Since the publication of the ISO Guide to the Expression

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1 DE SILVA: JOURNAL OF AOAC INTERNATIONAL VOL. 86, NO. 5, SPECIAL GUEST EDITOR SECTION Uncertainty of Analytical Determinations GALAPPATTI M.S. DE SILVA 4, PB Alwis Perera Mawatha, Katubedda, Sri Lanka Two examples of uncertainty calculation are described for determination of nitrates in solution (APHA method 4500B) and total suspended solids in water (APHA method 540D). Since the publication of the ISO Guide to the Expression of Uncertainty in Measurement (GUM; 1), a large number of papers have been published on the determination of test and measurement uncertainties. In the area of physical testing and calibration, the methods of determining uncertainties are well established. In the fields of chemical and biological testing, uncertainty determinations are still in an evolutionary stage. Several organizations have published guides for calculating test and measurement uncertainties. A useful one applicable to the area of chemical testing was published by EURACHEM and CITAC (). The present paper contains examples of uncertainty calculations from the chemical testing field based on the guidelines given in the GUM and EURACHEM/CITAC guides. Example 1: Determination of Nitrates in a Solution Method The determination of nitrates is a common test performed in many analytical test laboratories. The concentration of nitrates in a test solution is determined by the spectrophotometric method (3) described below. A stock solution is prepared by dissolving g potassium nitrate (KNO 3 ) in 1000 ml distilled water. The stock solution is preserved by adding ml chloroform. Preserved in this manner the stock solution is stable for at least 6 months. An intermediate solution is prepared by diluting 100 ml stock solution to 1000 ml with distilled water. A series of standardizing solutions is prepared by diluting 1,, 5, and 10 ml intermediate solution to 50 ml with distilled water. These calibration standards are measured for absorbance in a spectrophotometer at wavelengths of 0 and 75 nm. The corrected absorbance of each standard is determined from the equation: Corrected absorbance = absorbance at 5 nm x absorbance at 75 nm (1) Guest edited as a special report on Uncertainty of Measurement in Chemical and Microbiological Testing by John L. Love. Corresponding author s quantum@sltnet.lk. The test solution is prepared by adding 1 ml 1N hydrochloric acid to 50 ml filtered test sample. Five such test solutions are prepared. The absorbance of each test solution is measured at 5 and 75 nm in the spectrophotometer. Experimental Results Absorbance of standardizing solutions. The corrected absorbance readings obtained from measurements of the standardizing solutions are given in Table 1. A linear equation of theformy=mx+cisfitted to these data by performing a linear regression analysis. The values of m and c obtained and their corresponding standard errors are given in Table. The linear equation connecting concentration of nitrates to measured absorbance is thus: Absorbance (Y) = concentration, ppm (X) () Absorbance of test solution. The corrected absorbance readings of the test solution are given in Table 3. The mean absorbance of the test solution is units and the concentration of nitrates in the test solution is computed by Equation : Concentration, ppm = ( )/0.301 = ppm (3) The value stated above is complete only when the uncertainty is included. Also, the uncertainty is required in order to round off the value to the correct number of decimal places. Uncertainty Analysis The basic steps performed in the determination are shown in Figure 1. A set of standardizing solutions is measured with the spectrophotometer, and their corrected absorbances are recorded. A straight line is fitted to this data by performing a linear regression analysis. The absorbance of the prepared test solution is measured with the spectrophotometer. The concentration of the test solution corresponding to the measured absorbance is computed from the equation of the straight line. The cause and effect diagram of Figure shows the main contributory factors to the uncertainty of the measured result. The main contributory effects are absorbance measurements of the test solution and standardization of the spectrophotometer with standardizing solutions. Absorbance Measurements of the Test Solution The uncertainty of the absorbance measurements of the test solution consist of components: calibration uncertainty of

