Uncertainty of Measurement Results

Transcription

1 Quality Assurance and Quality Control Uncertainty of Measurement Results A measurement result is an information. Without knowledge of its uncertainty it is just a rumour. Albert Weckenmann

2 Overview Uncertainty definition; Uncertainty and Error; Why do we need uncertainty? When should you evaluate uncertainties of measurement results? Uncertainty sources and uncertainty components; Low of propagation of uncertainty; Procedure s steps for uncertainty evaluation; Examples; Spreadsheet method of Kragten for uncertainty calculation and assessment.

3 Uncertainty definition Basic documents GUM: Guide to the Expression of Uncertainty in Measurement (JCGM 100:2008), first published in 1993 by ISO in collaboration with BIPM, IEC, IFCC, ILAC, IUPAC, IUPAP and OIML; (freely available from EURACHEM/CITAC-Guide Quantifying Uncertainty in Analytical Measurement, 3 rd edition published in 2012 (QUAM:2012.P1); (freely available from EUROPEAN CO-OPERATION for ACCREDITATION Evaluation of the Uncertainty of Measurement in Calibration (EA-4/02 M: 2013); (freely available from

4 Quantity and Uncertainty/Error approach Quantity - property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference (VIM: 2012; 1.1); Measurand the quantity intended to be measured; (VIM: 2012; 2.3); Error Approach (Traditional Approach or True Value Approach) to determine an estimate of the true value that is as close as possible to that single true value. The deviation from the true value is composed of random and systematic errors. The objective of measurement in the Uncertainty Approach is not to determine a true value as closely as possible. Rather, it is assumed that the information from measurement only permits assignment of an interval of reasonable values to the measurand, based on the assumption that no mistakes have been made in performing the measurement.

5 Uncertainty - definition Non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used (VIM: 2012, 2.26) A parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand (GUM: 2008, 2.2.3) result = value ± uncertainty

6 Uncertainty and Error Y true value Error Y A -U +U measurand Y estimated value of the measurand Y A Correction for all recognized systematic effects

7 Why do we need uncertainty? It is an intrinsic part of every measurement result; It is required by ISO/IEC 17025: 2005 Accreditation; Testing laboratories shall have and shall apply procedures for estimating uncertainty of measurement; When estimating the uncertainty of measurement, all uncertainty components which are of importance in the given situation shall be taken into account using appropriate methods of analysis. It improves the knowledge about the measurement procedure; It allows comparison of results GUM does not require statistical tests unless you need it.

8 Why do we need uncertainty? The uncertainty of the result demonstrates the metrological QUALITY of the measurements (not measuring with the smallest achievable uncertainty): In laboratory documents in transparent way the measurement procedure; For end-user gives the result with proper confidence; Measurement uncertainty - is it a criterion for judgement and placing orders of testing laboratories? Measurement uncertainties should not be misused as a quality indicator for laboratories - otherwise there will be a tendency among some laboratories to report unrealistically low measurement uncertainties; There is no need to have lower uncertainties then requested from the customer.

9 Why do we need uncertainty? Customer needs Before calculating or estimating the measurement uncertainty, it is recommended to find out what are the needs of the customers; The main aim of the actual uncertainty calculations will be to find out if the laboratory can fulfill the customer demands; Customers often are not used to specify demands, so in many cases the demands have to be set in dialogue with the customer.

10 Why do we need uncertainty? A well documented uncertainty statement: Underpins your results and provides transparency; Identifies major uncertainty contributors - find out ways to improve the procedure; Demonstrates compliance with limits (legal or contractual). Your best defence in discussions! Repeating the measurement 2, 10 or 100 times gives you only repeatability or over time within-lab reproducibility (not total uncertainty)!

11 Why do we need uncertainty? Comparison of results 12.5 Mean value 12.0 mg kg ± 1 s ± U Lab A Lab B Lab A Lab B Lab A Lab B

12 Why do we need uncertainty? Repeatability / Reproducibility / Precision Measurement precision: closeness of agreement between indications or measured quantity values obtained by replicate measurements on the same or similar objects under specified conditions (VIM: 2012, 2.15). Measurement repeatability: measurement precision under repeatability conditions of measurement (VIM: 2012, 2.21); Measurement reproducibility: measurement precision under reproducibility conditions of measurement (VIM: 2012, 2.25);

13 Why do we need uncertainty? Compliance against limits Example: QUAM:2012 A decision rule must be defined! Guidance can be found in EURACHEM/CITAC (2007) Use of uncertainty information in compliance assessment.

14 When should you evaluate uncertainties of measurement results? When a procedure is introduced inside your laboratory; When a critical factor changes in the procedure (instrument, operator, sample type ); During / together with procedure validation. An individual evaluation is not needed for every individual result produced!

