Download UNCERTAINTY OF MEASUREMENT

The Expression of
Uncertainty in Measurement
Bunjob Suktat
JICA Uncertainty Workshop
January 16-17, 2013
Bangkok, Thailand
Acceptance of the Measurement Results
Contents
•
•
•
•
•
•
•
Introduction
GUM Basic Concepts
Basic Statistics
Evaluation of Measurement Uncertainty
How is Measurement Uncertainty estimated?
Reporting Result
Conclusions and Remarks
Introduction
• Guide to the Expression of Uncertainty in
Measurement was published by the
International Organization for
Standardization in 1993 in the name of 7
international organizations
• Corrected and reprinted in 1995
• Usually referred to simply as the “GUM”
Guide to the Expression of Uncertainty
in Measurement (1993)
International Organisations
BIPM
- International Bureau of
Weights and Measures
http//: www.bipm.org
IEC
- International Electrotechnical
Commision
http//: www.iec.ch
IFCC
- International Federation of
Clinical Chemistry
http//: www.ifcc.org
IUPAP - International Union of
Pure and Applied Physics
http//: www.iupap.org
IUPAC - International Union of
Pure and Applied Chemistry
http//: www.iupac.org
ISO
- International Organisation for
Standardisation
http//: www.iso.ch
OIML
- International Organisation
for legal metrology
http//: www.oiml.org
Basic concepts
 Every measurement is subject to some
uncertainty.
 A measurement result is incomplete
without a statement of the uncertainty.
 When you know the uncertainty in a
measurement, then you can judge its
fitness for purpose.
 Understanding measurement uncertainty
is the first step to reducing it
Introduction to GUM
• When reporting the result of a measurement of a
physical quantity, it is obligatory that some
quantitative indication of the quality of the result
be given so that those who use it can assess its
reliability.
• Without such an indication, measurement results
can not be compared, either among themselves
or with reference values given in the
specification or standard.
GUM 0.1
Stated Purposes
• Promote full information on how
uncertainty statements are arrived at
• Provide a basis for the international
comparison of measurement results
Benefits
• Much flexibility in the guidance
• Provides a conceptual framework for
evaluating and expressing uncertainty
• Promotes the use of standard terminology
and notation
• All of us can speak and write the same
language when we discuss uncertainty
Uses of MU
•
•
•
•
QC & QA in production
Law enforcement and regulations
Basic and applied research
Calibration to achieve traceability to national
standards
• Developing, maintaining, and comparing
international and national reference standards and
reference materials
–GUM 1.1
Are these results different?
value
12.5
After uncertainty
evaluation
No uncertainty
evaluation
(only precision)
mg kg-1
12.0
11.5
11.0
10.5
R1
R2
R1
R2
R1
R2
En-score according to GUM
En 
xlab  xref
(ulab  uref )
2
2
“Normalized” versus ...
propagated combined uncertainties
Performance evaluation:
0 <|En|< 2 : good
2 <|En|< 3 : warning  preventive action
|En|> 3 : unsatisfactory  corrective
action
What is Measurement?
Measurement is
‘Set of operations having the object of
determining a value of a quantity.’
( VIM 2.1 )
Note: The operations may be performed
automatically.
Basic concepts
• Measurement
– the objective of a measurement is to determine
the value of the measurand, that is, the value of
the particular quantity to be measured
• a measurement therefore begins with
– an appropriate specification of the measurand
– the method of measurement and
– the measurement procedure
GUM 3.1.1
Principles of Measurement
DUT
Method
of
Comparison
Standard
Result
Basic concepts
• Result of a measurement
– is only an estimate of a true value and only
complete when accompanied by a statement
of uncertainty.
GUM 3.1.2
– is determined on the basis of series of
observations obtained under repeatability
conditions
GUM 3.1.4
• Variations in repeated observations are
assumed to arise because influence quantities
Gum 3.1.5
Influence quantity
• Quantity that is not the measurand but that
affects the result of measurement.
Example : temperature of a micrometer used to
measure length.
( VIM 2.7 )
What is Measurement Uncertainty?
• “parameter, associated with the result of a
measurement, that characterizes the dispersion of
the values that could reasonably be attributed to the
measurand” – GUM, VIM
• Examples:
– A standard deviation (1 sigma) or a multiple of it
(e.g., 2 or 3 sigma)
– The half-width of an interval having a stated level
of confidence
Uncertainty
• The uncertainty gives the limits of the range in
which the “true” value of the measurand is
estimated to be at a given probability..
Measurement result = Estimate ± uncertainty
(22.7 ± 0.5) mg/kg
The value is between 22.2 mg/kg and 23.2 mg/kg
Measurement Error
Measurement Error
Real Number System
Measured Value
True Value
Measured values are inexact observations of a
true value.
The difference between a measured value and
a true value is known as the measurement
error or observation error.
Basic concepts
• The error in a measurement