2 1078 DE SILVA: JOURNAL OF AOAC INTERNATIONAL VOL. 86, NO. 5, 003 Table 1. Concn of nitrates, ppm the spectrophotometer (obtained from the calibration certificate) and repeatability of absorbance readings. The standard uncertainty of the absorbance measurement is calculated by combining the calibration uncertainty of the spectrophotometer obtained from the certificate (0.01 absorbance units) and the standard deviation of the mean (SDOM) of test readings, absorbance units (Table 3) by the root sum square method as shown below. U = absorbance units (4) Y Absorbance readings of standard solutions Corrected absorbance Standardization of Spectrophotometer Readings Both the x and y coordinates (i.e., the concentrations and the absorbances of the standardizing solutions) of the data points have inherent uncertainties. These uncertainties are propagated to the uncertainties of the coefficients of the linear fit (m and c of Equation ) in addition to the other uncertainties that result from random effects. The determination of the uncertainties of the linear regression (those of m and c) accurately requires complex numerical techniques. Two possible methods are described by Cecchi (4) and Lira (5). In the present study a less accurate but simple procedure () is used to estimate the uncertainties of the linear regression. Uncertainties of the concentration of standardizing solutions. Figure 1 shows the steps followed in the preparation of a standardizing solution. The cause and effect analysis (Figure 3) indicates the main contributory factors. The uncertainty of the stock solution is propagated to the standardizing solutions through the intermediate solution and the transfer apparatus used for dilution, namely pipets and vessels. The evaluations of these components are explained below. (a) Uncertainty of stock solution. The stock solution is prepared by dissolving g reagent quality KNO 3 in 1000 ml distilled water. The uncertainty of the stock solution concentration is due to the uncertainties of the purity of KNO 3, uncertainty of mass determination and uncertainty of volume determination. Table. Linear regression coefficients Coefficients Standard error X variable (m) Intercept (c) (1) Purity of KNO 3. In the supplier s certificate, the purity of KNO 3 is quoted as % by mass. The purity as mass fraction is therefore Because there is no additional information about the value, a rectangular distribution is assumed. To obtain the standard uncertainty, the value of is divided by 3. Relative uncertainty due to purity = /3 = (5) () Mass. The weighing is done with a balance having an expanded uncertainty (k = ) of 3 mg. This includes the uncertainties due to repeatability and readability of the balance. Relative uncertainty of mass determination = 0.003/( ) = 0.00 (6) (3) Volume. The combined uncertainty of the volume measurement inclusive of calibration uncertainty of the vessel and repeatability of measurement is ml. The measured volume corrected to the temperature of the laboratory (5C) is ml. Relative uncertainty of volume determination = 0.035/ = (7) Combined uncertainty of stock solution. The combined relative uncertainty of the stock solution is computed as the root sum square of the above 3 uncertainties as shown below. Relative combined uncertainty of concn of stock solution = (8) Concn of stock solution = g/ml equivalent to 100 ppm nitrogen (b) Uncertainty of intermediate solution. The intermediate solution (Table 4) is prepared by diluting 100 ml stock solution to 1000 ml. The uncertainty of the intermediate solution consists of 4 main components: the relative uncertainty of the stock solution determined in the previous section (0.00), Table 3. solution Reading Corrected absorbance readings of the test Absorbance Mean Standard deviation Standard deviation of the mean 0.018/5 = 0.008

