M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
Mean Value and Standard
Deviation of a Random Sample
frequency
s qk
q
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
Parameters of a Normal
Distribution
Arithmetic mean value:
1 n
q   qk
n k 1
Experimental variance:
1 n
2
v qk  
qk  q

n  1 k 1
Experimental standard deviation;
sqk   vqk 
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
Variance of the Mean q
of a Random Sample
2
s
(q)
2
V(q)  s (q) 
n
sq 
distribution of
the mean
sqk 
distribution of the
single values of
one individual determination of q
mean
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3
M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
Systematic Effects of a
Measurement
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
5
Outdated and Non Practical Splitting
of Measurement Deviations
Measurement
deviation
Random
error
Systematic
error
Partly
corrected
Result
Error
type B
Error
type A
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
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
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
7
Modern and Practical Way of Dealing
With Measurement Uncertainty
measurement
deviation
measurement
systematic
deviation
random
deviation
known systematic
deviation
correction
measurement
value
remaining
deviation
unknown
systematic
deviation
measurement
uncertainty
measurement result
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000
M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
Concept Based on Observed
Quantities
Uncorrected mean value
of observations
Corrected mean value
of observations
1. C orrection
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
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
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.
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
10
Example:Time Correlation of a
Measured Quantity q
q
t
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
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
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
12
Uncertainty of the Experimental
Standard Uncertainty
Numbers of
measurements n
2
Uncertainty of the
Uncertainty / %
76
3
4
52
42
5
10
36
24
20
30
16
13
50
10
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
13
Calculation of the
Measurement Uncertainty
Specification
Identify Uncertainty
Sources
Quantify Uncertainty
Components
Convert to Standard
Deviations
Calculate Combined
Uncertainty
Re-evaluate
Yes
Re-evaluating?
End
No
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
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
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
Step 2:
Identify Uncertainty Sources
Cause and effect diagram
First stage
parameter 1 parameter 2
measurand
parameter 3 parameter 4
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
Step 2:
Identify Uncertainty Sources
Cause and effect diagram
further stages
parameter 1
parameter 2
2 level influence
3 level influence
1 level influence
measurand
parameter 3
parameter 4
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
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
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
18
Step 3 and 4: Quantifying the
Uncertainty Components and
Conversion into Standard Uncertainty
Example: Usual tolerances for some
volumetric pipettes
content
[mL]
colour code
tolerance
[mL]
1
blue
0.007
5
white
0.015
10
red
0.020
25
blue
0.030
50
red
0.050
100
yellow
0.080
waiting time 15s
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
19
Step 3 and 4:
Quantification and Conversion
Triangular distribution
Standard uncertainty for a triangular
distribution within the limits a- and a+
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
Step 3 and 4:
Quantification and Conversion
Triangular distribution
Centre of the interval
a  a
q
2
Variance
2
(
a

a
)
u2(q)   
24
With a+-aa2
u (q) 
6
2
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
Step 3 and 4:
Quantification and Conversion
Rectangular distribution
Standard distribution for a rectangular
distribution within the limits a- and a+
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
Step 3 and 4:
Quantification and Conversion
Rectangular distribution
Centre of interval
a a
q
2
Variance
2
a
u2 (q) 
3
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
23
Step 5: Calculation of the
Combined Standard Uncertainty
2
 f  2
 u x i 
u y    
i1  x i 
2
c
N
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
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
24
Step 5: Calculation of the
Combined Standard Uncertainty
1. Rule: Addition and subtraction
y = p+q-r+...
Uc yp, q, r,...

u2 (p)  u2 (q)  u2 (r )  ...
2. Rule: multiplication and division
y = p q ...
2
2
 u(p)   u(q) 
Uc ( y )
 
  
  ....
y
 p   q 
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
25
Step 5: Calculation of the
Combined Standard Sncertainty
Example:
op
y
qr
Substitution: z = o + p
n=q+r
Calculation of the combined
standard uncertainty for z and n
according to rule 1:
u (z)  u2 (o)  u2 (p)
u (n)  u2 (p)  u2 (r )
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
26
Step 5: Calculation of the
Combined Standard Uncertainty
Example:
op
y
qr
Summary of the intermediate results
according to rule 2:
uc ( y )

y

2
 u( z) 
 u(n) 





z
n




u2 (o)  u2 (p)
o  p 
2

2
u2 (q)  u2 (r )
q  r 2
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
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
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27
M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
Example
A solution of sodium hydroxide (NaOH) is
standardized against the titrimetric standard potassium hydrogen phthalate (KHP)
What is its value of the uncertainty?
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
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
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
30
Example Step 1:
Specification
cNaOH
1000  m KHP  PKHP