– Measured value – True value.
– This is not known because:
GUM 2.2.4
• The true value for the measurand
– This is not known
– The result is only an estimate of a true value and only
complete when accompanied by a statement of
uncertainty.
GUM 3.2.1
Random & Systematic Errors
• Error can be decomposed into random
and systematic parts
• The random error varies when a
measurement is repeated under the same
conditions
• The systematic error remains fixed when
the measurement is repeated under the
same conditions
Random error
• Result of a measurement minus the mean
result of a large number of repeated
measurement of the same measurand.
( VIM 3.13 )
Random Errors
• Random errors result from the fluctuations in
observations
• Random errors may be positive or negative
• The average bias approaches 0 as more
measurements are taken
Random error
• Presumably arises from unpredictable
temporal and spatial variations
• gives rise to variations in repeated
observations
• Cannot be eliminated, only reduced.
GUM 3.2.2
Systematic Errors
Mean result of a large number of
repeated measurements of the same
measurand minus a true value of the
measurand.
( VIM 3.14 )
Systematic Errors
• A systematic error is a consistent deviation in
a measurement
• A systematic error is also called a bias or an
offset
• Systematic errors have the same sign and
magnitude when repeated measurements are
made under the same conditions
• Statistical analysis is generally not useful, but
rather corrections must be made based on
experimental conditions.
Systematic error
• If a systematic error arises from a recognized
effect of an influence quantity
– the effect can be quantified
– can not be eliminated, only reduced.
– if significant in size relative to required
accuracy, a correction or correction factor
can be applied to compensate
– then it is assumed that systematic error is
zero.
GUM 3.2.3
Basic concepts
Systematic error
• It is assumed that the result of a measurement
has been corrected for all recognised significant
systematic effects
GUM 3.2.4
Measurement Error
Systematic
error
Random error
Correcting for Systematic Error
• If you know that a substantial systematic error exists and
you can estimate its value, include a correction
(additive) or correction factor (multiplicative) in the
model to account for it
•
Correction - Value that , added algebraically to the
uncorrected result of a measurement , compensates for
an assumed systematic error
(VIM 3.15)
• Correction Factor - numerical factor by which the
uncorrected result of a measurement is multiplied to
compensate for systematic error.
[VIM 3.16]
Uncertainty
•
The result of a measurement after
correction for recognized systematic
effects is still only an estimate of the
value of the measurand because of the
uncertainty arising;
– from random effects and
– from imperfect correction of the result for
systematic effects
GUM 3.3.1
Classification of effects and uncertainties
• Random effects
•
•
•
•
Unpredictable variations of influence quantities
Lead to variations in repeated measurements
Expected value : 0
Can be reduced by making many measurement
• Systematic effects
•
•
•
•
Recognized variations of influence quantities
Lead to BIAS in repeated measurements
Expected value : unknown
Can be reduced by applying a correction which
carries an uncertainty
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Error versus uncertainty
• It is important not to confuse the terms error and
uncertainty
• Error is the difference between the measured
value and the “true value” of the thing being
measured
• Uncertainty is a quantification of the doubt about
the measurement result
• In principle errors can be known and corrected
• But any error whose value we do not know is a
source of uncertainty.
Blunders
• Blunders in recording or analysing data
can introduce a significant unknown
error in the result of a measurement.
• Measures of uncertainty are not
intended to account for such mistakes
GUM 3.4.7
Basic Statistics
Population and Sample
• Parent Population
The set of all possible
measurements.
• Sample
Samples
Handful of
marbles
from the
bag
A subset of the population measurements actually
made.
Population
Bag of Marbles
Slide 7
Histograms
• When making many
measurements, there is often
variation between readings.
Histogram plots give a visual
interpretation of all
measurements at once.
• The x-axis displays a given
measurement and the height of
each bar gives the number of
measurements within the given
region.
• Histograms indicate the
variability of the data and are
useful for determining if a
measurement falls outside of
“specification”.
For a large number of experiment replicates the results
approach an ideal smooth curve called the GAUSSIAN or
NORMAL DISTRIBUTION CURVE
Characterised by:
The mean value – x
gives the center of the
distribution
The standard
deviation – s
measures the width
of the distribution
Average
• The most basic statistical tool to analyze a series of
measurements is the average or mean value :
“Sum of”
Individual measurement
x

