CHEM 137.1 Measurement uncertainty

Introduction to
Estimation of
Measurement Uncertainty

1 - JCGM (2012) International vocabulary of metrology — Basic and general concepts and associated terms (VIM) JCGM
200:2012
Definitions
Uncertainty : non-negative parameter characterizing the dispersion
of the quantity values being attributed to a measurand, based on
the information used1
Measurand: quantity intended to be measured1
Error: measured quantity value minus a reference quantity
value1

Water
Essential in sustaining life
Presence of minerals
Presence of toxic metals
HEAVY METALS
(Pb, As, Hg, Cd, etc.)
Therefore, it is important to determine their levels,
specifically if the water is used for human
consumption.
How much is present? Is the level still safe for drinking?

Pb2+
Pb2+
Pb2+
Pb2+
Measurand : Lead content in water (ug/L)
5
6
7
CPb (ug/L)
C true = 5.7 ug/L
C measured = 6.1 ug/L
ERROR = Δ = Cmeasured - Ctrue

 The TRUE value will always remain
to be unknown.
 It is difficult to achieve the TRUE
value.
 Procedure should be optimized in
such a way that the result will be
close to the true value.
 Error cannot be used to
characterize the quality of the
MEASURAND.

TRUE VALUE lies within this RANGE
with HIGH PROBABILITY.
Uncertainty Range
Uncertainty = U = ½ uncertainty range
Result:
CPb = (6.1 + 0.5) ug/L
Instead of giving the TRUE value, what is
being reported is a range in which the
true value lie with high probability
The concentration of Lead in the water
sample is between 5.6 ….. 6.6 ug/L
IS THE WATER SAFE FOR HUMAN
CONSUMPTION?

Importance of MEASEUREMENT UNCERTAINTY
Analysis of Pb in water sample
2 Laboratories participated in the analysis
9
10
11
CPb (ug/L)
LABORATORY A
LABORATORY B
C A = 9.2 ug/L
C B = 9.8 ug/L
Why is measurement uncertainty
important?
 It is an important quality characteristic
of the measurement result.
 It is necessary in comparing results

2 Types of uncertainty estimates
A Type
- uncertainty estimate is obtained from
statistical treatment of repeated measurement
results
- Treatment is usually a calculation of the
standard deviation

2 Types of uncertainty estimates
B Type
- All uncertainty estimates that are obtained
without the use of repeated measurements
- Concentration of standard solutions
- Data from the documentation of instruments
- Educated guesses, expert opinions

A and B Type uncertainty estimates:
Relation to random and systematic errors
Effect Type A estimation Type B estimation
Random The usual way
Possible if the data on the
magnitude of the effect
are available without
performing repeated
measurements
Systematic
Only possible if the effect
will change into a random
effect in the long term
The usual way

QUANTIFYING UNCERTAINTY COMPONENTS
Uncertainty Components
Standard uncertainties
Must be converted to

RECTANGULAR DISTRUBUTION
TRIANGULAR DISTRIBUTION

OTHER DISTRUBUTION FUNCTIONS
The Student Distribution vs Normal Distribution

PIPET
Use to deliver exact VOLUME
Of liquid
V
CALIBRATION
TEMPERATURE
REPEATABILITY
 Calibration
 Repeatability
 Temperature

Standard uncertainties
When expressed as standard deviations
u(V, REP)  standard uncertainty of the volume due to
REPEATABILITY
u(V, CAL)  standard uncertainty of the volume due to
CALIBRATION
u(V, TEMP)  standard uncertainty of the volume due to
TEMPERATURE
 Calibration
 Repeatability
 Temperature

u(V, TEMP) in detail
u(V, TEMP) = V x ΔT x γ
√3
Where:
V = volume
ΔT= max. possible deviation from 20oC = +4oC
γ = thermal coefficient of water = 0.00021/oC
Note : √3 is used to convert the rectangular distribution to normal distribution

THE COMBINED STANDARD UNCERTAINTY, uc(V)
uc(V) = 𝑢 𝑉, 𝑅𝐸𝑃 2 + 𝑢 𝑉, 𝐶𝐴𝐿 2 + 𝑢 𝑉, 𝑇𝐸𝑀𝑃 2
For volume measurement using a pipet

