establishing traceability and estimating … expdesign...establishing traceability and estimating...

Establishing traceability and estimating measurement uncertainty in physical,

chemical and biological measurements

Experimental Design8.2.2013

Doc. Martti Heinonen

[email protected]

2MH 2013

Outline1. Establishing traceability in measurements

2. Estimating uncertainty

2.1 Objectives and level of modelling

2.2 What should be covered by the analysis

3. Physical measurements - example

4. Chemical and biological measurements 4.1 Sampling

4.2 Example

1 Establishing traceability

4MH 2013

Traceability Tower

SI – System of Units

DU

KC

Primary standard

DU

KC

Secondary standard

DU

KC

Reference standard

DU

KC

Calibration standard

DU

KC

Measuring instrument

DU

KC

Measurement

Mea

sure

men

tunc

erta

inty

• The tower is collapsed if anypart of it is missing orincomplete

i.e.

there is no traceability unless alllevels include all the characteristics of traceability

• At any level the measurementuncertainty can´t be smallerthan levels below.

5MH 2013

Characteristics of Unbroken Traceability

For each calibration of the chain:

• Uncertainty estimation

• Documented and generally acknowledged procedures, documented results

• Competence

• Calibration is valid for the application.(interval of calibrations, conditions etc.)

SI SI –– System of UnitsSystem of Units

PU VC

2 Estimating uncertainty

2.1 Objectives and level of modeling

2.2 What should be covered by the analysis?

2.3 Benefits from uncertainty analysis

7MH 2013

6 steps to evaluating uncertainty

1) Measurement model: List essential input quantities (i.e. parameters xi having a significant effect on the result) and build up a mathematical model (function) showing how they are related to the final result: y = f(x1,,…, xi)

2) Standard uncertainty:Estimate the standard uncertainty of each input quantity (xi)

3) Use the model in uncertainty calculations:Determine the uncertainty due to standard uncertainty of each input quantity (xi): ui(y) = ci u(xi)

4) Correlation:Determine correlation between the input quantities (if relevant).

5) Calculate the combined standard uncertainty

6) Calculate the expanded uncertainty.

2.1 Objectives and level of modelling

9MH 2013

Objectives of modelling

• to describe how the measurement result is calculated from input data (incl. measurement values, the data often include information from earlier measurements, specifications, calibration certificates etc. )

• to show how various factors affect the result

• to provide a tool for calculating the estimate and the uncertainty

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0 10 20 30 40 50 60

Temperature (°C)

Ref

- In

d (°

C) 2005

200620072008

GresolDCalindx tttttt δδδδ ++++= tx= (21,6 ± 0,3) °C

10MH 2013

Level of modelling

• The model is never complete; approximations are needed:high accuracy ⇒ more details in the modellow accuracy ⇒ rough approximations in the model

• A measurement model is never identical with a presentation of a physical phenomenon:• All data used in the calculation of the estimate and the uncertainty

should have corresponding input quantities in the model

• All input quantities of the model should be connected to data actually used in the calculations

11MH 2013

GresolDCalindx tttttt δδδδ ++++=

display calibrationcertificate long-term

instabilitymonitoring

displayearlier

measurements(validated

specifications)

Source of data:

12MH 2013

niCind RHRHRHRH δδ ++=

CalcGRDCIndwded

wGdRdDdCdIndwcec RHtttttepp

etttteppRH δ

δδδδδδδδδδ

+⋅++++++++++

= rh%100)()(

)()(

e tS

t CS S t C S t C

S t C

ws ( ) exp[/ ,

( / , ) ( / , )

ln( / , )]

= ⋅° +

+ + ° + + ° +

+ ° +

1 Pa 12 3 4

2

5

273 15273 15 273 15

273 15

Completeness of the model:

2.2 What should be covered by the analysis?

14MH 2013

GresolDCalindx tttttt δδδδ ++++=

display calibrationcertificate long-term

instabilitymonitoring

displayearlier

measurements(validated

specifications)

Source of data:

15MH 2013

Issues to be considered

• Measurement target/object

• Measurement method

• Measurement device / equipment

• Measurement conditions / environment

• Measurer

16MH 2013


• Measurement target/object – examples:• gradients

• homogeneity

• disturbance due to the measurement

• temporal variations

17MH 2013


• Measurement method – examples:• reproducibility

• variations affecting the target and/or the equipment

• representativity (number of measurements, measurement time, sampling frequency etc.)

• errors in sampling

• thermoelectric effects, non-ideal contacts, trace gases in tubing, contaminations, etc.