x
i
n
The average of the three
values 10, 15and 12.5 is
given by:
Number of measurements
10  15  12.5
x
 12.5
3
Deviation
Deviation = individual value – avg value
di  xi  x
Need to calculate an average or “standard” deviation
To eliminate the possibility of a zero deviation, we square di
Standard Deviation
• The average amount that each measurement
deviates from the average is called standard
deviation (s) and is calculated for a small number
of measurements as:
s
 (x  x)
i
n 1
2
Sum of deviation squared
xi = each measurement
x = average
n = number of measurements
Note this is called root mean square: square root of the mean of the squares
Standard Deviation
Standard Deviation
For example, calculate the standard deviation of the
following measurements: 10, 15 and 12.5 (avg = 12.5)
s
 (x  x)
i
n 1
2

10  12.52  15  12.52  12.5  12.52
n 1
(2.5) 2  (2.5) 2
12.5


 2.5
2
2
The values deviate on average plus or minus 2.5 :12.5 ± 2.5
10.0
12.5
15.0
Other ways of expressing the precision of the data:
• Variance
Variance = s2
• Relative standard deviation
s
RSD 
x
• Percent RSD or Coefficient of Variation (CV)
s
% RSD   100
x
Standard Deviation of the Mean
The uncertainty in the best measurement is given
by the standard deviation of the mean (SDOM)
s

n
Gaussian Distribution
• Given a set of repeated measurements which
have random error.
• For the set of measurements there is a mean
value.
• If the deviation from the mean for all the
measurements follows a Gaussian probability
distribution, they will form a “bell-curve”
centered on the mean value.
• Sets of data which follow this distribution are
said to have a normal (statistical) distribution of
random data.
POPULATION DATA
For an infinite set of data,
n→∞
x → µ
and
population mean
s→σ
population std. dev.
The experiment that produces a small
standard deviation is more precise .
Remember, greater precision does not
imply greater accuracy.
Experimental results are commonly
expressed in the form:
mean  standard deviation
_
xs
The Gaussian curve equation:
1
(x μ )2 /2σ 2
y
e
σ 2π
1
σ 2π
= Normalisation factor
It guarantees that the area under
the curve is unity
The Gaussian curve whose area is unity is called a
normal error curve.
µ = 0 and σ = 1
Relative frequency, dN / N
Normal Error Curve
m
-1
• 68.3% of
measurements will
fall within ±  of the
mean.
+1
-2
• 95.5% of
measurements will
fall within ± 2 of the
mean.
+2
-3
+3
xi
• 99.7% of
measurements will
fall within ± 3 of the
mean.
EXAMPLE
Replicate results were obtained for the measurement
of a resistor. Calculate the mean and the standard
deviation of this set of data.
Replicate
ohms
1
752
2
756
3
752
4
751
5
760
xi

x
_
Replicate ohms
n
752  756  752  751  760

 754.2
5
s
 x i
2
752
2
756
3
752
4
751
5
760
n 1


 x
1
752  754.22  756  754.22  752  754.22  751  754.22  760  754.22
5  1
 2.22  1.82   2.22   3.22  5.82
4
 3.77
s
3.77


 1.69
n
5
NB DON’T round a
std dev. calc until
the very end.
x  754.2
s  3.77
  1.69
Also:
s
RSD 
x
3.77

