M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W....

1M. Rösslein,

B. WampflerEvaluation of Uncertainty in Analytical Measurements

B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000

Mean Value and Standard Deviation of a Random Sample

freq

uen

cy

q

s q k

2M. Rösslein,



n

1kkq

n

1q

Experimental variance:

Experimental standard deviation;

Arithmetic mean value:

Parameters of a Normal Distribution

v qn

q qk kk

n

1

12

1

s q v qk k

3M. Rösslein,



Variance of the Mean of a Random Sample

n

)q(s)q(s)q(V

22

q

distribution of the mean

distribution of thesingle values of one individual de-termination of q

kqs

qs

mean

4M. Rösslein,



Systematic Effects of a Measurement

5M. Rösslein,



Outdated and Non Practical Splitting of Measurement Deviations

Measurementdeviation

Systematic error

Random error

Partlycorrected

ResultError

type A

Errortype B

6M. Rösslein,



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

7M. Rösslein,



Modern and Practical Way of Dealing With Measurement Uncertainty

measurement

systematicdeviation

measurement result

unknown systematicdeviation

random deviation

correction remainingdeviation

measurementdeviation

known systematicdeviation

measurementvalue

measurementuncertainty

8M. Rösslein,



Concept Based on Observed Quantities

Uncorrected mean value of observations

Corrected mean value of observations

1. Correction 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

9M. Rösslein,



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.

10M. Rösslein,



Example:Time Correlation of a Measured Quantity q

t

q

11M. Rösslein,



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

12M. Rösslein,



Uncertainty of the ExperimentalStandard Uncertainty

Numbers ofmeasurements n

Uncertainty of theUncertainty / %

2 76

3 52

4 42

5 36

10 24

20 16

30 13

50 10

13M. Rösslein,



Calculation of the Measurement Uncertainty

Specification

Calculate Combined Uncertainty

Identify Uncertainty Sources

Quantify Uncertainty Components

Convert to Standard Deviations

Re-evaluating?

Re-evaluate

EndYes

No

14M. Rösslein,



Step 1: Specification of the Measurand

Complete equation for the measurand

Description of the scope of the measurement

Correction for the known systematic effects

15M. Rösslein,



Step 2: Identify Uncertainty Sources

parameter 2parameter 1


measurand

Cause and effect diagram

First stage

16M. Rösslein,





parameter 4 parameter 3

measurand

1 level influence

2 level influence

3 level influence


further stages

17M. Rösslein,



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


Cause and effect diagramFinal stages

18M. Rösslein,



c o n te n t[m L ]

c o lo u r -c o d e

to le ra n c e[m L ]

1 b lu e 0 .0 0 7

5 w h ite 0 .0 1 5

1 0 re d 0 .0 2 0

2 5 b lu e 0 .0 3 0

5 0 re d 0 .0 5 0

1 0 0 ye llo w 0 .0 8 0

Example: Usual tolerances for some volumetric pipettes

waiting time 15s

Step 3 and 4: Quantifying the Uncertainty Components and

Conversion into Standard Uncertainty

19M. Rösslein,



Step 3 and 4: Quantification and Conversion

Triangular distribution

Standard uncertainty for a triangular distribution within the limits a- and a+

20M. Rösslein,




6

a)q(u

22

Triangular distributionCentre of the interval

Variance

With a+-a-

2

aaq

24

)aa()q(u

22

21M. Rösslein,




Rectangular distribution

Standard distribution for a rectangular distribution within the limits a- and a+

22M. Rösslein,




2

aaq

3

a)q(u

22

Rectangular distribution

Centre of interval

Variance

23M. Rösslein,



i2N

1i

2

i

2c xu

x

fyu

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

24M. Rösslein,




1. Rule: Addition and subtraction

y = p+q-r+...

2. Rule: multiplication and division

y = p q ...

