Mean Value and Standard Deviation of a Random Sample

1. Mean Value and Standard Deviation of a Random Sample

2. Parameters of a Normal Distribution Arithmetic mean value: Experimental variance: Experimental standard deviation;

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. Systematic Effects of a Measurement

5. Outdated and Non Practical Splitting of Measurement Deviations Measurement deviation Random error Systematic error Partly corrected Error type B Error type A Result

6. Recommendations of the CIPM (1980) Goal: Comparability of results and unproblematic further processing of quoted uncertainties New definition of types of measurement uncertainties: a) Uncertainties determined with statistical methods b) Uncertainties which cannot be determined by a statistical mean

7. measurement deviation measurement systematic deviation random deviation known systematic deviation unknown systematic deviation remaining deviation correction measurement uncertainty measurement value measurement result Modern and Practical Way of Dealing With Measurement Uncertainty

8. Uncorrected mean value Corrected mean value of observations of observations orrection of all known 1. C systematic effects 2. Incorporation of the uncertainty of the correction Standard deviation Summarized measurement of the uncorrected uncertainty of the corrected mean value mean value Concept Based on Observed Quantities

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. q t Example:Time Correlation of a Measured Quantity q

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. Numbers of Uncertainty of the n measurements Uncertainty / % 2 76 3 52 4 42 5 36 10 24 20 16 30 13 50 10 Uncertainty of the Experimental Standard Uncertainty

13. Calculation of theMeasurement Uncertainty

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. parameter 1 parameter 2 measurand parameter 3 parameter 4 Step 2: Identify Uncertainty Sources • Cause and effect diagram • First stage

16. parameter 1 parameter 2 2 level influence 3 level influence 1 level influence measurand parameter 3 parameter 4 Step 2: Identify Uncertainty Sources • Cause and effect diagram • further stages

17. Step 2: Identify Uncertainty Sources Cause and effect diagram Final stages • 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

18. Step 3 and 4: Quantifying the Uncertainty Components and Conversion into Standard Uncertainty • Example: Usual tolerances for some volumetric pipettes • waiting time 15s

19. Step 3 and 4: Quantification and Conversion • Triangular distribution • Standard uncertainty for a triangular distribution within the limits a- and a+

20. Step 3 and 4: Quantification and Conversion • Triangular distribution • Centre of the interval • Variance • With a+-a-

21. Step 3 and 4: Quantification and Conversion • Rectangular distribution • Standard distribution for a rectangular distribution within the limits a- and a+

22. Step 3 and 4: Quantification and Conversion • Rectangular distribution • Centre of interval • Variance

23. Step 5: Calculation of the Combined Standard Uncertainty uc(y) combined standard uncertainty f functional relationship between influence quantities xi and the result y xi i-th influence quantity u(xi) standard uncertainty of the influence quantity xi N number of the influence quantities

24. 1. Rule: Addition and subtraction 2. Rule: multiplication and division Step 5: Calculation of the Combined Standard Uncertainty y = p+q-r+... y = p q ...

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. Step 5: Calculation of the Combined Standard Uncertainty

27. Quoting the Measurement Uncertainty • 1. m = 100.2147 g with (a combined standard uncertainty) uc = 0.35 mg • 2. m = 100.02147(35) g • 3. m = 100.02147(0.00035) g • 4. m = ( 100.02147 0.00035) g

28. Example A solution of sodium hydroxide (NaOH) is standardized against the titrimetric stan-dard potassium hydrogen phthalate (KHP) What is its value of the uncertainty?

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 CNaOH is about 0.1 mol/L

30. Example Step 1: Specification • cNaOH: concentration of NaOH [mol/L] • mKHP: initial weight des KHP [g] • PKHP: purity of the titre KHP [factor] • VTit: consumption of NaOH solution [mL] • FKHP: molecular weight of KHP[g/mol]

31. m(KHP) P(KHP) c(NaOH) V(Tit) F(KHP) Example Step 2: Sources of Uncertainty • Cause and effect diagram • First stage

32. P(KHP) m(KHP) intercept linearity calibration repeatability c(NaOH) calibration repeatability temperature endpoint V(Tit) F(KHP) repeatability bias Example Step 2: Sources of Uncertainty • Cause and effect diagram • further stages

33. repeatability P(KHP) m(KHP) linearity calibration repeatability c(NaOH) calibration repeatability temperature endpoint V(Tit) F(KHP) Bias repeatability Example Step 2:Sources of Uncertainty • Cause and effect diagram • Validation of data

34. repeatability P(KHP) m(KHP) linearity calibration c(NaOH) calibration temperature endpoint Bias V(Tit) F(KHP) Example Step 2: Sources of Uncertainty • Cause and effect diagram • Final stage

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. 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. 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. Example Steps 3 and 4: Quantification and Conversion • Consumption of NaOH solution • Standard uncertainty

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. Example Steps 3 and 4: Quantification and Conversion • Molecular weight of KHP • sum formula: C8H5O4K

41. Example Steps 3 and 4: Quantification and Conversion • Molecular weight of KHP • Standard uncertainty • Assumption: triangular distribution

42. Example Steps 3 and 4: Quantification and Conversion • Molecular weight of KHP • Standard uncertainty of FKHP

43. Example Steps 3 and 4: Quantification and Conversion 0.1%

44. Example Step 5: Combined Standard Uncertainty • List of the calculated values:

45. Example Step 5: Combined Standard Uncertainty • Concentration of NaOH

46. Example Step 5: Combined Standard Uncertainty

47. Example Step 5: Combined Standard Uncertainty • Calculation

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