15 When should you evaluate uncertainties of measurement results? How often re-evaluation of uncertainty is needed? In case the laboratory needs to complains with regulation follows the respective document; In case of no guidelines available: (i) Use a common sense for setting the frequency; (ii) Monitor the data from control chart.

16 Uncertainty sources Sampling Storage conditions Instrument effects Reagent purity Assumed stoichiometry Measurement conditions Sample effects Computation effects Blank correction Operator effects Random effects What is usually taken into account when calculating a traditional Confidence Interval (CI)?

17 Uncertainty components Uncertainty type Uncertainty components are formally grouped into two categories based on their method of evaluation, A and B. Type A evaluation (of uncertainty) - method of evaluation of uncertainty by the statistical analysis of series of observations; Type B evaluation (of uncertainty) - method of evaluation of uncertainty by means other than the statistical analysis of series of observations; (previous experiments, literature data, manufacturer s information, expert s estimate). (GUM: 2008, 0.7, 2.3.2, 2.3.3)

18 Uncertainty components Standard uncertainty - u s When expressed as a standard deviation, an uncertainty component is known as a standard uncertainty ( ). (QUAM:2012; 2.3.1) Uncertainty of the result of a measurement expressed as a standard deviation. (GUM: 2008; 2.3.1)

19 Uncertainty components Combined (standard) uncertainty - u c For a measurement result y, the total uncertainty, termed combined standard uncertainty and denoted by u c (y), is an estimated standard deviation equal to the positive square root of the total variance obtained by combining all the uncertainty components, however evaluated, using the law of propagation of uncertainty or by alternative methods (e.g. the spreadsheet method of Kragten). (QUAM:2012; 2.3.2) For example: In general: =(,,,, ) y

20 Uncertainty components Combined (standard) uncertainty - u c Law of propagation of uncertainty: =(,,,, ) =. +2., "...., where, = - sensitivity coefficients; ( ),( ) - standard uncertainties of and ; - correlation coefficient between and ;

21 Uncertainty components Combined (standard) uncertainty - u c Law of propagation of uncertainty: =(,,,, ) When all variables,,,, are independent: = % % When the function involves only sum or difference (e.g.): = ,, ". = + $ = ( ) +( ) +( $ ) When the function involves only product or quotient (e.g.): = $ = ( ) + ( ) + ( $) $

22 Uncertainty components Expanded uncertainty - U U is obtained by multiplying the combined standard uncertainty u c (y), with a coverage factor k; The expanded uncertainty U provides an interval within which the value of the measurand is believed to lie with a higher level of confidence (QUAM:2012; 2.3.3); The choice of the factor k is based on the level of confidence desired: k = 2 corresponded to P = 95%; k = 3 corresponded to P = 99%; For normal distribution: µ ± 2σ µ ± 3σ

23 ± ( ),*.+, (Re)-calculating standard uncertainty (standard deviation) Confidence Interval (CI) The used level of confidence (α) and degree of freedom (f = N-1) are known. ±-. -.= ( ),*.+, / =+= -.., ( ),*

24 (Re)-calculating standard uncertainty (standard deviation) Rectangular distribution p(x) 0 = 1 22, 2<<2 0, otherwise Rectangular distribution is usually described in terms of: the average value and the range (±a); One can only assume that it is equally probable for the value to lie anywhere within the interval; x / =+= 2 3 Certificates or other specification give limits where the value could be, without specifying a level of confidence (or degree of freedom). Example: Concentration of calibration standard is quoted as (1000 ± 2) mg/l.

25 (Re)-calculating standard uncertainty (standard deviation) Triangular distribution p(x) 0 = 2+ 2, 2<<0 2 2, 0<<2 0, otherwise x / =+= 2 6 Distribution used when it is suggested that values near the centre of range are more likely than near to the extremes (Values close to mathematical expectation are more likely than near the boundaries); Example: Volumetric glassware. If any time in doubt, use the rectangular distribution!

26 Procedure s steps for uncertainty evaluation EURACHEM/CITAC Guide In GUM: 2008 ( 8) is presented summary of procedure for evaluating and expressing uncertainty; Approach for uncertainty estimation is also suggested in QUAM: 2012 (a procedure containing 4 steps): Step 1 Step 2 Measurand = quantity intended to be measured. (VIM: 2012; 2.3) Measurand = the particular kind of quantity to be measured, usually the concentration or mass fraction of an analyte (QUAM: 2012; 5.2).