Different ways of calculating the
COMBINED STANDARD UNCERTAINTY, uc(Y)
OUTPUT QUANTITY (Y)
INPUT QUANTITIES
 Pb content in water
 N content of plant sample, etc
measured indirectly by the
 Volume of sample
 Mass of sample
 Absorbance, etc.
determine standard uncertainty of each input
quantity to obtain the
Standard Uncertainty of the OUTPUT QUANTITY
COMBINED STANDARD UNCERTAINTY
uc(Y)
OUTPUT QUANTITY (Y)
measured indirectly by the

EXAMPLE : Titration
INPUT QUANTITIES
 Volume of sample solution
 Volume of titrant used
 Titrant concentration
OUTPUT QUANTITY (Y)  Concentration of analyte, Cs
FOR a 1:1 stoichiometry
Cs = VT x CT
Vs
MEASUREMENT MODEL

uc(Y) = 𝑢 𝑋1
2 + 𝑢 𝑋2
2 + 𝑢 𝑋 𝑛
2
If the MODEL is :
Y = X1 – X2 + ……..+ Xn
then:

If the MODEL is :
Y = X1 .X2
X3 . X4
then:
uc(Y) =Y . (
𝑢 𝑥1
𝑋1
)2+ (
𝑢 𝑥2
𝑋2
)2 +(
𝑢 𝑥3
𝑋3
)2+(
𝑢 𝑥4
𝑋4
)2
uc(Y) = ( (
𝑢 𝑥1
𝑋1
)2+ (
𝑢 𝑥2
𝑋2
)2 +(
𝑢 𝑥3
𝑋3
)2+(
𝑢 𝑥4
𝑋4
)2)
Y
RELATIVE
STANDARD
UNCERTAINTY
of the OUTPUT
QUANTITY

If the MODEL is :
Y = f (X1, X2,…, Xn)
then:
uc(Y) =(√[
𝜕𝑌
𝜕𝑋1
u(X1)]2 + [
𝜕𝑌
𝜕𝑋2
u(X2)]2 +…+ [
𝜕𝑌
𝜕𝑋𝑛
u(Xn)]2
Uncertainty
component of
the respective
input quantity
Partial
derivative of
output quantity
w/ respect to the
input quantity

Comparing uncertainty components
RESULT FROM PIPETTING
uc(V) = 0.006 2 + 0.017 2 + 0.005
uc(V) = 0.019 mL
highest contributor to
uncertainty
Vol. delivered = (10.000-0.019 … 10.000 + 0.019) mL

THE EXPANDED UNCERTAINTY, U
U(Y) = uc(Y) x k
Coverage factor
95%: k = 2
99%: k = 3
Combined standard uncertainty

Presenting MEASUREMENT RESULT
Volume of liquid delivered (V)
V = (10.000 + 0.038) mL, k = 2, norm

Measurement Procedure
HOMOGENIZATION
EXTRACTION
PURIFICATION
CONCENTRATION
ANALYSIS

Uncertainty Sources
in chemical analysis
Sampling Reduction
Homogenization
Extraction
Purification
Concentration
Instrumental Analysis
Data Treatment Result
Calibration with
analyte
Non-representativeness
SAMPLE PREPARATION

Reduction
Homogenization
Extraction
Purification
Concentration
Result
SAMPLE PREPARATION
DECOMPOSITION
LOSSES DURING PURIFICATION
INCOMPLETE EXTRACTION
VOLATILIZATION
ADSORPTION
CONTAMINATION
Uncertainty Sources

Sampling
Data Treatment
INSTRUMENTAL SOURCES
(Noise, Drift, etc.)
INCOMPLETE SELECTIVITY
Ex. Integration of peaks
Calibration with
analyte
INSTRUMENTAL SOURCES
CONCENTRATION/PURITY OF
CALIBRANTS
Uncertainty Sources

Sampling Reduction
Homogenization
Extraction
Purification
Concentration
Calibration with
analyte
SAMPLE PREPARATION
WEIGHING
VOLUMETRIC
UNCERTAINTY
Uncertainty Sources

Examples of Measurement
Uncertainty Evaluation

EXAMPLES OF MU EVALUATION
 evaluation of the uncertainty associated with a
volumetric operation
 evaluation of the uncertainty associated with a
weighing operation
 evaluation of the uncertainty associated with
instrumental quantification

Examples of MU EVALUATION
Evaluation of the uncertainty associated with a
 Single volumetric operation
Possible sources of uncertainty:
1. Uncertainty associated with the calibration
of the equipment, u(V,CAL)
- quantified considering the reported tolerance
(rectangular distribution)
2. Uncertainty associated with the
repeatability of the manipulation of the
equipment, u(V,REP)
- repeatability of a gravimetric test performed with
water in an adequate balance (‘deliver and weigh’)

