercepts. As this robustregression is less sensitive to outliers, it may be verified that theobtained slope is within the uncertainty of the slope deduced from thenon-weighted regression (see Appendix A). A lack of coherence isrelated to an abnormal value (outlier) of one of the intensities. Moresophisticated median-based robust regressions are available but morecomplex to use. In any case, it does not seem that it is possible tocompute confidence bands for this type of regression.

7. Examples

An Excel spreadsheet has been developed and is available onrequest to compute the various regressions previously described, i.e. anon-weighted linear regression, weighted regressions with differentweighting factors (1/s, 1/s2, 1/y, 1/y2), and a non-weighted quadraticregression. The median-based regression is also included. Moreover, a

Table 5Data in mg L−1 using ICP-AES (Si I 216 nm) with 5 replicates y1−y5, the correspondingmean of the replicates, yi, the standard deviation, s, and the relative standard deviation,RSD.

x y1 y2 y3 y4 y5 yi s RSD (%)

0 34 37 35 38 30 35 2.92 8.400.01 59 62 65 56 66 62 4.20 6.830.02 93 82 83 91 85 87 5.03 5.790.05 182 177 194 186 182 184 6.34 3.440.1 347 356 333 348 354 348 9.02 2.590.2 658 650 676 660 664 662 9.53 1.440.3 941 941 943 940 952 943 4.93 0.520.4 1260 1263 1263 1292 1267 1269 13.10 1.030.5 1560 1577 1560 1570 1577 1569 8.49 0.541 3169 3158 3193 3140 3201 3172 25.15 0.792 6429 6475 6465 6473 6450 6458 18.96 0.295 15,958 15,900 15,761 15,974 15,921 15,903 84.33 0.53 Fig. 8. Calibration of Si (Si I 212.4 nm) in solution by ICP-AES using data in Table 5:

comparison of the prediction bands obtained by using a non-weighted regression(black line) and a 1/y2 regression (gray line).

Table 7Computed concentration with the associated calibration %uncertainty when appropri-ate, for an intensity y= 3172 and y= 184, that should lead to a concentration of 1 and0.05 mg L−1, respectively. Values used for bracketing are given into parenthesis.

Regression Computedconcentrationfor an intensityy = 3172

%uncertainty Computedconcentrationfor an intensityy = 184

%uncertainty

Non-weighted 0.9915 1.2 0.0521 24.5Weighted (1/y) 0.9926 1.7 0.0488 8.1Weighted (1/y2) 1.0174 5.2 0.0490 5.9Weighted (1/s) 0.9905 1.6 0.0516 16.4Weighted (1/s2) 0.9956 2.5 0.0503 12.7Quadratic 0.9872 1.7 0.0539 27.9Median 0.9997 0.05022 points 0.9885 0.0470Bracketing 0.9959 (0.5 and 2) 0.0499 (0.02 and 0.1)

516 J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

computation is also performed by selecting the highest and the lowestvalues of the concentration range (so-called two-point regression).Bracketing is also used by selecting the two closest concentrations tothe computed one.

Computation will provide the b0, b1, b2 (when appropriate), sy/x, r2,xm and ym, sb0, sb1 (see Appendix A) and sb2. When a backgroundcorrection is performed, it will be verified whether the zero iscontained within the uncertainty on the intercept, ±t∙sb0. If not, thecause may be related to a wrong correction or to a possiblecontamination. The spreadsheet is also used to compare the quadraticand the linear regressions.

At last, for any value of the intensity, the computation will returnthe concentration and the associated uncertainty due to theregression for a given level of confidence using non-simplifiedrelationships. As mentioned above, the uncertainty computation isnot available for the median-based regression, and is obviouslymeaningless for the two-point one or for the bracketing. A practicalrange can be determined by indicating a maximum accepteduncertainty, e.g. 10%, for which the range will be returned. Thisproceduremay also be used to determine a limit of quantitation (LOQ)related to the uncertainty, e.g. for an uncertainty of 50%, by looking atthe lowest value of the range. Note the LOQ related to the confidencebands [15] is also displayed.

Various graphs are available, including the absolute and relativeresiduals, and the percentage uncertainty. The plot of the absoluteresiduals may be useful to verify a trend such as a curvature.

Examples were selected to illustrate the role of the uncertainty asthe quality criterion, the influence of the various weighting proce-dures, and the way of dealing with curvatures.