754
s
%RSD   100
x
3.77

 100
754
Variance  s
 3.77 
2
2
Student's t-Distribution
• If the sample size is not large enough, say
n ≤ 30.
• Then the distribution of x is not normal.
• It has a distribution called Student’s tdistribution.
t = (x – m)/(s/n).
Student's t-Distribution
• The Student's t-distribution was
discovered by W. S. Gosset in
1908.
• He used the pseudonym
‘Student’ to avoid getting fired for
doing statistics on the job!!
Student's t-Distribution
• The shape of the Student's t-distribution is
very similar to the shape of the standard
normal distribution.
• The Student's t-distribution has a (slightly)
different shape for each possible sample
size.
• They are all symmetric and unimodal.
• They are all centered at 0.
Student's t-Distribution
• They are somewhat broader than normal
• distribution, reflecting the additional
uncertainty resulting from using s in place
of .
• As n gets larger and larger, the shape of
the t-distribution approaches the standard
normal.
Degrees of Freedom
• If the sample size is n, then t is said to
have n – 1 degrees of freedom.
• We use df to denote degrees of freedom.
Student's t-Distribution
for 95% Confident level
Student's t-Distribution
• When  is estimated from the sample standard
deviation , s
s
2
(
x

x
)
 i
n 1
• The distribution for the mean x follows a
t- distribution with degrees of freedom, n − 1
x m
t
s n
CONFIDENCE INTERVAL
The confidence interval is the expression stating that
the true mean, µ, is likely to lie within a certain
distance from the measured mean, x
The confidence interval is given by:
ts
mx
n
Where t is the value of student’s t taken from the table
Use of t-Table
95% confidence interval; n = 11
s
m : x  2.2281
11
Degrees of Freedom
1
2
.
.
10
0.80
3.0777
1.8856
.
.
1.3722
0.90
6.314
2.9200
.
.
1.8125
0.95
0.98
12.706
4.3027
.
.
2.2281
31.821
6.9645
.
.
2.7638
.
.
.
.
.
.
.
.
.
.
100

1.2901
1.282
1.6604
1.6449
1.9840
1.9600
2.3642
2.3263
0.99
63.657
9.9250
.
.
3.1693
.
.
2.6259
2.5758
bunjob_ajchara
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Example:
The mercury content in fish samples were determined
as follows: 1.80, 1.58, 1.64, 1.49 ppm Hg. Calculate the
50% and 90% confidence intervals for the mercury
content.
Find x = 1.63
s = 0.131
_
ts
μ  x
n
50% confidence:
t =0.765 for n-1 = 3

0.765 0.131
μ  1.63 
4
μ  1.63  0.05
There is a 50% chance that the true
mean lies between 1.58 and 1.68
ppm Hg.
x  1.63
s  0.131
1.78
90% confidence:
90%
t = 2.353 for n-1 = 3
_
ts
μ  x
n
1.68
50%
1.63
1.58