 Single volumetric operation- possible
sources of uncertainty:
3. Uncertainty associated with the
temperature effect, u(V, temp)
Combination of the sources of uncertainty:
uc(V) = 𝑢 𝑉, 𝑅𝐸𝑃 2 + 𝑢 𝑉, 𝐶𝐴𝐿 2 + 𝑢 𝑉, 𝑇𝐸𝑀𝑃 2
U(V, TEMP) = V x ΔT x γ
√6
Where:
V = volume
ΔT= max. possible deviation from 20oC = +4oC
γ = thermal coefficient of water = 0.00021/oC

 Volumetric dilution: dilution of Vi to Vf; Df = Vf/Vi
The temperature effect is cancelled
𝑢(𝐷𝑓)
𝐷 𝑓
= (
𝑢(𝐷𝑓, 𝑉𝑖)
𝑉𝑖
)2 + (
𝑢(𝐷𝑓, 𝑉𝑓)
𝑉𝑓
)2
𝑢(𝐷𝑓)
𝐷 𝑓
=
𝑢 𝑉𝑖, 𝐶𝐴𝐿 2 + 𝑢 𝑉𝑖, 𝑅𝐸𝑃 2
𝑉𝑖
2
+
𝑢 𝑉 𝑓, 𝐶𝐴𝐿 2 + 𝑢 𝑉 𝑓, 𝑅𝐸𝑃 2
𝑉 𝑓
2

weighing operation
 Single weighing operation
2 sources of uncertainty:
1. Repeatability of operation, u(m,REP)
- estimated by the standard deviation
of successive weighing operations
2. Balance calibration, u(m,CAL)
- uncertainty associated with the
calibration function (“sensitivity” and
“linearity” of the balance response;
available at the calibration certificate)
𝑢(𝑚) = 𝑢 𝑚, 𝑅𝐸𝑃 2 + 𝑢(𝑚, 𝐶𝐴𝐿)2

weighing operation
 Weighing by difference
m = m(gross) - m(tare)
The repeatability and linearity of the balance
affects this operation twice and the sensitivity
components is cancelled.
Sensitivity can be neglected
because weighing was done on
the same balance over a very
narrow range.
𝑢(𝑚) = 2[𝑢 𝑚, 𝑅𝐸𝑃 2] + 2[𝑢(𝑚, 𝐶𝐴𝐿)2]

 The uncertainty quoted in the calibration certificate of the
balance can be used
(the uncertainty in the calibration data has already taken into
account all the possible contributions to the uncertainty)
Evaluation of the uncertainty associated with
a weighing operation

2 independent sources of uncertainty:
1. interpolation of the sample signal in the calibration curve
following the regression model, uinter
2. preparation of the calibration standards, ustd
(usually negligible)
Evaluation of the uncertainty associated with instrument
quantification

The standard uncertainty, uinter , of the interpolation (LSM
regression model):
sy= standard deviation of y
m= slope
k= no. of replicate measurements of the unknown
n= no. of data points for the calibration line
= mean value of y for the points on the calibration line
xi = individual values of x for the points on the calibration line
= mean value of x for the points on the calibration line

quantification
42
 uncertainty associated with the preparation of calibration
standards
- relative standard uncertainty associated with the preparation of
calibration standards, (ustd/Cstd): estimated in excess by the
relative standard uncertainty associated with the content of the
calibration standard with the lowest concentration (larger
standard uncertainty)
- This uncertainty results from the combination of the uncertainty
associated with the weighing and dilutions of the reference
substance

quantification
43
Combination of uinter with ustd:
cinter = interpolated sample content = Q
relative standard uncertainty
c

Example: Evaluation of uncertainty associated with an instrumental
measurement
1. Description of the analytical method
Determination of ammonium in drinking water (Phenate method): An
intensely blue compound, indophenol, is formed by the reaction of
ammonia, hypochlorite, and phenol catalyzed by sodium nitroprusside.
The blue compound formed is quantified by UV-Vis at 640 nm.
44
Sample
aliquot
Dilution and formation of
indophenol
Spectrometric
quantification
Preparation of
calibration
standards
Calibration of the
spectrometer

Determination of ammonium in drinking water (Phenate method)
45
Sample
aliquot
Dilution and formation of
indophenol
Spectrometric
quantification
Preparation of
calibration
standards
Calibration of the
spectrometer
V1 V2
Q
measurement