7.1. Si calibration in solution using ICP-AES

The first example is given using data in Table 5. It is almost an idealcase with elements in solution in ICP-AES in the range 0–5 mg L−1, i.e.without a matrix. Calibration was based on five replicates and a

Table 6Various calibration parameters computed from Table 5 data.

b0 b1 b2 sy/x r2

Non-weighted 18.3 3181 32.91 0.99995Weighted (1/y2) 32.8 3086 3.26 0.99802Weighted (1/s2) 24.9 3161 3.27 0.99949Weighted (1/y) 29.4 3166 7.09 0.99978Weighted (1/s) 19.7 3183 9.51 0.99983Median-based 26.2 31472-points 34.8 3174Quadratic 11.2 3207 −5.55 32.79 0.99996

background correction was performed. Table 6 gives the majorparameters based on a 95% level of confidence in the case of the Si I212.4 nm line. It may be seen that the r2 value is always better than0.999. Although the quadratic regression provides an impressive r2

value better than 0.9999, the three tests indicate that the quadraticregression is inadequate. We have −17.63bb2b12.08, i.e. zero iscontained within the limits, and Eqs. (56) and (57) provide 1.0079and 0.0079 values, respectively, which are below the F value, 3.14.

It is interesting to note that, although the data are almost perfectlyaligned, the comparison with the median-based regression indicatesthat at least one value seems to be an outlier. Actually, when perfectlyaligned, a small departure from the straight line may lead to anapparent aberrant value.

Both the non-weighted linear and quadratic regression includezero within the uncertainty on the intercept, which would indicatethat background correction was efficiently performed. However, each

xm ym t∙sb0 sb1

0.798 2557.6 24.5 Includes 0 6.90.0071 54.8 2.6 Does not include 0 43.50.109 368.4 13.9 Does not include 0 22.50.045 173.1 6.8 Does not include 0 14.80.24 791.7 18.0 Does not include 0 13.2

Non-cohérentsb2

29.2 Includes 0 5.3

Fig. 9. Calibration of Si in solution by ICP-AES (Si I 212.4 nm) using data in Table 5: %uncertainties computed in the range 0.001–10 mg L−1 for various regressions. Blackline: non-weighted regression; dark gray: 1/s regression; light gray: 1/s2 regression;dashed black line: 1/y regression; dashed light gray line: 1/y2 regression.

Table 8Lowest determinable concentrations that can be determined with an uncertainty of 20%and 50% due to the various regressions.

Uncertainty 20% 50%

Weighted (1/y2) 0.006 0.002Weighted (1/s2) 0.040 0.020Weighted (1/y) 0.020 0.006Weighted (1/s) 0.050 0.020Non-weighted 0.070 0.030

517J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

weighted regression fails to include zero. It is often the case as theconfidence bands are very narrow at low concentrations as seenbefore and confirmed in Fig. 8 for concentrations below 0.1 mg L−1.

When the intensity corresponding to the 1 mg L−1 standard isused (y=3172), it may be seen in Table 7 the computed concentra-tions along with the associated concentration uncertainties. Com-puted concentrations are very close to 1 mg L−1, with uncertaintiesbelow 2%, except for the 1/y2 or 1/s2 based regressions, for which thedeparture from high data is stronger as expected. Note that theregressions based on y or s do not provide similar values. It would bethe case if s would have been fully proportional to the concentration,which is not the case (Table 5). Five replicates were used, but it is notsufficient to obtain a fair estimation of the standard deviation, and aweighting as a function of the intensity is more appropriate. The use ofonly two concentrations, i.e. the highest and the lowest ones, leads toa 6% bias, but obviously with no uncertainty due to the regression.Similarly, bracketing based on the 0.5 and 2 mg L−1 concentrationsleads to a small bias.

When the intensity corresponding to the 0.05 mg L−1 standard isused (y=184), the uncertainty obtained when carrying out a non-weighted linear or quadratic regression is very high. The lowest

Table 9Data in % using LA-ICP-AES (Mn II 257 nm) with 5 replicates y1–y5, the corresponding mean

x y1 y2 y3 y4

0.292 283,471 336,786 340,867 284,40.328 395,508 361,061 388,809 355,90.333 366,297 375,875 375,795 375,40.472 594,826 621,687 583,688 613,00.604 669,194 660,102 645,332 633,60.618 773,101 768,162 756,344 760,30.671 802,244 802,699 778,810 788,60.701 838,345 828,171 900,937 864,40.838 962,630 947,292 894,600 882,51.025 1,198,624 1,278,812 1,321,367 1,221,81.59 1,648,622 1,678,457 1,726,513 1,694,51.72 1,864,334 1,911,624 1,896,014 1,903,6

uncertainties are obtained for the y-based weighted regressions(Table 7). Bracketing based on the 0.05 and 0.1 concentrations stilllead to a small bias. Uncertainties over a concentration range 0.001–10 mg L−1, are shown in Fig. 9. It may be seen the change around therange 0.1–1 mg L–1 with improvement in the uncertainties below thisrange for weighted regressions, and, slight degradations above.

Different LOQs can be determined from the calibration graph [15].When the confidence bands are used, 0.026, 0.0047 and 0.002 mg L−1

are obtained using a non-weighted and 1/y and 1/y2 based weightedregressions, respectively. As expected, LOQs are improvedwhen usingweighted regressions, because of closer confidence bands. Analternative for the determination of the LOQs is by selecting a givenlevel of uncertainty due to regression. Results are given in Table 8,according to the various regressions and for two levels of uncertainty,20 and 50%. Using these various approaches, the LOQs are in the range0.002–0.07 mg L−1, i.e. more than an order of magnitude. Obviously,these computations are without object with a two-point regression.