3 DE SILVA: JOURNAL OF AOAC INTERNATIONAL VOL. 86, NO. 5, Figure 1. Nitrates determination procedure. the calibration uncertainty of the 100 ml vessel, the calibration uncertainty of the 1000 ml vessel, and the uncertainties due to repeatability of transfer to the vessels. The calibration uncertainties of the 100 and 1000 ml vessels are computed from their tolerances assuming a rectangular distribution. These are ml for the 100 ml vessel and 0.46 ml for the 1000 ml vessel (taken from ISO 104; 7). The components arising from repeatability are estimated as 0.6 and 0.8 ml for the 100 and 1000 ml vessels, respectively. Thus, the combined relative uncertainties of the vessels are and 0.001, respectively. Again, the combined relative uncertainty is computed by the root sum square method as shown in Table 4. (c) Uncertainties of standardizing solutions. Each standardizing solution is prepared by diluting the intermediate solution with pipets of 1,, 5, and 10 ml capacity, and the vessel of 50 ml capacity. The contributory factors to the uncertainties of standardizing solutions are the uncertainty of the intermediate solution (Table 4) and those of the dilution factors. The uncertainties of the dilution factors are obtained by considering the uncertainties of the pipets and vessels used to prepare them. (1) Uncertainties of pipets and vessels. The uncertainties arising from the use of a pipet or vessel consist mainly of components: the variations that result in the transfer operation (repeatability) and the calibration uncertainty of the pipet or vessel. The repeatability of transfer is estimated experimentally by weighing a number of pipetings and calculating the standard deviation of the resulting volumes. The standard uncertainty of repeatability is computed as an SDOM, i.e., the standard deviation is divided by the square root of the number of repetitions (SDOM = standard deviation/n, where n = number of repetitions). The calibration uncertainties of pipets and vessels are obtained from their calibration certificates. In the absence of a calibration certificate for the pipet or vessel, a Type B estimation could be made from the specified tolerance corresponding to the class of the pipet or vessel. For example, the specification tolerances given in ISO 835- (6) for Class A pipets and ISO 104 (7) for one mark volumetric flasks are given in column 3 of Table 5. The calibration uncertainty of a pipet or vessel could be computed from these tolerances assuming a rectangular distribution. Another uncertainty that arises in the use of pipets and vessels is due to the difference in temperature during calibration and use. Usually volumetric glassware is calibrated at a reference temperature of 0C. However, it may not be used at this temperature. When not used at 0C, a correction to the volume of the pipet or vessel can be computed by using the coef- Figure. Cause and effect diagram for uncertainty of nitrates concentration. Figure 3. Cause and effect diagram for uncertainty of standardizing solutions.

4 1080 DE SILVA ET AL: JOURNAL OF AOAC INTERNATIONAL VOL. 86, NO. 5, 003 Table 4. Uncertainty of intermediate solution Concn ppm Factor ( ) = xi 3.6, x i 5., and = 0.9 we obtain var(x pred ) = ppm. The combined standard uncertainty (U x ) due to the linear regression and uncertainty of the measured absorbance Y is therefore given by U x = (0.0018) = ppm ficient of volume expansion of glass and the temperature difference. In the present case, this correction is negligible as the dilution factor is a ratio of the volume of the vessel to that of the pipet. The combined relative uncertainty of the pipets and vessels is computed by the root sum square method shown in Table 5. The uncertainty of standardizing solutions is obtained by combining the relative uncertainties of the intermediate solution (0.0064; Table 4), and those of the pipets and vessel (Table 5) using the root sum square method. This computation is given in Table 6. The uncertainty of each standardizing solution is obtained by multiplying the relative uncertainty of the dilution factor by the corresponding concentration of the standardizing solution (Table 7). Uncertainty of nitrate concentration of test solution. To obtain the uncertainty of the nitrate concentration of the test solution, the following equation given in Appendix E of ref. is applied: var(x pred) = var(y obs) / m S 1 m n i i xpred x x x / n In the above equation, var(x pred ) is the quantity we want to determine, namely the square of the standard uncertainty of the nitrates concentration, X; var(y obs ) is the square of the uncertainty of the measured absorbance Y, U y ; m is the slope of the regression line; X i are values of the concentrations of 4 standardizing solutions (column 1 of Table 1), and is the mean of the 4 X i values. n is the number of data points and is equal to 4. S, the sum of squares of the residuals, is obtained from the least-squares analysis and has a value of By substituting the following values in Equation 9, Uy = 0.013, m = 0.301, n =4 (9) In order to obtain the total combined uncertainty, we also have to add the contribution arising from the uncertainties of the standardizing solutions. This is estimated by using the following approximate formula (ref. ): U U /N (10) stds In this formula, U ss is the uncertainty of the standardizing solutions and N the number of standardizing solutions used in the regression. In the present case,n=4andthevalues of U ss are given in the third column of Table 7. The value of U stds calculated from the above formula is ppm (( )= ppm) The total combined standard uncertainty of the nitrates concentration is computed by combining the standard uncertainty U x and the contribution from the uncertainties of the standardizing solutions. ss Combined standard uncertainty = ( ) = ppm (11) Expanded Uncertainty The expanded uncertainty is obtained by multiplying the combined standard uncertainty by a coverage factor (k). The coverage factor is usually taken as the t-value (value of the t distribution; 1) corresponding to the number of effective degrees of freedom of the combined standard uncertainty at a given level of confidence. A confidence level of 95%, as recommended by the GUM, is widely used presently. However, the estimation of the effective degrees of freedom on a sound statistical basis is somewhat difficult except for the simplest determinations. Table 5. Uncertainties of pipets and vessels Volume, ml Repeatability, ml Tolerance, ml Tolerance/3, ml Combined uncertainty, ml Relative uncertainty ( ) = /1 = ( ) = / = ( ) = /5 = ( ) = /10 = (vessel) ( ) = /50 = (vessel) ( ) = /100 = (vessel) ( ) = /1000 = 0.001

5 DE SILVA: JOURNAL OF AOAC INTERNATIONAL VOL. 86, NO. 5, Table 6. Uncertainties of dilution factors (standardizing solutions) Concn, ppm Factor Relative uncertainty ( ) = ( ) = ( ) = ( ) = In most cases, the coverage factor lies between and 3, corresponding to approximately 95 and 99% confidence levels, respectively. It is now common practice to use a coverage factor of to compute the expanded uncertainty. When this is done, the expanded uncertainty is said to have a confidence level of approximately 95%. The expanded uncertainty = = ppm (1) This may be rounded off to 1 significant figure, 0.1 ppm. Reporting the Result The values of the result, the expanded uncertainty, and the coverage factor used (or confidence level) are reported. The result is rounded off to the same number of decimal places as are found in the expanded uncertainty. In the present case, the value ppm is rounded off to 0.1 ppm, giving 1.8 ppm. Thus, concentration of nitrates (as nitrogen) = ppm, withk=(orconfidence level of approximately 95%). Example : Determination of Total Suspended Solids Method The total of suspended solids in a test solution is determined by weighing the residue left on a filter paper when the test solution is filtered with a vacuum suction apparatus. The details of the method are given in ref. 8. A measured volume of the test solution is pipetted out onto a preweighed filter paper. Excess solution is removed with a vacuum suction apparatus. The filter paper is dried in an oven at a temperature of C until constant mass is reached. Theory The total suspended solids (T) is given by the equation: T = (M M1)/V = M/V (13) where M1 = mass of filter paper and M = mass of filter paper with residue. Experimental Values The experimental values in the determination of total suspended solids of a sample using the above procedure are as follows: Mass of residue = 0. g = 00 mg Volume of solution transferred (volume of pipet) = 0 ml = 0.0 L Calibration uncertainty (expanded) of the balance = 1mgatk= Calibration standard uncertainty of the pipet = Class A tolerance/3 = 0.1/3 ml Applying the above experimental values, T = 00/0.0 = mg/l (14) Uncertainty Analysis The cause and effect diagram of Figure 4 shows the main contributory factors to the uncertainty of suspended solids. These arise from the determination of mass and volume. (a) Mass determination. The uncertainty of the mass determination (U M ) is mainly due to the calibration uncertainty of the balance used. This includes the repeatability and the readability of the balance and is obtained from the calibration certificate of the balance. Table 7. Uncertainties of standardizing solutions Concn, ppm Relative uncertainty (from Table 6) Uncertainty, ppm Uncertainty/4, ppm = = = = Figure 4. Cause and effect diagram for uncertainty of suspended solids.