VTit  FKHP
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]
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
Example Step 2:
Sources of Uncertainty
Cause and effect diagram
First stage
P(KHP)
m(KHP)
c(NaOH)
V(Tit)
F(KHP)
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
32
Example Step 2:
Sources of Uncertainty
Cause and effect diagram
further stages
P(KHP)
m(KHP)
linearity intercept
calibration
repeatability
c(NaOH)
calibration
repeatability
temperature
endpoint
repeatability
bias
V(Tit)
F(KHP)
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
33
Example Step 2:
Sources of Uncertainty
Cause and effect diagram
Validation of data
repeatability
P(KHP)
m(KHP)
linearity
calibration
repeatability
c(NaOH)
calibration
repeatability
temperature
endpoint
V(Tit)
repeatability
F(KHP)
Bias
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
34
Example Step 2:
Sources of Uncertainty
Cause and effect diagram
Final stage
repeatability
P(KHP)
m(KHP)
linearity
calibration
c(NaOH)
calibration
temperature
endpoint
Bias
V(Tit)
F(KHP)
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
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
umKHP  
0.15
3
 0.087
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
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
u ( VCal ) 
0.05
6
 0.02 mL
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
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
u (VTemp ) 
25  2.1 10 4  4
6
 0.009 mL
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
Example Steps 3 and 4:
Quantification and Conversion
Consumption of NaOH solution
Standard uncertainty
uv Tit   u v Cal   u v Temp 
2
2
 0.02  0.009  0.02mL
2
2
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
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
u PKHP 
0.0014
6
 5.7104
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
Example Steps 3 and 4:
Quantification and Conversion
Molecular weight of KHP
sum formula: C8H5O4K
element
atomic
weight
published
tolerance
C
12.011
± 0.001
H
1.00794
± 0.0007
O
15.9994
± 0.0003
K
39.0983
± 0.0001
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40
M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
Example Steps 3 and 4:
Quantification and
Conversion
Molecular weight of KHP
Standard uncertainty
Assumption: triangular distribution
element
C8
molecular
weight
96.088
standard uncertainty
single
element
all elements
0.00041
0.0033
H5
5.0397
0.00029
0.0015
O4
K
63.9976
39.0983
0.00012
0.000041
0.00048
0.000041
FKHP  204.2236 g / mol
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
42
Example Steps 3 and 4:
Quantification and
Conversion
Molecular weight of KHP
Standard uncertainty of FKHP
u(FKHP ) 
3.3  10   1.5  10 
4.8  10   4.1 10 
3
2
3
2
4
2
5
2

 0.0037
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
Example Steps 3 and 4:
Quantification and
Conversion
Repeatability
The repeatability of the whole
procedure is
0.1%
according to the validation data
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
44
Example Step 5:
Combined Standard Uncertainty
List of the calculated values:
Parameter
Description
Value
Uncertainty
m KHP
Initial weight KHP
0.511 g
8.7•10-5 g
VTit
Consumption of
NaOH
24.49 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
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
45
Example Step 5:
Combined Standard Uncertainty
Concentration of NaOH
c NaOH 
1000  mKHP  PKHP
1000  0.511  1

VTit  FKHP
24.49  204.2236
 0.10217 mol  L1
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M. Rösslein,
B. Wampfler
Evaluation of Uncertainty in Analytical
Measurements
46
Example Step 5:
Combined Standard Uncertainty
2
 u(m KHP )   u(VTit ) 


  
 mKHP   VTit 
2
2
 u(PKHP )   u(FKHP ) 
uc (c NaOH )
  

  
c NaOH
 PKHP   FKHP 
 u(c NaOH ; R ) 

 
 c NaOH 
2
2
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
47
Example Step 5:
Combined Standard Uncertainty
Calculation
2
 0.000087   0.022 

 

 0.511   24.49 
2
2
 0.00057   0.0037 
 
 

 1.0   204 .2236 
 1.0  10 3 2

2

 1.5  10 3
uc c NaOH   1.5  10 3  c NaOH
 1.5  10 3  0.10217 mol  L-1
 1.5  10  4 mol  L-1
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M. Rösslein,
Evaluation of Uncertainty in Analytical
Measurements
B. Wampfler
48
Step 5:
Combined Standard Uncertainty
Calculation
1.3
1
2
37.3
3
15.0
4 0.0
5
46.3
6
100
0
20
40
60
80
100
Ratio of the relative variances / %
1.
2.
3.
4.
5.
6.
Initial weight KHP
Consumption of NaOH solution
Purity of KHP
Relative molecular mass of KHP
Repeatability
Combined standard uncertainty of the standardized
sodium hydroxide solution
B. Neidhart, W. Wegscheider (Eds.): Quality in Chemical Measurements © Springer-Verlag Berlin Heidelberg 2000