It may be already deduced that, when using a large concentrationrange, the lower part of the concentration range cannot be used whennon-weighted regressions are selected. When the target concentra-tion is expected within a narrow range, bracketing is highlyappropriate, with no uncertainty due to regression.

7.2. Mn calibration in steel using LA-ICP-AES

Laser ablation as the introduction system for ICP-AES was selectedbecause it usually leads to some scattering of the data, either due touncertainties on the standards or to different ablation efficienciesrelated to variation in the material physical properties. Standardsfrom the former French Institute for Steel Research (IRSID) were used(Table 9) in this example. They originally were established to be usedin spark emission spectrometry.

It may be already seen that the r2 coefficient is always below 0.98regardless of the regression (Table 10). This corresponds to asignificant scattering of the data (Fig. 10). In this case predictionbands can be easily distinguished from the regression line, whichwould be not possible in the example of Section 7.1. When using theintensity y=864,236 related to the 0.701% concentration (Table 11),the uncertainties are around 10%, which is significantly higher thanthose obtained with solutions. Using a lower intensity, y=379,218related to the 0.328% concentration, the lowest uncertainty isobtained when using the 1/y2 weighted regression (Table 11).

Note that the test on b2 indicates an adequate quadratic regression,whereas the other two tests on sy/x indicate an adequate linearregression: −168,535bb2b−7652, and Eqs. (56) and (57) lead to1.461 and 0.461, respectively, for an F value of 3.14. In other words,the difference does not seem to be significant.

Actually, the steel standards were of different types. If only thenon-alloyed steels are kept, 0.472, 0.618, 0.701 and 1.025% in Table 9,results are significantly improved (Table 12). Linear regression isadequate irrespective of the test, and the r2 coefficient is always above

of the replicates, yi, the standard deviation, s, and the relative standard deviation, RSD.

y5 yi s RSD (%)

69 343,071 317,733 30,905 9.7306 394,807 379,218 19,193 5.0649 381,921 375,068 5595 1.4908 598,793 602,400 15,042 2.5060 640,875 649,833 14,512 2.2314 763,197 764,224 6572 0.8662 812,834 797,050 13,333 1.6710 889,319 864,236 31,413 3.6334 1,021,812 941,774 56,139 5.9611 1,255,049 1,255,133 48,092 3.8322 1,729,501 1,695,523 33,937 2.0043 1,882,029 1,891,529 18,712 0.99

Fig. 10. Calibration of Mn in steel by LA-ICP-AES (Mn II 257.6 nm) based on Table 10data, showing the scattered data, the non-weighted regression and the correspondingprediction bands.

Table 10Various calibration parameters computed from Table 9 data.

b0 b1 b2 sy/x r2 xm ym t∙sb0 sb1

Non-weighted 56,315 1,072,448 54,479 0.98942 0.766 877,810 69,358 Includes 0 35,070Weighted (1/y2) 217 1,149,510 2.71 0.98463 0.441 508,127 50,651 Includes 0 45,421Weighted (1/s2) 30,528 1,114,172 3.48 0.98607 0.538 630,043 57,296 Includes 0 41,880Weighted (1/y) 25,372 1,109,090 381.90 0.98715 0.567 654,071 59,054 Includes 0 40,012Weighted (1/s) 47,913 1,082,534 383.74 0.98970 0.643 743,692 58,643 Includes 0 34,919Median-based 29,987 1,117,411 Coherent2-points -4080 1,102,098 sb2Quadratic −69,653 1,410,241 −168,535 45,064 0.99348 133,621 Includes 0 71,124

Table 11Computed concentration with the associated calibration %uncertainty when appropri-ate, for an intensity y=864,236 and y=379,218 that should lead to a concentration of0.701 and 0.328%, respectively. Values used for bracketing are given into parenthesis.

Regression Computedconcentrationfor an intensityy = 864,236

%uncertainty Computedconcentrationfor an intensityy = 379,218

%uncertainty

Non-weighted 0.753 8.0 0.301 22.2Weighted (1/y) 0.756 7.9 0.319 14.2Weighted (1/y2) 0.752 8.2 0.330 9.2Weighted (1/s) 0.754 8.4 0.306 19.7Weighted (1/s2) 0.748 11.1 0.313 21.5Quadratic 0.725 7.7 0.331 19.1Median 0.747 0.3132 points 0.788 0.348Bracketing 0.654 (0.671 and

0.838)0.359 (0.292 and

0.333)

518 J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

0.99, even 0.999. For an intensity y=1,255,133 corresponding to1.025%, uncertainties are significantly lowered (Table 13). Similarimprovements are observed (Table 13) for y=602,400 (0.472%). Forboth intensities, bias is very small.