18MH 2013


• Measurement device / equipment – examples:• calibration & reference standard

• drift (zero & full scale)

• resolution

• sensitivity

• non-linearity

• interaction of different parameters

19MH 2013


• Measurement conditions / environment – examples:• ambient temperature

• ambient humidity

• ambient air velocity

• vibrations

• electric noise

• lighting

• magnetic field

20MH 2013


• Measurer – examples:• skills: ability to obtain repeatable & reproducible &

comparable results

• heat and moisture

• contamination

2.3 Benefits from uncertainty analysis

22MH 2013

Improving quality

• Modelling = analysis of measurement

• Mathematical model shows explicitly how different factors affect the result and measurement process

• Uncertainty analysis (including the modelling) brings out• weak points

• factors in which efforts should be focused to get most effective improvement in quality

• Benefitting in risk analysis

23MH 2013

Innovating and planning

• By modelling you can test different methods before setting up an actual measurement system• efficient and low-cost

• You can study the effects of both methods and devices on the measurement quality on the basis of existing data.

• You can judge if a new approach has potential to achieve the required accuracy level.

3 Physical measurements - Example

25MH 2013

Example 3.1: Fluid temperature in a pipe

131.481 Ω

Measurement set-up:-Fluid: water-Pt-100 thermometer

-immersed 5 cm-in a thermometer well

-DMM for resistance measurements

26MH 2013

131.481 Ω

Step 1: Measurement model- How we get temperature from the resistance?→ Calibration equation

- What are the input quantities?- What is the measurement model?

Example 3.1 continuing:

27MH 2013

Example 3.1 – Input quantities

• Measurement target/object• Error due to temperature gradients in the pipe δtG• Fluid: Water & liquid in the thermometer well

⇒ temperature difference between the fluid and the tip of the thermometer is negligible⇒ error due to different time constants is negligible

• Measurement method• Thermometer is immersed only partly

⇒ error due to heat conduction along the probe δtcond(disturbance due to the thermometer is negligible)

• Error in resistance measurement (wires etc.) δRni

28MH 2013


• Measurement device / equipment

• Errors in the calibration equation of the Pt-100: δtcal

• Drift of the Pt-100 since the last calibration: δtdrift

• Reading of the DMM: IR• Calibration correction of the DMM: δRcal

• Drift of the DMM since the last calibration: δRdrift

• Error due to non-ideal resolution of the DMM: δRresol

• Error due to non-linearity of the DMM: δRnL

29MH 2013


• Measurement conditions / environment• Effect environmental conditions on the DMM: negligible

• Effect environmental conditions on the PRT & wiring: included in δtcond and δRni

• Measurer: • permanent set-up & automatic measurement

⇒ the effect of a measurer is negligible

30MH 2013

Example 3.1 – Measurement model

Gdriftcondcali

ninLresoldriftcalRi

i ttttRRRRRIat δδδδδδδδδ ++++⎥⎦

⎤⎢⎣

⎡+++++= ∑

=

)(4

0

31MH 2013


131.481 Ω

Step 2: Standard uncertainty of the input quantities- How we estimate?

32MH 2013


Gdriftcondcali

ninLresoldriftcalRi


⎤⎢⎣

⎡+++++= ∑

=

)(4

0

Quantity, X i Estimate, x i Standard unc., u (x i ) Probability Notes on the determinationi Description Value unit Value unit distribution of the standard uncertainty

1reading of DMM 131.540 Ohm 0.0187 Ohm normal display

2calibration equation 0 °C 0.0075 °C normal

taken from the calibration certificate

3calibration of the DMM 0.057 Ohm 0.0005 Ohm normal


4drift of the Pt100 0 °C 0.0071 °C rectangular

comparing last two calibration results

5drift of the DMM 0 ohm 0.0023 Ohm rectangular


6resolution of the DMM 0 ohm 0.0003 Ohm rectangular display

7temp. gradients 0 °C 0.1155 °C rectangular

from specification (or thermal modelling)

8heat flow along the Pt100 0.22 °C 0.0685 °C rectangular

earlier measurement data, e.g. with calibration

9recording method 0 Ohm 0.0000 Ohm validation results

10non-linearity of the DMM 0 °C 0.0040 Ohm rectangular

calculated from the calibration certificate

33MH 2013


131.481 Ω

Step 3: Effect on the combined uncertainty

34MH 2013


Gdriftcondcali

ninLresoldriftcalRi


⎤⎢⎣

⎡+++++= ∑

=

)(4

0

Quantity, X i Estimate, x i Standard unc., u (x i ) Probability Sensitivity coeff., c i Notes on the determinationi Description Value unit Value unit distribution Value unit of the standard uncertainty

