In the Laboratory
Exercise in Quality Assurance: A Laboratory Exercise Jens E. T. Andersen Department of Chemistry, Technical University of Denmark, Kemitorvet Building 207, DK-2800 Kgs. Lyngby, Denmark;
[email protected] In recent years there has been additional focus on quality assurance in analytical chemistry, and the effort must be supported by teaching and presentation of some of the novel tools of statistics (1–6). It has been long recognized that linear calibration is not as simple as anticipated when uncertainties are taken into consideration (2–4). Frequently, the familiar coefficient of correlation is associated with quality of analytical results but it does not provide much information on the uncertainty of an unknown. Thus, to estimate uncertainties on measurements, more delicate tools of statistics are required (7–9). It is imperative that the predicted uncertainties of calibrations correspond exactly to the uncertainties obtained by repetitive measurements of unknowns. However, as shown by the present analysis, the uncertainties of calibrations by the law of propagation of errors (LPE) correspond well to the uncertainty obtained by repetitive determinations of unknowns. If these two uncertainties did not correspond, it would become difficult to convince the student about the reliability of the method. The uncertainty of calibrations may be estimated by including in the LPE the term of covariance (10, 11). However, as shown by Salter and de Levie (5), covariance may be omitted in the expression of calibration uncertainty when regression parameters are truly independent, which may be accomplished in practice by applying to calculations the Microsoft Excel Solver in favor of least-squares linear regression, thus simplifying considerably the estimation of uncertainties. To the broad majority of students, the mathematical procedures are cumbersome but we present a laboratory exercise that seemed to arouse an interest among students and promoted enthusiasm and understanding. There are several schools of quality assurance (3–7, 10–13) and each has its own advantages and drawbacks. It is the aim of this article to simplify and suggest a general approach to the problem of estimating uncertainty, which also promotes understanding of statistics. Initially, a number of straightforward procedures ought to be followed if the investigation of quality is going to be successful. First, the number of measurements must be high, preferably close to one hundred, according to the central-limit theorem of statistics (4). Second, the concept of uncertainty should be recognized as the true measure of quality; coefficients of correlation are unimportant (14). It should also be recognized that quality to the analytical chemist means that the true uncertainty (3, 4) should be attached to the measurement; that is, quality of measurement was obtained when the predicted uncertainty corresponded to the observed uncertainty. The uncertainty of measurement is represented by standard deviation that approaches an inherently true value that is specific for the detector, given that a high number of repetitions were performed. Thus, a low uncertainty is not necessarily a token of quality. Experimental Overview In the experiment, the students have the opportunity to obtain competence in writing a standard operating procedure,
in constructing the uncertainty budget (3, 4), and in estimating the quality of their own measurements. The experiments are straightforward, and all solutions are prepared for the students in advance of the measurements. The students are organized in groups of two or three and they are instructed and supervised by teachers during the measurements. The purpose of this particular experiment is to determine the concentration of iron in an unknown by two methods of spectrophotometry. In the first experiment, the measurements were performed by conventional batch measurements using a cuvette and solutions at room temperature (not thermostatted). In the second experiment, the same iron concentration of the same unknown was determined by flow-injection analysis (FIA) utilizing the same spectrophotometric detector (15). Thus, the students performed the measurements under identical conditions using the same solutions that were prepared from a certified standard. The chemical reaction applied to the analysis was the wellknown thiocyanate reaction (15), where the iron(III) ions form a complex with thiocyanate (16):
− + Fe3 (aq) + 6SCN (aq)
[Fe(SCN)6]3 −(aq)
(1)