Table 12Various calibration parameters computed from Table 10 data related by keeping non-alloye

b0 b1 b2 sy/x

Non-weighted 36,552 1,186,003 6588Weighted (1/y2) 44,587 1,174,045 0.28Weighted (1/s2) 51,219 1,156,167 0.42Weighted (1/y) 40,345 1,180,574 43.26Weighted (1/s) 39,952 1,177,839 55.59Median-based 30,419 1,192,6602-points 45,276 1,180,348Quadratic 108,790 981,422 133,760 2846

7.3. Mg calibration in Al using LIBS

The two previous examples exhibited a linear behavior, but with adifferent scattering of the data. Use of LIBS is interesting as bothscattering and curvature may be observed as for the LIBS-baseddetermination of Mg in Al. Compared to an ICP-AES experiment basedon the use of standard solutions, and similarly to LA-ICP-AES, adegradation of the %RSDmay be seen (Table 14). Moreover, every testindicates that the quadratic regression is adequate: −426.3bb2b−207.6, and Eqs. (56) and (57) lead to 52.60 and 51.60, respectively,for an F value of 19.16.

Except for the quadratic regression, every r2 value is near 0.98(Table 15). Clearly, a curvature is present, which is confirmed by theplot of the absolute residuals (Fig. 11) for the non-weighted linear and1/y2 weighted regression. In contrast, the residuals are minimizedwith the quadratic regression.

When the intensity corresponding to the 1% standard is used(y=10,663), it may be seen that the uncertainties are very high,regardless of the regression (Table 16). Because of the adequatequadratic regression, the bias provided by the non-weighted linearregression is significant. As bracketing is not very sensitive to thecurvature, it provides a negligible bias. Obviously, points used for thebracketing should not be aberrant.

Even at high concentrations, uncertainties are always above 20%regardless of the regression. The lowest concentrations that can bedetermined with an uncertainty of 50% are in the range 0.7–3% for thevarious regression models. The same range is found when using theconfidence bands.

Note that it would be interesting to process data from variousexperiments involving different types of standard materials to verifyto which extent LIBS can be considered as a truly quantitativemethod.

7.4. Mg calibration in ICP-AES with self-absorption

Whena resonance line is used above a concentration of a fewmg L−1,it usually leads to a self-absorption phenomenon in ICP-AES, i.e. to acurvature at high concentrations. It may seem appropriate to use aquadratic regression. An example is given in Table 17, where the Mg I285.2 nm resonant line was used up to a concentration of 100 mg L−1.

Firstly, the r2 coefficient is always greater than 0.999 (Table 18).Then, every test indicates an adequate quadratic regression, which isconfirmed by the plot of the absolute residuals (Fig. 12): −5.65bb2b

d steels.

r2 xm ym t.sb0 sb1

0.99962 0.704 871,498 51,238 Includes 0 16,2550.99939 0.613 764,159 55,812 Includes 0 20,4460.99846 0.605 750,384 84,093 Includes 0 32,0670.99951 0.654 812,618 53,915 Includes 0 18,4310.99912 0.623 773,931 67,751 Includes 0 24,681

Coherentsb2

0.99996 301,582 Includes 0 42,904

Table 14Data in % using LIBS (Mg I 518.3 nm) with 6 replicates y1–y6, the corresponding mean ofthe replicates, yi, the standard deviation, s, and the relative standard deviation, RSD.

x y1 y2 y3 y4 y5 y6 yi s RSD (%)

0.5 5752 6241 6806 7014 7308 5913 6506 629.6 9.681 10,584 10,955 9908 11,226 10,547 10,754 10,663 447.2 4.192.5 22,416 21,939 24,415 22,676 23,049 20,864 22,560 1180.9 5.235 40,976 41,864 38,332 41,570 43,791 39,453 40,998 1916.6 4.6710 58,637 68,765 64,947 63,533 63,707 62,673 63,711 3284.2 5.15

Fig. 11. Calibration of Mg in Al by LIBS (Mg I 518.3 nm) using data in Table 17. Absoluteresiduals as a function of the concentration. Black square: non-weighted regression;gray square: 1/y2 regression; empty square: quadratic regression.

Table 16Computed concentration with the associated calibration %uncertainty when appropri-ate for an intensity y = 10,663 that should lead to a concentration of 1%.

Regression Computed concentration %uncertainty

Non-weighted 0.777 163.2Weighted (1/y) 1.004 69.1Weighted (1/y2) 1.060 35.3Weighted (1/s) 0.955 79.0Weighted (1/s2) 1.009 41.6Quadratic 0.996 22.4Median 1.0002 points 1.190Bracketing 1.008 (0.5 and 2.5)

Table 13Computed concentration with the associated calibration %uncertainty when appropri-ate for an intensity y = 1,255,133 and y = 602,400 that should lead to a concentrationof 1.025 and 0.472%, respectively.