1reading of DMM 131.540 Ohm 0.0187 Ohm normal 2.622 °C/Ohm display

2calibration equation 0 °C 0.0075 °C normal 1


3calibration of the DMM 0.057 Ohm 0.0005 Ohm normal 2.622 °C/Ohm


4drift of the Pt100 0 °C 0.0071 °C rectangular 1


5drift of the DMM 0 ohm 0.0023 Ohm rectangular 2.622 °C/Ohm


6resolution of the DMM 0 ohm 0.0003 Ohm rectangular 2.622 °C/Ohm display

7temp. gradients 0 °C 0.1155 °C rectangular 1


8heat flow along the Pt100 0.22 °C 0.0685 °C rectangular 1


9recording method 0 Ohm 0.0000 Ohm 2.622 °C/Ohm validation results

10non-linearity of the DMM 0 °C 0.0040 Ohm rectangular 1


35MH 2013


131.481 Ω

Step 4: Correlations:- Input quantities can be considered independent on each other

36MH 2013


131.481 Ω

Step 5: Combined standard uncertainty

Step6: Expanded uncertainty

37MH 2013


Gdriftcondcali

ninLresoldriftcalRi


⎤⎢⎣

⎡+++++= ∑

=

)(4

0

Quantity, X i Estimate, x i Standard unc., u (x i ) Probability Sensitivity coeff., c i Uncertainty contribution, u i Notes on the determinationi Description Value unit Value unit distribution Value unit Value unit of the standard uncertainty

1reading of DMM 131.540 Ohm 0.0187 Ohm normal 2.622 °C/Ohm 0.049 °C display

2calibration equation 0 °C 0.0075 °C normal 1 0.008 °C


3calibration of the DMM 0.057 Ohm 0.0005 Ohm normal 2.622 °C/Ohm 0.001 °C


4drift of the Pt100 0 °C 0.0071 °C rectangular 1 0.007 °C


5drift of the DMM 0 ohm 0.0023 Ohm rectangular 2.622 °C/Ohm 0.006 °C


6resolution of the DMM 0 ohm 0.0003 Ohm rectangular 2.622 °C/Ohm 0.001 °C display

7temp. gradients 0 °C 0.1155 °C rectangular 1 0.115 °C


8heat flow along the Pt100 0.22 °C 0.0685 °C rectangular 1 0.069 °C


9recording method 0 Ohm 0.0000 Ohm 2.622 °C/Ohm 0.000 °C validation results

10non-linearity of the DMM 0 °C 0.0040 Ohm rectangular 1 0.004 °C


combined standard uncertainty: 0.144 °CEstimate: 82.13 °C Expanded uncertainty: 0.287 °C

Measurement r 82.1 °C ± 0.3 °C

38MH 2013


131.481 Ω

Measurement result:

(82.0 ± 0.3) °C

39MH 2013


131.481 Ω

Conclusions on factors affecting the result:

- temperature gradients dominate the uncertainty

4 Chemical and biological measurements

4.1 Sampling

4.2 Example

4.1 Sampling

42MH 2013

Objectives of sampling

• The objective is to get reliable information on the whole target of interest• Most significant error sources: taking samples + handling

and analysis of the samples

• Often several samples are needed to be taken in steps

• Sampling is often the dominating uncertainty component

43MH 2013

Methods of sampling• Sampling methods:

• random samples

• systematic samples (periodicity)

• representative sample

• combined sample

• divided sample (sample is too large for the analysis)

• layered sample

44MH 2013

Sampling plan

• The sampling plan should state:• size and number of samples

• locations, times/dates, sampling method

• handling of samples

• sample container; cleaning, closing, storage

• labelling and records

• requirements for the analysis

• environmental conditions during sampling

• other requirements (e.g. authorized persons)

45MH 2013

Sources of uncertainty in sampling

• inhomogeneity of the material

• shape of particles

• instability of the material in time

• layers in the material

• weighting error

• number of samples

• errors in handling the samples

• errors in the analysis

46MH 2013

Uncertainty of sampling• Deviation of samples

• When analysed nb pieces from total number N of pieces and from each piece we take nw samples and we carry out na analysis with each samples, then the standard deviation of the mean is:

awba

wbw

b

bb nnnnnNn

nN 11)( 222 σσσσ ++−

⋅=

• σw = deviation of analysed samples

• σb = deviation of pieces

• σa = deviation of analysis

• nt = total number of analyses =nbnwna

• for homogenious material: σw = 0

• if all samples are analysed: N = nb

47MH 2013

Further information on sampling

• Minimum number of sampling• σs = standard deviation

• R = permitted maximum error

• t = student-t factor

• Literature• EN ISO/IEC 17025:2005, General requirements for the competence of testing and calibration laboratories.