6 108 DE SILVA: JOURNAL OF AOAC INTERNATIONAL VOL. 86, NO. 5, 003 Table 8. Uncertainty budget for total suspended solids Source Symbol Type Calculation Standard uncertainty Balance U M B 1/ mg 0.5 mg Calibration of pipet B 0.1/3 ml L Repeatability of transfer A 0. ml L Volume change of solution due to temp. variation B ( )/3 ml L Total uncertainty of volume U V B 10 6 ( ) L L (b) Volume determination. There are 3 main uncertainty components associated with the volume determination: the calibration uncertainty of the pipet, repeatability of transfer, and uncertainty arising from temperature variations in the laboratory. The combined uncertainty of the volume determination is U V. (c) Uncertainty budget. The uncertainty budget is given Table 8. (d) Uncertainty of measured value. To obtain uncertainty (U T ) of the measured value (T), the law of propagation of uncertainties is applied (Taylor theorem) to Equation 13 as shown below: U T U M T T UV M V (15) The partial differentials are obtained by differentiating Equation 13 with respect to M and V, respectively. They are: T M V V and T M Substituting in Equation 15, we obtain: U U 1 U M T M V V V 1 (16) V (17) We can now find the combined standard uncertainty of T by substituting the above values in Equation UT Expanded Uncertainty U T = g/l = mg/l g/l The expanded uncertainty is obtained by multiplying the combined standard uncertainty by the coverage factor. Expanded uncertainty = = 13.8 mg/l. Round this off to 14 mg/l (rounded off to the nearest whole number). Reporting the Result The result is reported as: Total suspended solids: mg/l at a coverage factor of k = (or confidence level of approximately 95%). Spreadsheet Method Calculation of the above uncertainty by using the spreadsheet method is shown in Appendix A. Acknowledgments The author thanks Subadra Jayasinghe and Gamini Jayasinghe of the Industrial Technology Institute (ITI) of Sri Lanka, and the referees for reading the manuscript and making valuable suggestions. References (1) ISO/BIPM/IEC/IUPAC/IUPAP/OIML (1995) Guide to the Expression of Uncertainty in Measurement, International Organization for Standardization, Geneva, Switzerland () EURACHEM-CITAC (1995) Quantifying Uncertainty in Analytical Measurement, Laboratory of the Government Chemist, London, UK (3) APHA (1998) Standard Methods for the Examination of Water and Wastewater, 0th Ed., Method 4500 B, American Public Health Association, Washington, DC (4) Cecchi, G.C. (1991) Meas. Sci. Technol., (5) Lira, I. (00) Evaluating the Measurement Uncertainty, Institute of Physics Publishing Ltd., London, UK (6) ISO 835- (1981) Volumetric Glassware Graduated Pipets, International Organization for Standardization, Geneva, Switzerland (7) ISO 104 (1998) One Mark Volumetric Flasks, International Organization for Standardization, Geneva, Switzerland (8) APHA (1998) Standard Methods for the Examination of Water and Wastewater, 0th Ed., Method 540D B, American Public Health Association, Washington, DC Appendix A: Spreadsheet Method, Example The spreadsheet method of computing the combined standard uncertainty as applicable to Example is illustrated in the spreadsheet given in Table A1. The basic principle of this method is the computation of the partial differential coefficients of Equation 15 numerically. A detailed explanation of the method is given in ref.. (1) The input values of Equation 13 corresponding to measured mass and volume are entered in cells B6 and B7, respectively.

7 DE SILVA: JOURNAL OF AOAC INTERNATIONAL VOL. 86, NO. 5, Table A1. Spreadsheet for uncertainty calculation A B C D 1 M V 3 4 Uncertainty E 04 5 Value 6 M V E 0 8 T E Diff E U c E 0 11 U c () The uncertainties of these quantities are entered in cells C4 and D4. (3) In row 6, cell C6 contains the value of M incremented by its uncertainty. Similarly in row 7, cell D7 contains the value of V incremented by its uncertainty. Cells C7 and D6 contain the unincremented values of V and M. (4) In row 8, the spreadsheet equation corresponding to Equation 13 is entered, i.e., cell B8 contains the equation = B6/B7. Similarly, C8 and D8 contain the equations = C6/C7 and = D6/D7. (5) In row 9, the differences, C9 B9 and D9 B9 are computed, i.e., cells C9 and D9 contain the equations = C9 B9 and = D9 B9, respectively. (6) In row 10, cells C10 and D10 contain the squares of cells C9 and D9, respectively, i.e., = C9^ and = D9^. (7) In row 10, cell B10 contains the sum of cells C10 and D10, i.e., = sum (C10 + D10). In row 11, cell B11 contains the square root of the cell B10, i.e., = SQRT(B10). This is the required combined standard uncertainty. It can be seen that the spreadsheet method gives nearly the same result as that of the analytical method, the difference being only 1 mg/l.