Regression Computedconcentrationfor an intensityy = 1,255,133

%uncertainty Computedconcentrationfor an intensityy = 602,400

%uncertainty

Non-weighted 1.027 2.5 0.477 4.2Weighted (1/y) 1.029 3.2 0.476 4.0Weighted (1/y2) 1.031 3.9 0.475 3.7Weighted (1/s) 1.032 4.6 0.478 4.0Weighted (1/s2) 1.041 6.7 0.477 3.6Quadratic 1.025 4.2 0.473 11.2Median 1.027 0.4802 points 1.025 0.472Bracketing 1.025 (0.618 and

1.025)0.471 (0.472 and

0.701)

519J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

−3.33, and Eqs. (56) and (57) lead to 17.12 and 16.12, respectively,for an F value of 4.95.

Finally, the use of a 1/s weighting is inappropriate as a linearregression of s against the concentration leads to negative s values(Table 17). At high values, i.e. y=563,500 for 100 mg L−1 (Table 19),the quadratic regression provides the smallest bias and uncertainty.Such a regression seems then justified. However, whenmoving to lowconcentrations, such as y = 5969 for 1 mg L−1, a 7% bias is obtainedalong with a large uncertainty (Table 19). Clearly, a quadraticregression is not adequate at low concentrations. This can be easilyexplained as self-absorption and the corresponding curvature areonly observed at high concentrations, and the graph remains linear atlow and mid concentrations. It may be concluded that a singleregression cannot cover the whole range of concentrations when self-absorption is present. The problem may be overcome by selecting anon-resonant line, sensitivity being not a crucial factor at highconcentrations, by using bracketing or by cutting the graph in severalparts. Possibility of reprocessing is then an interesting feature ininstrument softwares, as rejection of standards allows the computa-tion in the range of interest.

Table 15Various calibration parameters computed from Table 14 data.

b0 b1 b2 sy/x r2

Non-weighted 5977 6029 3678 0.981Weighted (1/y2) 3411 6839 2.10 0.981Weighted (1/s2) 3836 6765 2.02 0.980Weighted (1/y) 4212 6428 89.74 0.981Weighted (1/s) 4561 6388 88.53 0.981Median-based 3183 74802-points 3495 6022Quadratic 1625 9386 −317.1 507.07 0.999

7.5. Na calibration in ICP-AES with axial viewing

A different type of curvature may be observed in some cases, suchas the determination of alkali elements using axial viewing in ICP-AES.Data are given in Table 20 for Na I 589 nm. Curvature seems to extendall over the concentration range and exhibits a convex form (Fig. 13),in contrast to self-absorption. Obviously, each test indicates anadequate quadratic regression with r2 N 0.999 (Table 21): 4115 b b2 b4910, and Eqs. (56) and (57) lead to 16.66 and 15.66, respectively, foran F value of 3.39.

For a high concentration, y = 495,530 related to 3 mg L−1, thequadratic regression provides the lowest bias and uncertainty, whilethe other regressions leads to significant biases (Table 22). However,similar to self-absorption, at low concentrations such as y = 4230related to 0.03 mg L−1, the quadratic regression is in no way adequatebecause of a large bias and an unacceptable uncertainty (Table 22).Note that, similarly to self-absorption, the 1/s regression cannot beused at low concentrations because of a negative s values due to the slinear regression. Although a curvature seems to occur all along the

xm ym t∙sb0 sb1

82 3.80 28,888 7759 Includes 0 473.619 0.88 9422 2390 Does not include 0 546.785 1.20 11,966 3029 Does not include 0 545.698 1.69 15,064 4468 Includes 0 502.828 1.96 17,111 4958 Includes 0 509.4

Coherentsb2

77 2083 Includes 0 25.4

Table 18Various calibration parameters computed from Table 17 data.

b0 b1 b2 sy/x r2 xm ym t∙sb0 sb1

Non-weighted 1513.0 5688 4435 0.99967 30.19 173,223 4951 Includes 0 42.4Weighted (1/y2) 13.5 5850 5.56 0.99950 0.00002 13.6 0.80 Does not include 0 53.4Weighted (1/s2) 27.2 5933 7.71 0.99957 0.106 657.1 42.1 Includes 0 50.3weighted (1/y) 15.4 5736 118.37 0.99974 0.016 107.4 69.9 Includes 0 38.0Weighted (1/s) 92.1 5763 106.58 0.99975 0.557 3304.5 318.8 Includes 0 37.2Median-based 172.2 5780 Coherent2-points 13.5 5635 sb2Quadratic −573.0 6097 −4.49 1071.92 0.99998 1369.1 Includes 0 0.45

Table 17Data inmg L−1 using ICP-AES (Mg I 285.2 nm)with 5 replicates y1–y5, the correspondingmean of the replicates, yi, the standard deviation, s, and the relative standard deviation, RSD.A linear regression slin of s was added.

x y1 y2 y3 y4 y5 yi s RSD (%) slin

0 14.58 13.22 14.68 9.575 15.57 14 2.36 17.47 −145.980.5 3055 3056 3060 3048 3060 3056 4.92 0.16 −127.571 5980 5935 5985 5949 5995 5969 25.54 0.43 −109.165 28,950 29,210 29,050 29,090 29,080 29,076 93.17 0.32 38.1310 58,700 58,820 58,850 58,840 59,020 58,846 114.37 0.19 222.2550 290,600 292,300 292,900 294,200 292,000 292,400 1313.39 0.45 1695.1875 432,200 435,900 430,700 433,200 432,600 432,920 1904.47 0.44 2615.76100 558,600 559,200 565,700 567,700 566,300 563,500 4266.73 0.76 3536.34

Fig. 12. Calibration of Mg is solution by ICP-AES (Mg I 285.2 nm) using data in Table 20.Absolute residuals as a function of the concentration. Black square: non-weightedregression; gray square: 1/y2 regression; empty square: quadratic regression.