2.3530.131
μ  1.63 
4
μ  1.63  0.15
There is a 90% chance that the true
mean lies between 1.48 and 1.78 ppm
1.48
Evaluation of Measurement
Uncertainty
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Terms specific to the GUM
• Standard uncertainty,
– the uncertainty of the result of a measurement
expressed as a standard deviation
GUM 2.3.1
• Type A evaluation (of uncertainty)
– method of evaluation of uncertainty by the statistical
analysis of a series of observations
GUM 2.3.2
• Type B evaluation (of uncertainty)
– method of evaluation of uncertainty by means other than
the statistical analysis of series of observations
GUM 3.2.3
Terms specific to the GUM
• Combined standard uncertainty
– the standard deviation of the result of a
measurement when the result is obtained from the
values of a number of other quantities.
– It is obtained by combining the individual standard
uncertainties (and covariances as appropriate),
using the law of propagation of uncertainties,
commonly called the "root-sum-of-squares" or
"RSS method.
GUM 2.3.4
Terms specific to the GUM
• expanded uncertainty
– quantity defining an interval about the
result of a measurement that may be
expected to encompass a large fraction of
the distribution of values that could
reasonably be attributed to the measurand.
GUM 3.2.5
• coverage factor, k
– numerical factor used as a multiplier of
combined standard uncertainty in order to
obtain expanded uncertainty
GUM 3.2.6
Process of Uncertainty Estimation
• Specify Measurand
• Identify all Uncertainty Sources
• Quantify Uncertainty Components
• Calculate Combined Uncertainty
Specify the Measurand
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The measurand?
Measurand = particular quantity subject to
measurement
[VIM 2.6 / GUM B.2.9]
Example:
the conventional mass of a 1kg
weight.
GUM 1.2
Measurement Model
• Define the measurand – the quantity subject to
measurement
• Determine a mathematical model, with input
quantities, X1,X2,…,XN, and (at least) one output
quantity,Y.
• The values determined for the input quantities
are called input estimates and are denoted by
x1,x2,…,xN.
• The value calculated for the output quantity is
called the output estimate and denoted by y.
Identify all Uncertainty
Sources
2. How is MU estimated?
78
ISO/IEC 17025
• 5.4.7.2
– attempt to identify all the components of
uncertainty
• 5.4.7.3
– All uncertainty components which are of
importance shall be taken into account
Sources of uncertainty
ISO/IEC 17025 5.4.7.3 Note 1:
Some sources contributing to the uncertainty:
– reference standards
– reference materials
– methods
– equipment
– environmental conditions
– properties and condition of the item to be tested
– the operator
Sources of MU
•
•
•
•
•
•
•
•
•
•
GUM 3.3.2
Incomplete definition of the measurand
Imperfect realisation of the definition of the measurand
Non-representative sampling
Effects of environmental conditions on the measurement
Personal bias in reading analogue instruments
Finite instrument resolution or discrimination threshold
Inexact values of measurement standards
Inexact values of constants obtained from external sources
Approximations incorporated into the measurement
Variations in repeated observations under apparently
identical conditions
2. How is MU estimated?
81
Causes for uncertainty
Measurement
standard
Measuring
methods
Calibration certificate
Secular change
Measuring
instrument
Measurement
results
Manufacturer’s
specification
Resolution
Measurement
environment
Measurer
Peculiarities in
readout
Number of
measurements
Dispersions in repetition
Sources of error and uncertainty in
dimensional calibrations
•
•
•
•
•
Reference standards and instrumentation
Thermal effects
Elastic compression
Cosine errors
Geometric errors
UKAS M3003 Dec 1999
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Sources of error and uncertainty in
electrical calibrations
•
•
•
•
Instrument Calibration
Secular Stability
Measurement Conditions
Interpolation of calibration
data
• Resolution
• Layout of apparatus
• Thermal emfs
• Loading and lead
impedance
• RF mismatch errors and
uncertainty
• Directivity
• Test port match
• RF Connector
repeatability
UKAS M3003 Dec 1999
bunjob_ajchara
84
Sources of error and uncertainty in
mass calibrations
•
•
•
•
•
Reference weight calibration
Secular stability of reference weights
Weighing machine / weighing process
Air buoyancy effects
Environment
UKAS M3003 Dec 1997
bunjob_ajchara
85
Quantify
Uncertainty Components
2. How is MU estimated?
86
The Measurement Model
• Usually the final result of a measurement is not
measured directly, but is calculated from other
measured quantities through a functional
relationship
• This is called function a “measurement model”
• The model might involve several equations, but
we’ll follow the GUM and represent it abstractly
as a single equation:
Y  f ( X 1 , X 2 ,..., X N )
Input and Output Quantities
• In the generic model Y = f(X1,…,XN), the
measurand is denoted by Y
• Also called the output quantity
• The quantities X1,…,XN are called input
quantities
• The value of the output quantity
(measurand) is calculated from the values
of the input quantities using the
measurement model
Input and Output Estimates
• When one performs a measurement, one
obtains estimated values x1,x2,…,xN for the input
quantities X1,X2,…,XN
• These estimated values may be called input
estimates
• The calculated value for the output quantity may
be called an output estimate
Measurement model
A measurand Y can be determined from N inputs
quantities X1, X2, X3 … XN
The model is written abstractly as
Y=f(X1,X2,…,XN) where X1,X2,…,XN are input
quantities and Y is the output quantity
Developing a Measurement
model
• Decide what is the measurand Y
– the quantity subject to measurement
• Decide what are the quantities X1, …, XN
influencing the measurement
– observed quantities, applied corrections,
material properties, etc
• Decide the relationship between Y and X1, …,
XN
– the model of the measurement
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Example: CALBRATION OF A HAND-HELD
DIGITAL MULTIMETER AT 100 V DC
The error of indication EX of the DMM to be calibrated is obtained from
where
Vi X - voltage, indicated by the DMM (index i means indication),
VS - voltage generated by the calibrator,
δ VI X - correction of the indicated voltage due to the finite resolution
of the DMM,
δ VS - correction of the calibrator voltage due to
(1) drift since its last calibration,
(2) deviations resulting from the combined effect of offset,
non-linearity and differences in gain,
(3) deviations in the ambient temperature,
(4) deviations in mains power,
(5) loading effects resulting from the finite input resistance
of the DMM to be calibrated.
EA-4/02:1999
Measurement model
An estimate of Y, denoted by y, is obtained from x1, x2,
x3 … xN, the estimates of the input quantities X1, X2,
X3 … XN,
Represent each input quantity Xi by
1. Best estimate xi as mean of distribution, and
2. Standard uncertainty u(xi) as s.d. of distribution
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Measurement Model
For each input quantity
1. Obtain knowledge of that quantity
2. Assign a probability distribution to each quantity
consistent with that knowledge
Often a Gaussian (normal) or a rectangular distribution
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Classification of uncertainty
components
• Type A components: those that are evaluated by
statistical analysis of a series of observations
• Type B components: those that are evaluated by
other means
– Both based on probability distributions
– standard uncertainty of each input estimate is
obtained from a distribution of possible values of input
quantity: both based on the state of our knowledge
– Type A founded on frequency distributions
– Type B founded on a priori distributions
Type A evaluations of uncertainty
Type A evaluations of
uncertainty are based on the
statistical analysis of a series
of measurements.
96
Type A Evaluation of Standard
Uncertainty
• For component of uncertainty arising from
random effect
• Applied when multiple independent
observations are made under the same
conditions
• Data can be from repeated measurements,
control charts, curve fit by least-squares
method etc
• Obtained from a probability density function
derived from an observed frequency
distribution (usually Gaussian
bunjob_ajchara
97
Type A Evaluation
Arithmetic mean
• Best estimate of the expected value of a
input quantity -
1 n
q   qk
n k 1
Type A Evaluation
Experimental standard deviation
Distribution of the quantity
Type A Evaluation
Experimental standard deviation of the mean
• spread of the distribution of the means __
s(q k )
s( q ) 
n
Type A Evaluation
• Type A standard uncertainty
u( xi )  s(q)
• degrees of freedom
 i  n 1
Example
A digital multimeter is used to measure a high value
resistor and the following readings are recorded.
The standard uncertainty, u, is therefore 0.008 83 kΩ.
Type A Evaluation
Pooled Experimental Standard Deviation
•
For a well-characterized measurement under
statistical control, a pooled experimental standard
deviation Sp that characterizes the measurement
may be available.
– The value of a measurand q is determined
from n independent observations and
– The standard uncertainty is
u ( q)  s p
n
Type A Evaluation
Example:
A previous evaluation of the repeatability of
measurement process (10 comparisons between
standard and unknown) gave an experimental
standard deviation
sWR   8.7mg
If 3 comparisons between standard and unknown
were made this time (using 3 readings on the
unknown weight), this is the value of n that is
used to calculate the standard uncertainty of the
measurand
sWR  8.7
u WR   s WR 

