1. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 1 Mean Value and Standard Deviation of a Random Sample

2. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 2 Experimental variance : Experimental standard deviation; Arithmetic mean value: Parameters of a Normal Distribution

3. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 3 Variance of the Mean of a Random Sample distribution of the mean distribution of the single values of one individual de- termination of q mean

4. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 4 Systematic Effects of a Measurement

5. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 5 Outdated and Non Practical Splitting of Measurement Deviations Measurement deviation Systematic error Random error Partly corrected Result Error type A Error type B

6. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 6 Recommendations of the CIPM (1980) New definition of types of measurement uncertainties: a) Uncertainties determined with statistical methods b) Uncertainties which cannot be determined by a statistical mean Goal: Comparability of results and unproblematic further processing of quoted uncertainties

7. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 7 Modern and Practical Way of Dealing With Measurement Uncertainty measurement systematic deviation measurement result unknown systematic deviation random deviation correction remaining deviation measurement deviation known systematic deviation measurement value measurement uncertainty

8. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 8 Concept Based on Observed Quantities Uncorrected mean value of observations Corrected mean value of observations 1. C orrection of all known systematic effects 2. Incorporation of the uncertainty of the correction Standard deviation of the uncorrected mean value Summarized measurement uncertainty of the corrected mean value

9. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 9 Definition of Measurement Uncertainty A parameter, associated with the result of a measurement, that characterises the dispersion of the values that could reasonably be attributed to the measurand.

10. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 10 Example:Time Correlation of a Measured Quantity q t q

11. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 11 Determining Measurement Uncertainty Non-Statistically Possible sources of information:  previous measurement data  experience with the sample and the measurement technique being used  information quoted by the manufacturer  data based on calibrations or certificates  uncertainties taken from manuals

12. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 12 Uncertainty of the Experimental Standard Uncertainty Numbers of measurements n Uncertainty of the Uncertainty / % 276 352 442 536 1024 2016 3013 5010

13. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 13 Calculation of the Measurement Uncertainty

14. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 14 Step 1: Specification of the Measurand  Complete equation for the measurand  Description of the scope of the measurement  Correction for the known systematic effects

15. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 15 Step 2: Identify Uncertainty Sources parameter 2 parameter 1 parameter 4parameter 3 measurand Cause and effect diagram First stage

16. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 16 Step 2: Identify Uncertainty Sources parameter 2parameter 1 parameter 4parameter 3 measurand 1 level influence 2 level influence 3 level influence Cause and effect diagram further stages

17. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 17 Reduction of the diagram after its creation:  Cancelling effects: remove both  Similar effect, same in time: combine into a single input  Different instances re-label Step 2: Identify Uncertainty Sources Cause and effect diagram Final stages

18. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 18 Example: Usual tolerances for some volumetric pipettes waiting time 15s Step 3 and 4: Quantifying the Uncertainty Components and Conversion into Standard Uncertainty

19. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 19 Step 3 and 4: Quantification and Conversion Triangular distribution Standard uncertainty for a triangular distribution within the limits a - and a +

20. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 20 Step 3 and 4: Quantification and Conversion Triangular distribution Centre of the interval Variance With a + -a -

21. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 21 Step 3 and 4: Quantification and Conversion Rectangular distribution Standard distribution for a rectangular distribution within the limits a - and a +

22. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 22 Step 3 and 4: Quantification and Conversion Rectangular distribution Centre of interval Variance

23. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 23 Step 5: Calculation of the Combined Standard Uncertainty u c (y)combined standard uncertainty ffunctional relationship between influence quantities x i and the result y x i i-th influence quantity u(x i )standard uncertainty of the influence quantity x i Nnumber of the influence quantities

24. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 24 Step 5: Calculation of the Combined Standard Uncertainty 1. Rule: Addition and subtraction y = p+q-r+... 2. Rule: multiplication and division y = p q...

25. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 25 Step 5: Calculation of the Combined Standard Sncertainty Example: Substitution: z = o + p n = q + r Calculation of the combined standard uncertainty for z and n according to rule 1:

26. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 26 Step 5: Calculation of the Combined Standard Uncertainty

27. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 27 Quoting the Measurement Uncertainty 1. m = 100.2147 g with (a combined standard uncertainty) u c = 0.35 mg 2. m = 100.02147(35) g 3. m = 100.02147(0.00035) g 4. m = ( 100.02147  0.00035) g

28. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 28 Example What is its value of the uncertainty? A solution of sodium hydroxide (NaOH) is standardized against the titrimetric stan- dard potassium hydrogen phthalate (KHP)

29. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 29 Step 1: Specification Procedure: 1) weigh approx. 0.5 g KHP (standard) 2) add water and stir until the KHP is dissolved 3) titrate with caustic soda solution C NaOH is about 0.1 mol/L

30. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 30 Example Step 1: Specification c NaOH :concentration of NaOH [mol/L] m KHP :initial weight des KHP [g] P KHP :purity of the titre KHP [factor] V Tit :consumption of NaOH solution [mL] F KHP: molecular weight of KHP [g/mol ]

31. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 31 Cause and effect diagram First stage Example Step 2: Sources of Uncertainty F(KHP)V(Tit) c(NaOH) m(KHP) P(KHP)

32. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 32 Example Step 2: Sources of Uncertainty m(KHP)P(KHP) V(Tit) c(NaOH) F(KHP) repeatability calibration linearity calibration repeatability temperature endpoint biasrepeatability intercept Cause and effect diagram further stages

33. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 33 Example Step 2: Sources of Uncertainty m(KHP)P(KHP) V(Tit) c(NaOH) F(KHP) repeatability linearity calibration temperature endpoint Bias repeatability calibration Cause and effect diagram Validation of data

34. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 34 Example Step 2: Sources of Uncertainty Cause and effect diagram Final stage m(KHP)P(KHP) V(Tit) c(NaOH) F(KHP) calibration linearity calibration temperature endpoint Bias repeatability

35. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 35 Example Steps 3 and 4: Quantification and Conversion Weight of KHP Measured value of weight: 0.511 g Non-linearity (declaration): ± 0.15 mg Conversion to a standard deviation using a rectangular distribution

36. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 36 Example Steps 3 and 4: Quantification and Conversion Consumption of NaOH solution Calibration of 50 mL piston burette Measured value of volume: 24.49 mL Declaration: 50 mL + 0.05 mL Conversion to a standard deviation using a triangular distribution

37. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 37 Example Steps 3 and 4: Quantification and Conversion Consumption of NaOH solution Expansion of the NaOH solution as a result of temperature variation  Variation of temperature : + 4  C  Expansion coefficient of water: 2.1  10 -4  C –1 Conversion to a standard deviation using a triangular distribution

38. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 38 Example Steps 3 and 4: Quantification and Conversion Consumption of NaOH solution Standard uncertainty

39. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 39 Example Steps 3 and 4: Quantification and Conversion Purity of the standard Declaration:99.87% - 100.14% Factor:1.000 ± 0.0014 Conversion to a standard deviation using a triangular distribution

40. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 40 Example Steps 3 and 4: Quantification and Conversion Molecular weight of KHP sum formula: C 8 H 5 O 4 K

41. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 41 Example Steps 3 and 4: Quantification and Conversion Molecular weight of KHP Standard uncertainty Assumption: triangular distribution

42. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 42 Example Steps 3 and 4: Quantification and Conversion Molecular weight of KHP Standard uncertainty of F KHP

43. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 43 Example Steps 3 and 4: Quantification and Conversion 0.1%

44. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 44 Example Step 5: Combined Standard Uncertainty List of the calculated values:

45. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 45 Example Step 5: Combined Standard Uncertainty Concentration of NaOH

46. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 46 Example Step 5: Combined Standard Uncertainty

47. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 47 Example Step 5: Combined Standard Uncertainty Calculation

48. M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000 48 Step 5: Combined Standard Uncertainty Calculation 1.Initial weight KHP 2.Consumption of NaOH solution 3.Purity of KHP 4.Relative molecular mass of KHP 5.Repeatability 6.Combined standard uncertainty of the standardized sodium hydroxide solution