520 J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

concentration range, a single regression is not sufficient. There areonly a few sensitive lines for alkali elements and at least Na and Kprovide a similar behavior. Once again, bracketing seems to be apossibility of solving the problem.

Table 19Computed concentration with the associated calibration %uncertainty when appropri-ate for an intensity y=563,500 and y=5969 that should lead to a concentration of 100and 1 mg L−1, respectively.

Regression Computedconcentrationfor an intensityy = 563,500

%uncertainty Computedconcentrationfor an intensityy = 5969

%uncertainty

Non-weighted 98.80 1.7 0.78 153.6Weighted (1/y) 98.23 2.0 1.04 11.4Weighted (1/y2) 96.33 3.5 1.02 3.5Weighted (1/s) 97.76 2.0 1.02Weighted (1/s2) 94.96 5.5 1.00 15.7Quadratic 99.87 0.5 1.07 31.5Median 97.47 1.002 points 100.00 1.06Bracketing 99.69 (50 and 100) 1.00 (0 and 1)

7.6. Optimisation of a calibration graph

A calibration graph was obtained by using Ni 216 nm in ICP-AES inthe range 0.01–10 mg L−1 (Table 23). Using directly the graph, anintensity of y = 1556 corresponding to 0.1 mg L−1 leads to resultsgiven in Table 24. Values of the bias are small, and uncertainties areacceptable for the 1/y and 1/y2 regressions. However, examination ofthe absolute residuals reveals that a slight curvature is observed forthe two highest concentrations. Besides, the %RSD is very high for thelowest concentration. Removing the 0, 6.5 and 10 mg L−1 concentra-tions leads to results in Table 24. Practically no bias is observed, andthe uncertainties are below 1%. It is good evidence of the quality of thestandard preparation and the stability of the instrument during thecalibration process. It is certainly the limits of what can be obtainedfor a calibration, and the uncertainty due to regression becomesnegligible in the uncertainty budget. Note that it is assumed that thereis no uncertainty on standard concentrations when using the LSM. Atthis level of uncertainty, the one on standard preparationmay becomeof concern.

8. Conclusions

Because a large dynamic range may be obtained in atomic emis-sion spectrometry (and mass spectrometry), it seems possible toconstruct and use a linear calibration graph over several orders ofmagnitude. This might be justified when the expected analyteconcentration is unknown. In this case, a non-weighted regression ismainly useful in the upper mid part of the range. In the lower part, aweighted regression, better based on the intensity weighting factor, ismore appropriate. When using a large range of concentration, itappears that the use of uncertainty is certainly more efficient than theuse of the coefficient of correlation or determination for the selectionof the most appropriate regression. Several thresholds may beselected. For instance, an uncertainty below 2% may be consideredas adequate, with a warning in the range 2–5%, and a rejection above5%. Obviously, these limits depend on both the concentration leveland the analytical problem and have to be decided by the analyst.Requirements will certainly not be the same for precious metals as forenvironmental samples.

Table 20Data in mg L−1 using ICP-AES (Na I 589 nm) with 5 replicates y1–y5, the corresponding mean of the replicates, yi, the standard deviation, s, and the relative standard deviation, RSD.

x y1 y2 y3 y4 y5 yi s RSD (%)

0 383 367 435 370 366 384 29.2 7.600.01 1661 1728 1671 1647 1696 1681 32.0 1.900.03 4215 4213 4238 4211 4273 4230 26.4 0.620.065 8773 8854 8824 8875 8899 8845 48.8 0.550.1 13,538 13,540 13,579 13,540 13,543 13,548 17.4 0.130.3 40,685 40,669 40,349 40,572 40,158 40,487 227.4 0.560.65 91,636 92,338 91,872 92,177 91,561 91,917 336.4 0.371 147,374 146,688 147,302 146,617 146,950 146,986 345.2 0.233 495,001 497,037 495,686 495,822 494,106 495,530 1082.6 0.226.5 1,221,888 1,204,032 1,207,588 1,203,415 1,207,992 1,208,983 7499.7 0.6210 1,973,822 1,975,936 1,974,322 1,991,388 1,975,052 1,978,104 7468.6 0.38

Fig. 13. Calibration of Na by ICP-AES using axial viewing (Na I 589 nm): convexcurvature of the graph.