...)r(u)q(u)p(ur,...q,p,yU 222c

....q

)q(u

p

)p(u

y

)y(U22

c

25M. Rösslein,



)r(u)p(u)n(u 22

Step 5: Calculation of the Combined Standard Sncertainty

rqpo

y

)p(u)o(u)z(u 22

Example:

Substitution: z = o + p

n = q + r

Calculation of the combined standard uncertainty for z and n according to rule 1:

26M. Rösslein,




Example:rq

poy

Summary of the intermediate results

according to rule 2:

222

2

22

22

c

rq

)r(u)q(u

po

)p(u)o(u

n

)n(u

z

)z(u

y

)y(u

27M. Rösslein,



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

28M. Rösslein,



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)

29M. Rösslein,



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

30M. Rösslein,



Example Step 1: Specification

KHPTit

KHPKHPNaOH FV

Pm1000c

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]

31M. Rösslein,



Cause and effect diagramFirst stage

Example Step 2: Sources of Uncertainty

F(KHP)V(Tit)

c(NaOH)

m(KHP) P(KHP)

32M. Rösslein,




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

33M. Rösslein,



Example Step 2:Sources of Uncertainty

m(KHP)P(KHP)

V(Tit)

c(NaOH)

F(KHP)

repeatability

linearity

calibration

temperature

endpoint

Biasrepeatability

repeatability

repeatability

calibration

Cause and effect diagram Validation of data

34M. Rösslein,





Final stage

m(KHP)P(KHP)

V(Tit)

c(NaOH)

F(KHP)

calibration

linearity

calibration

temperature

endpoint

Bias

repeatability

35M. Rösslein,



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

087.03

15.0mu KHP

36M. Rösslein,




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

mL02.06

05.0)V(u Cal

37M. Rösslein,





Expansion of the NaOH solution as aresult of temperature variation

Variation of temperature : + 4C Expansion coefficient of water:

2.110-4 C–1


mL009.06

4101.225)(Vu

4

Temp

38M. Rösslein,





Standard uncertainty

mL02.0009.002.0

vuvuvu

22

Temp2

Cal2

Tit

39M. Rösslein,



Example Steps 3 and 4:Quantification and Conversion

4KHP 107.5

6

0014.0Pu

Purity of the standard

• Declaration: 99.87% - 100.14%• Factor: 1.000 ± 0.0014


40M. Rösslein,




Molecular weight of KHP

sum formula: C8H5O4K

elementatomic weight

published tolerance

C 12.011 ± 0.001

H 1.00794 ± 0.0007

O 15.9994 ± 0.0003

K 39.0983 ± 0.0001

41M. Rösslein,




Molecular weight of KHPStandard uncertainty

Assumption: triangular distribution

mol/g2236.204FKHP

single element all elements

C8 96.088 0.00041 0.0033

H5 5.0397 0.00029 0.0015

O4 63.9976 0.00012 0.00048

K 39.0983 0.000041 0.000041

elementmolecular weight

standard uncertainty

42M. Rösslein,




0.0037

101.4108.4

105.1103.3)u(F

22

22

54

33

KHP

Molecular weight of KHP

Standard uncertainty of FKHP

43M. Rösslein,




Repeatability

The repeatability of the whole

procedure is

according to the validation data

0.1%

44M. Rösslein,



Example Step 5: Combined Standard Uncertainty

List of the calculated values:

Parameter Description Value Uncertainty

mKHP Initial weight KHP 0.511 g 8.7•10-5 g

VTitConsumption of

NaOH24.49 mL 0.022 mL

PKHP Purity KHP 1.0 5.7•10-4

FKHP Molecular mass 204.2236 0.0037 g/mol

Repeatability 1.0•10-3

45M. Rösslein,




Concentration of NaOH

1

KHPTit

KHPKHPNaOH

Lmol0.10217

204.223624.49

10.5111000

FV

Pm1000c

46M. Rösslein,




2

NaOH

R;NaOH

2

KHP

KHP

2

KHP

KHP

2

Tit

Tit

2

KHP

KHP

NaOH

NaOHc

c

)u(c

F

)u(F

P

)u(P

V

)u(V

m

)u(m

c

)(cu

47M. Rösslein,




Calculation

1-4

1-3

NaOH3

NaOHc

3

23

22

22

Lmol105.1

Lmol10217.0105.1

c105.1cu

105.1

100.1

2236.204

0037.0

0.1

00057.0

49.24

022.0

511.0

000087.0

48M. Rösslein,



Step 5: Combined Standard Uncertainty

Calculation

46.3

0.0

37.3

1.3

100

15.0

0 20 40 60 80 100

6

5

4

3

2

1

Ratio of the relative variances / %

1. Initial weight KHP

2. Consumption of NaOH solution

3. Purity of KHP

4. Relative molecular mass of KHP

5. Repeatability6. Combined standard uncertainty of the standardized

sodium hydroxide solution

M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W....

Documents

Transcript of M. Rösslein, B. Wampfler Evaluation of Uncertainty in Analytical Measurements B. Neidhart, W....