 5.0 mg
n
3
 
Type B Evaluation of Standard
Uncertainty
• Evaluation of standard uncertainty is usually
based on scientific judgment using all relevant
information available, which may include:
– previous measurement data,
– experience with, or general knowledge of the
behavior and property of relevant materials
and instruments,
– manufacturer's specifications,
– data provided in calibration and other reports,
and
– uncertainties assigned to reference data taken
from handbooks.
GUM 4.3.1
Type B Evaluations
• Normal distribution:
Ui
ui 
k
99.7%
68%
where Ui is the expanded
uncertainty of the
contribution and k is the
coverage factor (k = 2 for
95% confidence).
-4
-3
-2
-1
0
1
2
3
• Examples:
– expanded uncertainties from a calibration
certificate
95%
March 2006
Slide 106
4
Type B Evaluations
Normal distribution
Example
A calibration certificate reports the measured value of
a nominal 1kg OIML weight class F2 at approximately
95% confidence level as:
0.999999kg  10mg
U 10mg
u xi   
 5mg
k
2
Rectangular distribution
“It is likely that the value is somewhere in
that range”
Rectangular distribution is usually described in terms of: the
average value and the range (±a)Certificates or other
specification give limits where the value could be,without
specifying a level of confidence (or degree of freedom).
The value is
between the limits
2a(=  a)
a  a
The expectation
y  xa
1/2a
X
Rectangular distribution
Range = 2a ,
Semi-range = Range /2 = a
a
a
P=1/2a
A
B
Rectangular distribution
B
B
A
A
 2   x 2 Pdx  P  x 2 dx