2. Evaluation of uncertainty components
[‘Bottom-up’ Approach]
-Involves the quantification of all individual sources of uncertainty
responsible for the occurrence of random and systematic errors on
the measurement (…)
2.1 Definition of the measurand
-ammonium nitrogen in a 1L sample of drinking water
46
measurement

2.2 Identification of the sources of uncertainty
SAMPLING
- Sample non-representativeness
SAMPLE PREPARATION
- Inhomogeneity
- Separation of analyte is incomplete
- Analyte adsorbs
- Analyte or photometric complex decomposes
- Analyte volatilizes
- Incomplete reaction
- contamination
47
measurement
The result is expressed for
sample, sampling is not included
The sample is homogenous, the
analyte is not separated and does
not adsorb
Analyte or the photometric
complex can decompose or get
contaminated: Cdc

Preparation and dilution of solutions
Weighing
Calibration of instrument
- standard substance purity
- Solution preparation
Measurement of sample
- Interferences
- Repeatability of reading
- Drift of reading
- Memory effects 48
measurement
Absent in our sample

49
measurement

50
* If ammonium content in the diluted sample, Q, is 51.2 µg NH3-N /L,
and the water sample was diluted from 10 mL to 50 mL:
measurement
𝐶 𝑁𝐻3
−𝑁 =
𝑄 𝑥 𝑉2
𝑉1
=
51.2 𝑥 50.00
10.00
= 256 ug 𝑁𝐻3 − 𝑁/L

2.3 Quantification of the sources of uncertainty
51
measurement
Assumption:
umatrix ≈ 0
c

Given:
52
measurement
vol. pipet tolerance (mL) 0.03
vol. pipet repeatability (mL) 0.005
vol. flask tolearance (mL) 0.05
vol. flask repeatability (mL) 0.054
concentration of diluted sample (µg/L) 51.2
uncertainty in concentration determination due
to interpolation (LR)
0.978
The relative standard
uncertainty of the output
quantity
c

53
measurement
negligible
c

54
measurement

55
measurement
negligible
Given:
uinter = 0.978
Directly substitute this value in the
Relative standard uncertainty equation.
c

56
measurement
for samples with matrices
equivalent to the calibration
standards
- not applicable for waters with complex matrices

2.4 combination of the sources of uncertainty
57
measurement
measurement
c
c

measurement
𝑢 𝑐 𝐶 𝑁𝐻3
= 𝐶𝑁𝐻3
𝑥 0.01918
= 256 𝑥 0.01918 = 4.91008 𝑢𝑔/𝐿
Convert to combined standard uncertainty
c

measurement
= 𝐶𝑁𝐻3
𝑥 0.01918
= 256 𝑥 0.01918 = 4.91008 𝑢𝑔/𝐿
Convert to EXPANDED UNCERTAINTY, U
(95% K=2, 99% K=1)
𝑈 𝐶 𝑁𝐻3
= 4.91008
𝑢𝑔
𝐿
𝑥 2 = 9.82016 𝑢𝑔/𝐿

• 2.4 Expression of results with uncertainty
measurement
CNH3-N = (256 + 10) ug NH3-N/L (norm, K= 2)
𝐶 𝑁𝐻3
−𝑁 =
𝑄 𝑥 𝑉2
𝑉1
=
51.2 𝑥 50.00
10.00
= 256 ug 𝑁𝐻3 − 𝑁/L
𝑈 𝐶 𝑁𝐻3
= 9.82016 𝑢𝑔/𝐿
𝐶 𝑁𝐻3
−𝑁 =
51.2 𝑥 50.00
10.00
= 256 ug 𝑁𝐻3 − 𝑁/L
3SF’s 4 SF’s
4 SF’s
Report final answer to 3 SF’s. Measurement
uncertainty should have the same number of
decimal places as your Output value.

2.6 Examine the uncertainty budget
61
measurement
The percent contribution, p (%), of each of the uncertainty
components is estimated by:
=
0.0132
10
2
0.019182
𝑥 100 = 0.47%
=
0.0191 2
0.019182
𝑥 100 = 99.17%
=
0.0577
50
2
0.019182
𝑥 100 = 0.36%
V1 (0.47%)
V2 (0.36%)
Q (99.2%)

CHEM 137.1 Measurement uncertainty

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CHEM 137.1 Measurement uncertainty