The thiocyanate complex is intensively red colored and the method may be applied to the analysis of iron(III) ions in aqueous solution at low levels, as low as 0.05 mg L‒1. In the batch method the solutions were already colored because the iron(III) ions were mixed before measurement with the thiocyanate reagent but in the FIA experiment the iron(III) ions were injected into a flow of alkaline thiocyanate reagent that gradually reacted to produce the red complex of eq 1, peaking in intensity of the red color at the time of arrival at the detector. All students were expected to perform the same experiments repeatedly to obtain a high number of results, thus improving reliability (4). As an additional objective, the students were asked to decide by statistical arguments, which one of the two methods is the better choice for analysis of iron(III). The results were presented in a colloquium where the laboratory value of the unknown was revealed to the students after their own presentations. It is important that the students were not informed about the laboratory value of the unknown before the start of experiments. It should be noted that the laboratory value is by no means a “true” value but merely an additional value that adds to the pool of data collected by the students. Experimental Details Chemicals All standards and a sample denoted as the unknown were prepared by dilution of a certified standard solution of iron(III) nitrate in nitric acid (999 ± 2 mg L‒1, Merck) by the laboratory assistant. Standards for the batch experiments were prepared in 0.5 mM thiocyanate in 0.1 M sodium hydroxide at concentrations of 0.00, 5.00, 10.0, 20.0, 40.0, 50.0, 75.0, and 100 mg L‒1. These standards were freshly prepared, owing to the risk of pre-
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In the Laboratory Table 1. Spectrophotometric Determination of Iron(III) Concentration by Batch Measurements with Standard Deviations Calculated by Two Methods: Repetition and Calibration Line Repetition/ (mg L–1)
Number of Determinations
Calibration/ (mg L–1)
Number of Data in the Calibration Line
Group
Experiment
1
1
39.76 ± 0.05
10
39.8 ± 0.2
61
1
2
47.4 ± 0.1
40
47.4 ± 0.1
81
2
1
39.2 ± 0.3
3
39.2 ± 0.5
27
2
2
48.7 ± 0.2
6
49 ± 1
3
1
44.9 ± 0.1
67
3
2
40.4 ± 0.1
129
27
44.9 ± 0.2
629
40.41 ± 0.09
1005
Note: Each group performed the experiment twice on different days using a different number of repetitions. The uncertainties were estimated by the standard deviation of repetitions and compared to the standard deviations of a single calibration line (eq 4). All the results deviated significantly from the laboratory value (43.6 ± 0.2 mg L–1, N = 3). Two results (In bold) were found to correspond at the 95% level of confidence
cipitation of iron(III) hydroxide. The sample of the unknown was prepared from this stock solution to provide a signal that corresponded to an absorbance approximately at the middle absorbance value of the calibration line of batch experiments (40–50 mg L‒1). The reagent for FIA determinations contained 1 mM thiocyanate in 0.2 M sodium hydroxide, and it was prepared by dissolving solid KSCN (99% p.a., Merck) and adding sodium hydroxide (Merck, Titrisol) in doubly-distilled water at room temperature. Apparatus The measurements were performed by an HP8542A spectrophotometer with a diode-array detector using a single wavelength of 460 nm and a deuterium-source lamp. A quartz cell (1 cm) of 1 mm bore was used for the FIA experiments, and cuvettes (1 cm) were filled by the solution (freshly prepared) for the batch experiments. In the FIA experiments, the solutions were propelled by a peristaltic pump (Ismatec Reglo) at 20 rpm and pumping tubes of blue–yellow color code. The manifold was constructed by Microline tubings (i.d. 0.5 mm) and the length of the loop at the injection valve was 20 cm. The FIA manifold (15) was constructed as simple as possible with a single-line arrangement where the iron samples were injected by an injection valve into the flow of the complexing reagent. All measurements were performed at room temperature but without utilizing temperature control and thermostatting. Hazards Solutions of potassium thiocyanate should be stored in a fume hood because addition of acids releases hazardous gases. It can irritate the skin, eyes, and respiratory tract. Sodium hydroxide is corrosive and causes burns to any area of contact. Iron(III) nitrate is a strong oxidizer and contact with other material may cause fire. It is harmful if swallowed or inhaled and can cause irritation to skin, eyes, and respiratory tract. Discussion Precision After the first series of measurements, the three groups reported two determinations of the unknown (Table 1). The uncertainty was estimated by two independent methods that were supposed to be equal but, unexpectedly, they were found to deviate significantly (Table 1). 734
The results were obtained by using the calibration line to calculate the concentrations, where the repetitions of the unknown were performed by a series of measurements in association with the measurement of standards. All experiments thus exhibited an almost ideal linear calibration line (not shown) with very small residuals of absorbances and small standard deviations (SDs) on the unknowns. The concentration of the unknown, cu, was calculated by the inverse formula of the calibration line (6), cu =