Sections 5.2.5, 5.7

• Measurement uncertainty arising from sampling: A guide to methods and approaches, EURACHEM / CITAC Guide (2007)1st edition (can be downloaded at www.eurachem.org/guides)

• ISO 15189/2003: Medical laboratories -- Particular requirements for quality and competence

• ISO 3534 - 1,2(1993): Statistics

2

22

Rtn sσ

=

4.2 Example

49MH 2013

Preparation of a Cd calibration standard for AAS (Atomic absorption spectroscopy) 1)

• Description of the measurement:• Metal oxide contamination is removed with an acid mixture treatment.

• A volumetric flask of 100 ml is weighed with and without the purified metal inside.

• 1 ml of nitric acid (65 %m/m) and 3 ml ion-free water are added to the flask to dissolve the cadmium. Afterwards the flask is filled with ion-free water up to the mark and mixed by inverting the flask at least thirty times.

• The concentration is:

1) Based on the Example A1 presented in [1]

VPmcCd =

m = mass of metalP = purity of metal,

i.e. mass fractionV = volume of the liquid

of the calibration standard

50MH 2013

Preparation of a Cd calibration standard - continuing

• Step 1: Measurement model

Repeatability(dissolve efficiency)

Supplier’s certification

51MH 2013



( )fillingtempcal

effcertifresolreplinItareIgrossCd VVV

PPmmmmmV

Pmcδδ

δδδδ++

++++−==

])[(

Repeatability(dissolve efficiency)

Supplier’s certification

52MH 2013



• mIgross , mItare = balance readings with and without the metal

• δmlin = correction due to non-linearity of the balance

• δmrep , δmresol = correction due to non-ideal repeatability and resolution of the balance

• P = purity of the metal according to the supplier’s certificate

• δP = correction due to the non-ideal dissolving

• Vcal = the inner volume of the flask according to its calibration certificate

• δVtemp = thermal expansion of the measured volume

• δVtilling = error in filling water up to the mark

( )fillingtempcal


PPmmmmmV

Pmcδδ

δδδδ++

++++−==

])[(

53MH 2013


• Step 2: Standard uncertainty of the input quantities

Quantity, X i Estimate, x i Standard unc., u (x i ) Probability Notes on the determinationi Description Value unit Value unit distribution of the standard uncertainty

1balance readings with the metal 100.28 mg 0.030 mg normal

standard deviation of recorded readings

2balance readings without the metal 0 mg 0.010 mg normal

standard deviation of recorded readings after taring

3non-linearity of the balance 0 mg 0.0173 mg rectangular

According to calibration, the non-linearity is less than ±0.03 mg

4non-ideal repeatability of the balance 0 mg 0.0115 mg rectangular

According to repeated measurements in calibration of the balance, the repeatability is less than ±0.02 mg

5non-ideal resolution of the balance 0 mg 0.0058 mg rectangular

6 purity of the metal 0.9999 0.00006 rectangular

Manufacturer's certificate states that the purity of the metal is 0.9999 ± 0.0001

7 non-ideal dissolving 0 0.0001 rectangular

According to repeated preparations the efficiency is larger than 99.98 %

8 inner volume of the flask 100 ml 0.05 ml normal

The flask was calibrated by weighing with distilled water. The result was (100 ± 0.1) ml with k =2

9thermal expansion of the measured volume 0 ml 0.048 ml rectangular

Because the thermal expansion of glass is much smaller than water, only the water (2.1 x 10 -̂4 1/°C) is taken into account. Temperature variation is ± 4 °C at maximum.

