m. rösslein, b. wampfler evaluation of uncertainty in analytical measurements b. neidhart, w....
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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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Systematic Effects of a Measurement
5M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Outdated and Non Practical Splitting of Measurement Deviations
Measurementdeviation
Systematic error
Random error
Partlycorrected
ResultError
type A
Errortype B
6M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Example:Time Correlation of a Measured Quantity q
t
q
11M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Step 2: Identify Uncertainty Sources
parameter 2parameter 1
parameter 4parameter 3
measurand
Cause and effect diagram
First stage
16M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Step 2: Identify Uncertainty Sources
parameter 2parameter 1
parameter 4 parameter 3
measurand
1 level influence
2 level influence
3 level influence
Cause and effect diagram
further stages
17M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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 diagramFinal stages
18M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Step 3 and 4: Quantification and Conversion
Triangular distribution
Standard uncertainty for a triangular distribution within the limits a- and a+
20M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Step 3 and 4: Quantification and Conversion
6
a)q(u
22
Triangular distributionCentre of the interval
Variance
With a+-a-
2
aaq
24
)aa()q(u
22
21M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Step 3 and 4: Quantification and Conversion
Rectangular distribution
Standard distribution for a rectangular distribution within the limits a- and a+
22M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Step 3 and 4: Quantification and Conversion
2
aaq
3
a)q(u
22
Rectangular distribution
Centre of interval
Variance
23M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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 ...
...)r(u)q(u)p(ur,...q,p,yU 222c
....q
)q(u
p
)p(u
y
)y(U22
c
25M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
)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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Step 5: Calculation of the Combined Standard Uncertainty
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Cause and effect diagramFirst stage
Example Step 2: Sources of Uncertainty
F(KHP)V(Tit)
c(NaOH)
m(KHP) P(KHP)
32M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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
33M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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
35M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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
mL02.06
05.0)V(u Cal
37M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Example Steps 3 and 4: Quantification and Conversion
Consumption of NaOH solution
Expansion of the NaOH solution as aresult of temperature variation
Variation of temperature : + 4C Expansion coefficient of water:
2.110-4 C–1
Conversion to a standard deviation using a triangular distribution
mL009.06
4101.225)(Vu
4
Temp
38M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Example Steps 3 and 4: Quantification and Conversion
Consumption of NaOH solution
Standard uncertainty
mL02.0009.002.0
vuvuvu
22
Temp2
Cal2
Tit
39M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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
Conversion to a standard deviation using a triangular distribution
40M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Example Steps 3 and 4: Quantification and Conversion
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Example Steps 3 and 4:Quantification and Conversion
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Example Steps 3 and 4:Quantification and Conversion
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Example Steps 3 and 4:Quantification and Conversion
Repeatability
The repeatability of the whole
procedure is
according to the validation data
0.1%
44M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Example Step 5: Combined Standard Uncertainty
Concentration of NaOH
1
KHPTit
KHPKHPNaOH
Lmol0.10217
204.223624.49
10.5111000
FV
Pm1000c
46M. Rösslein,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Example Step 5: Combined Standard Uncertainty
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
Example Step 5: Combined Standard Uncertainty
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,
B. WampflerEvaluation of Uncertainty in Analytical Measurements
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
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