Au − β α
(2)
where the parameters are the absorbance of the unknown, Au and the slope, α, and intercept, β, from the equation of calibration. From a number of determinations, the SD of unknowns can be estimated by the conventional formula (3, 4, 12, 13). The uncertainty derived in this manner is denoted as the SD of repetition (Table 1) whereas the SD obtained by inserting the average value of the concentrations into eq 4 (below) is denoted as the uncertainty of calibration. Despite the careful instructions and uncomplicated experiments, the majority of results (Tables 1 and 2) deviated significantly, as judged by a Student’s t test (95% level of confidence). Accordingly, none of the groups seemed to agree about the concentration of the unknown, not even in the batch experiments (Table 1). The uncertainty of calibrations was represented by standard deviations, scal, as calculated by the modified law of propagation of errors (12), Nt
scal2 =
∑
i =1
∂c sx ∂xi i
2
(3)
Nt − 1
where Nt denotes the number of terms according to the number of independent parameters of the equation. In the case of a straight line, the number of terms equals three, Nt = 3. It can be shown (3, 4) that eq 3 provides an equation for calculation of the SD as a function of average concentration of the unknown, c–u,
scal
1 = α
( )
sA 2 + sβ2 + cu sα 2
2
(4)
where sα is the SD of the slope; sβ is the SD of the intercept; and sA is the SD of absorbance at a given average concentra-
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In the Laboratory Table 2. Spectrophotometric Determination of Iron(III) Concentration by FIA Measurements with Standard Deviations Calculated by Two Methods: Repetition and Calibration Line Repetition/ (mg L–1)
Number of Determinations
Calibration/ (mg L–1)
Number of Data in the Calibration Line
3
45.0 ± 0.6
25
60 ± 2
25
3
39.8 ± 0.6
27
6
45 ± 1
27
38.4 ± 0.1
3
38.4 ± 0.6
27
45.17 ± 0.09
3
45.2 ± 0.5
27
Group
Experiment
1
1
45.0 ± 0.1
1
2
59.7 ± 0.6
2
1
39.84 ± 0.08
2
2
44.78 ± 0.07
3
1
3
2
18
Note: Each group performed the experiment twice on different days using a different number of repetitions. The uncertainties were estimated by the standard deviation of repetitions and compared to the standard deviations of a single calibration line (eq 4). Two results of repetition (in bold) and three results of calibration (In bold) were found to agree within the 95% level of confidence.
A
B
0.6
0.5
Absorbance
Absorbance
0.5
0.6
0.4 0.3 0.2 0.1
0.4 0.3 0.2 0.1
0.0 0
20
40
60
80
100
120
Fe3∙ Concentration / (mg L∙1)
0.0 0
20
40
3∙
Fe
60
80
100
120
∙1
Concentration / (mg L
)
Figure 1. The calibration line for the determination of iron by (A) batch measurements and (B) FIA measurements. The data of all the calibration lines, obtained during different days, were pooled and used to construct a single calibration line. The spread of data thus represents the uncertainty of calibration, according to eq 4.
tion. However, the SDs of calibrations were larger than the SDs of repetitions, as shown in both batch experiments and FIA experiments, Tables 1 and 2, respectively. In addition, the concentrations determined by FIA had a larger spread (21.6 mg L‒1) than the concentrations of the batch method (9.8 mg L‒1). The relative standard deviations (RSDs) of batch experiments and of FIA experiments were determined by means of the uncertainty budget (3) as approximately 1% and 5%, respectively. The conventional limits of detections (LOD, ref 17) were determined from repetitions of blanks and estimated as 0.2 mg L‒1, which was a characteristic value of the individual batch experiments. The corresponding LODs of the FIA experiments were determined as 0.3 mg L‒1. With respect to the prediction of the uncertainty budget and the LOD (3, 4), the groups agreed about the values, in contrast to the divergence found between the results of the unknown (Tables 1 and 2). In summary, only a few experiments exhibited complete correspondence, as indicated by numbers in bold type in Tables 1 and 2. These apparent differences appeared as paradoxes to the students, and the origin of differences were not tangible at this stage of the exercise. In the discussion evolving at the colloquium, the prevailing view that was promoted by the groups was centered around a concern about their own skills and a concern about a tentative lack of their routine in chemical analysis. Also, the students were rightfully concerned about the lack of temperature control of the solutions during the analysis that may affect precision and to a minor extent also affect the accuracy.