10 filling error 0 ml 0.0200 ml normalstandard deviation from ten fillings and weighings

54MH 2013


Step 3: Effect on the combined uncertainty

Sensitivity coefficients:

• For mIgross , mItare δmlin , δmrep and δmresol :

• For P and δP :

• For Vcal, δVtemp and δVtilling :

mc

mcc CdCd =∂∂

=1

Pc

Pcc CdCd =∂∂

=2

Vc

Vcc CdCd −=∂∂

=3

55MH 2013


Step 3: Effect on the combined uncertaintyQuantity, X i Estimate, x i Standard unc., u (x i ) Probability Sensitivity coeff., c i Notes on the determination

i Description Value unit Value unit distribution Value unit of the standard uncertainty

1balance readings with the metal 100.28 mg 0.030 mg normal 9.999 1/l


2balance readings without the metal 0 mg 0.010 mg normal 9.999 1/l


3non-linearity of the balance 0 mg 0.0173 mg rectangular 9.999 1/l


4non-ideal repeatability of the balance 0 mg 0.0115 mg rectangular 9.999 1/l


5non-ideal resolution of the balance 0 mg 0.0058 mg rectangular 9.999 1/l

6 purity of the metal 0.9999 0.00006 rectangular 1002.8 mg/l


7 non-ideal dissolving 0 0.0001 rectangular 1002.8 mg/l


8 inner volume of the flask 100 ml 0.05 ml normal -10.03 g/l̂ 2


9thermal expansion of the measured volume 0 ml 0.048 ml rectangular -10.03 g/l̂ 2


10 filling error 0 ml 0.0200 ml normal -10.03 g/l̂ 2standard deviation from ten fillings and weighings

56MH 2013


• Step 4: Correlations:• Input quantities can be considered independent on

each other

( )fillingtempcal


PPmmmmmV

Pmcδδ

δδδδ++

++++−==

])[(

57MH 2013

Preparation of a Cd calibration standard - continuingStep 5 & 6: Combined standard and expanded uncertainty

Quantity, X i Estimate, x i Standard unc., u (x i ) Probability Sensitivity coeff., c i Uncertainty contribution, u i Notes on the determinationi Description Value unit Value unit distribution Value unit Value unit of the standard uncertainty

1balance readings with the metal 100.28 mg 0.030 mg normal 9.999 1/l 0.300 mg/l


2balance readings without the metal 0 mg 0.010 mg normal 9.999 1/l 0.100 mg/l


3non-linearity of the balance 0 mg 0.0173 mg rectangular 9.999 1/l 0.173 mg/l


4non-ideal repeatability of the balance 0 mg 0.0115 mg rectangular 9.999 1/l 0.115 mg/l


5non-ideal resolution of the balance 0 mg 0.0058 mg rectangular 9.999 1/l 0.058 mg/l

6 purity of the metal 0.9999 0.00006 rectangular 1002.8 mg/l 0.058 mg/l


7 non-ideal dissolving 0 0.0001 rectangular 1002.8 mg/l 0.116 mg/l


8 inner volume of the flask 100 ml 0.05 ml normal -10.03 g/l̂ 2 -0.501 mg/l


9thermal expansion of the measured volume 0 ml 0.048 ml rectangular -10.03 g/l̂ 2 -0.486 mg/l


10 filling error 0 ml 0.0200 ml normal -10.03 g/l̂ 2 -0.201 mg/lstandard deviation from ten fillings and weighings

combined standard uncertainty: 0.83 mg/lEstimate: 1002.70 mg/l Expanded uncertainty: 1.66 mg/l

58MH 2013


• Thus, the Cd content of the calibration standard prepared for an AAS was:

(1003 ± 2) mg/l

59MH 2013

References and literature[1] EURACHEM/CITAC Guide CG 4, Quantifying Uncertainty in Analytical measurement

(http://www.measurementuncertainty.org/mu/QUAM2000-1.pdf)

LITERATUREISO/IEC Guide 99-12:2007, International Vocabulary of Metrology — Basic and General Concepts and Associated Terms, VIM

International vocabulary of metrology — Basic and general concepts and associated terms (VIM), 3rd ed., JCGM 200:2008(can be downloaded at http://www.bipm.org/en/publications/guides/)

Metrology - in short, 3rd edition, EURAMET 2008, 84 p. (www.euramet.org)

JCGM 100:2008, Evaluation of measurement data – Guide to the expression of uncertainty in measurement, First edition, JCGM 2008 (http://www.bipm.org/en/publications/guides/)

European cooperation for Accreditation, EA-4/02 Expression of the Uncertainty of Measurement in Calibration, December 1999. (http://www.european-accreditation.org/n1/doc/ea-4-02.pdf)

UKAS M3003, The Expression of Uncertainty and Confidence in MeasurementS. A. Bell, A beginner's guide to uncertainty in measurement, Measurement Good Practice Guide No. 11 , (Issue 2), National Physical

Laboratory 2001, 41 p. (www.npl.co.uk)

MIKES has published several guides in Finnish on the uncertainty estimations in different fields.

establishing traceability and estimating … expdesign...establishing traceability and estimating...

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