521J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

A two-point procedure, e.g. the blank and the upper value, can alsobe used in the case of a linear behavior, provided that it waspreviously validated by amultipoint calibration.With only two points,there is obviously no more uncertainty due to regression.

However, when a curvature is present, it is difficult to cope with asingle quadratic regression that could be used all over the range.Cutting the curve in several parts is a possibility, whereas bracketingwith close values could solve the problem.

Regardless of the calibration procedure, needless to say that itmust be finally validated by using a certified reference material whenavailable, or at least a quality control sample. This requirement mightbe a problem with uncommon materials. Even when the material isavailable, its analyte concentrations should be preferably within thecalibration range.

Calibration is obviously a crucial step in quantitative analysis.Compared to the LSM-based calibration, more sophisticated calibra-tion methods have been described in atomic spectrometry [16–21].The LSM is very simple, and it is then rather surprising thatcommercially available instrument softwares provide limited infor-mation, with most of the time the only mention of the r2 coefficient.

Table 21Various calibration parameters computed from Table 20 data.

b0 b1 b2 sy/x r2

Non-weighted −21,310 195,110 35,307 0.997Weighted (1/y2) 370 144,283 25.98 0.979Weighted (1/s2) −566 147,968 35.88 0.984Weighted (1/y) −262 182,466 922.69 0.989Weighted (1/s) −3119 180,542 1137.77 0.988Median-based −936 157,3412-points 384 197,772Quadratic −4757 157,670 4115 8651.24 0.999

Numerous books, guidelines and norms clearly describe basicstatistics for calibration, and there are no difficulties in developingsimple programs to provide more information and more regressionpossibilities. In any case, softwares providing the possibility ofreprocessing, i.e. the selection of the most appropriate standards asa function of the analyte concentration, are most useful to take fullbenefit of calibration.

Appendix A. Uncertainty on the slope

The slope b1 is given by:

b1 =∑ xi−xmð Þ yi−ymð Þ

∑ xi−xmð Þ2 ðA1Þ

As:

∑ xi−xmð Þ yi−ymð Þ = ∑ xi−xmð Þyi−ym∑ xi−xmð Þ ðA2Þ

and:

∑ xi−xmð Þ = 0 ðA3Þ

b1 may be written in a different way:

b1 =∑ xi−xmð Þyi∑ xi−xmð Þ2 ðA4Þ

The coefficient ai may be defined as:

ai =xi−xmð Þ

∑ xi−xmð Þ2 ðA5Þ

Then b1 may be written in the form of a series [5]:

b1 = a1⋅y1 + … + an⋅yn ðA6Þ

Applying variances on b1:

s2b1 = a21⋅s2y1

+ ::: + a2n⋅s2yn

ðA7Þ

xm ym t∙sb0 sb1

33 1.97 362,790 28,362 Includes 0 336476 0.00096 509 122 Does not include 0 691299 0.059 8153 1248 Includes 0 609032 0.018 3000 2573 Includes 0 631945 0.102 15,283 6051 Includes 0 6506

Non-cohérentsb2

86 7775 Includes 0 345

Table 22Computed concentration with the associated calibration %uncertainty when appropri-ate for an intensity y = 495,530 and y = 4230 that should lead to a concentration of 3and 0.03 mg L−1, respectively.

Regression Computedconcentrationfor an intensityy= 495,530

%uncertainty Computedconcentrationfor an intensityy= 4230

%uncertainty

Non-weighted 2.65 8.4 0.131 175.4Weighted (1/y) 2.72 12.4 0.025 112.8Weighted (1/y2) 3.43 19.4 0.027 20.6Weighted (1/s) 2.76 14.2 0.041Weighted (1/s2) 3.35 23.0 0.032 71.0Quadratic 2.95 3.2 0.057 148.0Median 3.16 0.0332 points 2.50 0.019Bracketing 2.89 (1 and 6.5) 0.030 (0.01 and 0.065)

Table 23Data in mg L−1 using ICP-AES (Ni II 216.5 nm) with 5 replicates y1–y5, thecorresponding mean of the replicates, yi, the standard deviation, s, and the relativestandard deviation, RSD.

x y1 y2 y3 y4 y5 yi s RSD(%)

0 22 9 15 18 17 16 4.76 29.410.01 162 170 153 163 181 166 10.43 6.290.03 483 474 479 474 479 478 3.83 0.800.065 1008 1016 1016 1023 1024 1017 6.47 0.640.1 1541 1556 1553 1554 1575 1556 12.24 0.790.3 4615 4636 4624 4647 4596 4624 19.60 0.420.65 9942 9972 9939 9974 9927 9951 21.04 0.211 15,333 15,286 15,332 15,312 15,350 15,323 24.49 0.163 45,985 46,222 45,860 45,863 45,764 45,939 176.67 0.386.5 98,892 97,856 98,138 98,201 98,298 98,277 381.12 0.3910 149,206 149,449 149,487 149,675 148,795 149,322 338.80 0.23

Table B1Data for a quadratic regression.

x x2 y

5 25 293310 100 597450 2500 29,03275 5625 43,154100 10,000 56,214

522 J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

Under the assumption that the standard deviation is constantregardless of the concentration:

s2y = s2y1 = … = s2yn = s2y = x ðA8Þ

And the standard deviation sb1 on the slope is:

sb1 =sy=x

∑ xi−xmð Þ2� �1=2 ðA9Þ

Table 24Computed concentration with the associated calibration %uncertainty when appropri-ate for an intensity y = 1556 that should lead to a concentration of 0.1 mg L−1.