Because the batch experiments showed a higher sensitivity and, thus, exhibited a lower LOD, it was unanimously concluded by the groups that the batch method was superior to the FIA method with respect to overall performance. This conclusion was apparently supported by the results of Tables 1 and 2, which show an SD on results that was generally lower for the batch experiments. Accuracy In recognition of the incompatible results of the measurements of precision (above), it was worthwhile to consider another approach to the treatment of data. The concept of accuracy is the more important concept to analytical chemistry because it represents the proximity of the result to an unknown but consensus value (3, 4). Thus, in a second approach, the calibration data were pooled into a single depiction thus representing the overall distribution of data combined (Figures 1A and 1B). The general trend follows the expected behavior of an increasing spread of data as a function of concentration (3, 4). [The same general trend was excellently produced also by the FIA experiments (Figure 1B)]. The results as shown in Table 3 have SDs larger by an order of magnitude compared to the SDs of precision (Tables 1 and 2). The SD of batch–calibration showed an SD that was approximately 50% lower than the corresponding SD of repetition (Table 3) but this difference is much smaller than similar differences between groups during the measurements of precision (Tables 1 and 2) that were found to deviate
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In the Laboratory Table 3. The Values of the Unknowns Determined by the Two Methods Method
[Fe(III)]/ (mg L–1)
SD of Calibration/ (mg L–1)
SD of Repetition/ (mg L–1)
LOD/ (mg L–1)
Number of Repetitions
LLA/ (mg L–1)
Batch
42.4
2.3
3.2
0.4
255
0.5
FIA
47.6
4.3
5.5
0.2
36
1.4
Note: By pooling all calibration data (Figure1) that were obtained during different days, the value of the unknown could be calculated from a mutual calibration line of pooled calibration data, and the associated uncertainty was determined by eq 4. The values of the unknowns thus obtained were found to differ significantly, owing to the high number of repetitions influence on Student’s t test.
by more than an order of magnitude. An examination of the results in Table 3 by Student’s t test showed that the results deviated significantly but the average values as well as the SDs were in greater proximity than those of Tables 1 and 2. The data showed that a more pronounced correspondence was obtained, and that the SDs corresponded closely (eq 4), which further showed that the measurements were in statistical control. That is, nothing went wrong during the analysis performed by the students, in contrast to what was initially suspected during the discussion of the first series of results (Table 1). Accordingly, the results obtained by pooling all calibrations were determined satisfactorily at an RSD of 5–12% (Table 3). The high number of blanks used for the determinations in batch experiments (Table 1) provided a reasonable and realistic value of the LOD (Table 3). Since the number of blanks was limited to 26 for the determination of the LOD in FIA experiments (Table 3), the value of 0.2 mg L‒1 was determined at a lower degree of reliability of 84% (4), and it might accidentally have been underestimated. However, once the parameters of eq 4 had been established by the linear calibrations using, for example, Excel Solver, the concentration where the RSD exceeds 50% may be estimated. By assuming that the standard deviation of blanks is equal to the standard deviation of the intercept at zero concentration, that is, c = 0 ⇒ sA ~ = sβ (eq 4). This assumption leads to an expression of the lower limit of analysis (LLA), cmin, that is based on the total spread of all data contributing to the calibrations pooled (Figures 1A and 1B): sβ
2
c min = 1 − 2
sα α
2
α α < sα < for − 2
(5) α 2
The values of this equation are shown in Table 3 where the order is reversed of the lowest possible determination of an unknown, as compared to the values of the characteristic LODs. This discrepancy originates from the different definitions of the LOD and of the LLA, which depend on the SD of blanks and on the SD of intercept, respectively. As dependent on the value of the intercept and on the RSD on slope, the pre-factor of eq 5 may easily exceed a value of three that is conventionally used for calculating the LOD. Thus, the minimum concentration of eq 5 represents the more accurate concentration that may, in effect, be used for reliable determinations at an RSD lower than 50%. Based on these observations, it was confirmed that the batch method exhibited the prime overall performance (Table 3). Reliability All results presented above rely on standard deviations of a certain degree of validity that is determined by the number of 736
repetitions (4). It was noticed by the results of Table 1 that the number of repetitions ranged from 3 to 129, which indicates that there was a difference in reliability of the SD associated with the result of each experiment. Confidence ranges depend on a reliable determination of the SD. Thus, to estimate the reliability of the SD, R, the following formula may be considered (4):