27 Procedure s steps for uncertainty evaluation EURACHEM/CITAC Guide Step 2 Step 3 Express mathematically the relationship between the measurand y and the input quantities on which y depends: =(,,,, ) Should any correction factor be included? Input quantities ( ) may be quantities whose value and standard uncertainty are directly determined in the current measurement (evaluation of Type A, statistical analysis of series of observation) or brought into the measurement from external sources (evaluation of Type B). Step 4

28 Procedure s steps for uncertainty evaluation EURACHEM/CITAC Guide Step 4 (- ±?) mg/g, k = 2

29 Procedure s steps for uncertainty evaluation Understanding the measurement The evaluation of uncertainty is neither a routine task nor a purely mathematical one; it depends on detailed knowledge of the nature of the measurand and of the measurement. (GUM: 2008, 3.4.8) The state of the art of uncertainty evaluation is determined by the degree of understanding of measurement, that includes, critical thinking, professional skills and good understanding of the processes and chemistry involved!

30 Examples Volume of liquid (solution) Example 1. Calculate the combined uncertainty of the volume of water solution prepared in 1000 ml volumetric flask if it is known: In the certificate of the volumetric flask is declared: ( ± 0.2) ml at 20 0 C; The temperature in the laboratory varies within the interval: (20 ± 5) 0 C. The coefficient of volume expansion for water is 2.1x C -1 (the volume expansion of the flask itself is neglected); Repeatability - the uncertainty due to variations in filling is assessed to be 0.3 ml. Volumetric flask certificate

31 Examples Concentration of calibration standard Example 2. A calibration standard solution of Zn (0.102 mg/l) was prepared by transferring 1 ml of commercial reference material (ICP Multi VI mg/l Zn) into 1000 ml volumetric flask. The flask was filled to the mark with 0.1 % HNO 3 v/v. Calculate the combined uncertainty of the conc. of the obtained Zn calibration solution, if: In the certificate of the commercial RM is declared that the concentration of Zn is (102 ± 5) mg/l no information is supplied about the degrees of freedom and confidence level; The standard uncertainty of the pipetted volume is ml; For the standard uncertainty of the volumetric flask volume take the obtained value from the previous example (Problem 1). ICP Multi VI certificate CPAchem Muilti-element #2A certificate

32 Spreadsheet method of Kragten Applicable when: Either =,,,, is linear in ; or ( ) is small compared to ;

33 Method of Kragten Calculating the u c of solution volume Example 1. V flask (ml)= 1000 Variable mean u_s Ser. Temp. Repeat. Certificate uncertainty (ml)= 0.2 mlser Temp. variation ( 0 С)= 5 mltemp Fill repeatability (ml)= 0.3 mlrepeat V flask(ml)= diff diff^ sum Contribution, % 1.4% 79.2% 19.4% 1.4% 19.4% 79.2% Ser. Temp. Repeat.

34 Method of Kragten Calculating the u c of standard concentration Example 2. C1 (mg/l)= 102 Variable mean u_s C1 V1 pipette V2 flask C1 certificate uncertainty (mg/l)= 5 mg/lc V1 pipette (ml)= mlv1 pipette V1 u_s (ml)= mlv2 flask V2 flask (ml)= V2 u_s (ml)= 0.68 C2 (mg/l)= E E E-01 diff 2.89E E E-05 diff^2 8.33E E E E-06sum Contribution, % 96.9% 3.0% 0.1% 3.0% 96.9% C1 V1 pipette V2 flask

35 Method of Kragten Calculating the u c of C x - Bracket calibration Example 3. Calculating the combined uncertainty of the concentration of Zn in surface water (real sample determined by FAAS). mean u_s RSD% Variable mean u_s C3 C4 A3 A4 Aх C3 (mg/l)= mg/lc C4 (mg/l)= mg/lc A3 (signal)= signala A4 (signal)= signala Aх (signal)= signalaх % 1.7% Cx (mg/l)= diff= % diff^2= sum 20.1% C3 64.5% C3 Cx C4 A3 Ax A4 C4 A3 A4 Aх Cx Contribution, % 6.3% 64.5% 1.7% 7.3% 20.1% Advantages: The uncertainty of the concentration of the calibration standards is included; True homoscedastic model; = C 3 ( A 4 A X ) + C 4 ( A X A 3 ) A 4 A 3

36 Uncertainty of Measurement Results Conclusions Uncertainty estimation according to the GUM is a useful and accepted concept: It allows the analyst to combine prior knowledge and observations in a consistent and well defined way; It allows others (e.g. assessors) to understand what and how things were done; It doesn t requires to measure with smallest achievable uncertainty, but with the most realistic one.

37 Uncertainty of Measurement Results Conclusions Uncertainty adopted and accepted by: National Metrology Institutes and BIPM; ISO/IEC 17025: 2005 (required for accreditation); IUPAC, OIML and accreditation community such as EA and ILAC; CEN is incorporating these concepts.

38 Uncertainty of Measurement Results Thank you for your attention!