Regression Computedconcentration

%uncertainty Computedconcentration afterremoving the 0.00,6.5 and 10 mg L−1

concentrations

%uncertainty

Non-weighted 0.090 42.0 0.100 0.9Weighted (1/y) 0.102 5.2 0.101 0.7Weighted (1/y2) 0.101 1.4 0.100 0.8Weighted (1/s) 0.100 10.7 0.100 0.5Weighted (1/s2) 0.101 3.5 0.100 0.3Quadratic 0.101 4.1 0.100 1.2Median 0.100 0.1012 points 0.103 0.101Bracketing 0.100 (0.03 and 0.1) 0.100 (0.03 and 0.1)

When the standard deviation is not constant, which is usually thecase in atomic spectrometry, a weighting factor wi must be used(Section 3.2) and sb1 becomes:

sb1 =sy= x

∑wi xi−xmð Þ2� �1=2 ðA10Þ

Appendix B. Quadratic regression using a spreadsheet

The Excel «LINEST» function can also be used to compute theparameters of a quadratic regression, by including not only the x and yvalues (Table B1), but also the x2 values as in Table B1 when thesoftware requires the x values.

The function will return values as in Table B2.With the corresponding parameters given in Table B3.

Appendix C. Matrix approach for a quadratic regression

Still using the same data (Table 4), it is possible to define twomatrices X and Y:

X =

1 5 251 10 1001 50 25001 75 56251 100 10000

0BBBB@

1CCCCA

Y =

29335974290324315456214

0BBBB@

1CCCCA

Y=XB, with B the regression parameter matrix. The residuals aregiven by Y−XB.

Table B2Values returned by the Excel spreadsheet using the LINEST function.

−0.42906 606.4744 −83.41840.078 7.994 146.9190.9999764 158.9742,318 22,138,764,929 50540.11

Table B3Parameters corresponding to the data of Table B2. dof stands for degree of freedom.

b2 b1 b0sb2 sb1 sb0r2 sy/x

dofSSreg SSres

523J.-M. Mermet / Spectrochimica Acta Part B 65 (2010) 509–523

The third column of the matrix X is simply the square of x. Thethree parameters b0, b1 and b2 are obtained by computing:

XTX� �1

XTY�

¼b0b1b2

0@

1A

These calculations can be performed by using a spreadsheet suchas Excel, with the functions TRANSPOSE, MINVERSE and MMULT. Weobtain:

XT¼1 1 1 1 15 10 50 75 100

25 100 2500 5625 1000

0@

1A

XTX =1 1 1 1 15 10 50 75 100

25 100 2500 5625 10000

0@

1A

1 5 251 10 1001 50 25001 75 56251 100 10000

0BBBB@

1CCCCA

then:

XTX =5 240 18250

240 18250 154800018250 1548000 1:38Eþ 08

0@

1A

and:

XTX� �1

=0:854181 �0:03438 0:000273�0:03438 0:002529 �2:4E� 050:000273 �2:4E� 05 2:39E� 07

0@

1A

XTY =1 1 1 1 15 10 50 75 100

25 100 2500 5625 10000

0@

1A

29335974

290324315456214

0BBBB@

1CCCCA

=137306:4103839518:78Eþ 08

0@

1A

which leads to:

XTX� �1

XTY�

¼0:854181 �0:03438 0:000273�0:03438 0:002529 �2:4E� 050:000273 �2:4E� 05 2:39E� 07

0@

1A 137306:4

103839518:78Eþ 08

0@

1A

=�83:4184606:4744�0:42906

0@

1A =

b0b1b2

0@

1A

The b0, b1 and b2 values are obviously identical to those previouslyobtained.

It is also possible to compute SStot, SSres and SSreg and to deduct r:

r =SSregSStot

� �1=2

SStot = YTY–n:ym

SSres = YTY−bTXTY

SSreg = bT XTY�

−n:ym = SStot−SSres

Here n = 5 and:

bT¼ b0 b1 b2ð Þ

We have:

SStot = 2138815469

SSres = 50540

SSreg = 2138764929

Then r2 = 0.9999764.The uncertainty, t∙su, may be computed to obtain the confidence

bands.

t⋅su = t⋅sy= x1p

+1n

+ XT0 XTX�1�

X0

� � �1=2

with X0 the matrix [1 x0 x20], x0 being the concentration for which the

uncertainty has to be computed. Uncertainty on concentrations can beobtained using a linear interpolation (see Section 4.1).

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