R ≅ 1 −
1
2 ( N − 1)
(6)
According to this formula, the reliability is determined uniquely by the number of repetitions, N. The results of Table 1 were thus determined at reliabilities between 0.5 (50%) and 0.94 (94%), which adds to the general picture of the quality of measurement. Conclusion It was demonstrated that students at the undergraduate level who initially believed that they all measured different concentrations of iron in the sample of an unknown, at the end of the exercise obtained improved correspondence between determinations by accepting a higher uncertainty of the methodologies. The simultaneous correspondence between both the results and between the uncertainties was obtained by applying a slightly modified version of the law of propagation of errors, according to eqs 3 and 4 (12). It was imperative for the success of the exercise that the laboratory value of the concentration of the unknown was kept secret from the students before the colloquium, which ensured a truly unbiased treatment of results. All results were accepted and no outliers were removed from the data set. This approach showed that the influence of outliers on the final result may be eliminated only by increasing the number of repetitions. It may thus be concluded that correspondence of instrumental investigations may be achieved only by maintaining a high number of repetitions most favorably obtained by independent measurements over several days. The concept of precision was related to the performance of each individual group while the concept of accuracy could be related to the combined effort of the groups. Although the accuracy (Table 3) was considerably lower than precision (Tables 1 and 2), it was a pleasing experience for the students to see that all results converged within the limitations of the methodology. The introduction of a lower limit of analysis (eq 5) provided a concentration that was based on the uncertainty on the intercept, and thus took into account the spread of data around the pooled calibration line, which is not represented in the determination of the conventional LOD. It was also demonstrated that the uncertainty budget underestimated the observed uncertainty, and for the batch results it provided an
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In the Laboratory
unrealistically low value close to that of the LOD. The relatively high uncertainty on the final result was explained as an inherent property of the detector. Accordingly, this indicates that the procedure may be applied as a general method that allows determination of the uncertainty induced by detectors of the spectrophotometers, which is difficult to determine by other means of measurement. If the result of the unknown were to be reported to a customer, the right value would be given by the consensus value, as given by the average of the three measurements (Table 3 and laboratory value) with the uncertainty given by the pooled uncertainty, that is cu = 44.5 ± 3.3 mg L‒1. The RSD (7.4%) of this result corresponds well to the coefficient of variation (9.1%) predicted by the Horwitz curve (18). Acknowledgments Many thanks are due to Anders C. Raffalt for preparing the experimental setup and to the students who took part in the work of the course on Analysis and Chromatography (DTU no. 26316), spring 2007: Tanja Thorskov Bladt, Thomas HjulerSørensen, Karen M. D. Jørgensen, Maria C. Malmos, Mette A. Nielsen, Mina Rahmati, Sara P. Rasmussen, Hao Yan, and Valdemar Sporring. Literature Cited 1. Konieczka, P. Crit. Rev. Anal. Chem. 2007, 37, 173. 2. Coleman, D.; Vanatta, L. Am. Lab. 2004, 5, 40. 3. Eurachem/CITAC Guide, Quantifying Uncertainty in Analytical Chemistry, 2nd ed.; Ellison, S. L. R., Rosslein, M., Williams, A. Eds.; 2000; http://caeal.ca/assessor_training/at01_eurachem_uncertainty.pdf (accessed Jan 2009).
4. Guide to the Expression of Uncertainty in Measurement, 1st ed.; International Organization for Standardisation: Geneva, Switzerland, 1995; ISBN 92-67-10188-9. 5. Salter, C.; de Levie, R. J. Chem. Educ. 2002, 79, 268. 6. Centner, V.; Massart, D. L.; de Jong, S. F. J. Anal. Chem. 1998, 361, 2. 7. Quintar, S. E.; Santagata, J. P.; Villegas, O. I.; Cortinez, V. A. J. Chem. Educ. 2003, 80, 326. 8. Ogren, P.; Davis, B.; Guy, N. J. Chem. Educ. 2001, 78, 827. 9. Tellinghuisen, J. Appl. Spectrosc. 2000, 54, 431. 10. Salter, C. J. Chem. Educ. 2000, 77, 1239. 11. Meyer, E. F. J. Chem. Educ. 1997, 74, 1339. 12. Andersen, J. E. T. Chemia Analityczna 2007, 52, 715. 13. Danzer, K.; Currie, L. A. Pure and Appl. Chem. 1998, 70, 993. 14. Van Loco, J.; Elskens, M.; Croux, C.; Beernaert, H. Accreditation Quality Assurance 2002, 7, 281. 15. Tripathi, J. N.; Chikhalikar, S.; Patel, K. S. J. Autom. Meth. Man. Chem. 1997, 19, 45. 16. Aggarwal, S. G. J. de Phys. IV 2003, 107, 1155. 17. Huber, W. Accred. Qual. Assur. 2003, 8, 213. 18. Horwitz, W. Anal. Chem. 1982, 54, 67A.
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