1
 1  2
 1  3 B
2
P
;      x dx    X / 3 A
2a
 2a  A
 2a 
B   a; A  a
B
3
3
2






1

a

a
a


 2   


3  3
 2a   3
a

3
Example of Rectangular distribution
Example
• From the previous example, if the
Maximum Permissible Error (MPE)
according to OIML class F2 (±16 mg) is
used; then
uB 
a
3

16 mg
3
 9 . 23 mg
Example of Rectangular
distribution
Handbook
• A Handbook gives the value of coefficient of
C
linear thermal expansion of pure
copper at 20
 20 Cu   16.52 106 / C
and the error in this value should not exceed,
 0.40 10 6 / C
assuming rectangular distribution the
standard uncertainty is:
semi  range ; a  0.40 10 6
0.40 106
u xi  
 0.23 106 / C
3
Example of Rectangular distribution
• Manufacturer’s Specifications
A voltmeter used in the measurement process has the
accuracy of ± 1 % of full scale on 100 V. range
semi - range ( a ) = 1 V
a 1V
u xi  

 0.6 V
3
3
Example of Rectangular distribution
Resolution of a digital indication
•If the resolution of the
digital device is δx, the
value of X can lie with
equal probability anywhere
in the interval X - δx /2
to X + δx /2 and thus
described by a rectangular
probability distribution
with the width δx
6
1 2 3 4 5
4
Range   x
Semi - range 

x
u( x ) 
i
2 3
 0.29 x
x
2
Example of Rectangular distribution
•Digital indication
•A digital balance
having capacity of 210g
and the least significant
digit 10 mg. The
standard uncertainty
contributed by this
balance is:
0,01
u xi  
g
2 3
 2.9 mg
Example of Rectangular distribution
Hysteresis
The indication of instrument may differ by a fixed
and known amount according to whether successive
reading are rising or falling.
If the range of possible readings from that is dx

x
uxi  
2 3
 0.29 x
U-shaped distribution
• When the measurement result
has a higher likelihood of being
some value above or below the
median than being at the
median.
-2 ai
-ai
0
ai
2ai
-2 ai
-ai
0
ai
2ai
ai
ui 
2
• Examples:
– Mismatch (VSWR)
– Distribution of a sine
wave
March 2006
Slide 117
Example of U-Shaped distribution
• A mismatch uncertainty associated with the calibration of
an RF power sensor has been evaluated as having
semi-range limits of 1.3%. Thus the corresponding
standard uncertainty will be
UKAS M3003
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Triangular distribution
Distribution used when it is
suggested that values near
the centre of range are more
likely than near to the
extremes
y  xa
2a (=a)
1/a
Assumed standard deviation:
s  a 1 / 6
X
Example of Triangular distribution
Values close to x are more likely than near the boundaries
Example: A tensile testing machine is used in a testing
laboratory where the air temperature can vary
randomly but does not depart from the nominal value
by more than 3°C. The machine has a large thermal
mass and is therefore most likely to be at the mean air
temperature, with no probability of being outside the
3°C limits. It is reasonable to assume a triangular
distribution, therefore the standard uncertainty for
its temperature is:
UKAS M3003
In case of doubt, use the rectangular distribution
Which is better A or B?
It should be recognized that a
Type B evaluation of a standard
uncertainty can be as reliable as a
Type A evaluation, especially in a
measurement situation where a
Type A evaluation is based on a
comparatively small number of
statistically independent
observation.
GUM 4.3.2
Calculate
Combined Standard Uncertainty
combined standard uncertainty
• Components of standard uncertainty of measurand
y=f(x1,x2,x3……xN) are combined using the “ Law of
Propagation of Uncertainty” or “Root Sum of
Square :RSS”
y  f(x1 ,x2 ,...,xN )
2
N 1 N
 f  2
f f
 u (xi )  2 
uc (y)   
u ( xi )u ( x j )r ( xi , x j )
i 1  xi 
i 1 j i 1xi x j
N
2
N
2
N 1
N
  ci u(xi )  2  ci c j u ( xi )u ( x j )r ( xi , x j )
i 1
i 1 j i 1
r xi , x j  is the correlatio n between input quanties xi , x j
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ci is the sensitivit y coefficient
f
xi
123
Combined Standard Uncertainty, uc
The relationship between the measurand, Y,
and A, B and C is written most generally as
Y = f(A,B,C).
 f
  f
  f

u c  y    u (a)    u (b)    u (c) 
 a
  b
  c

2
2
2
u(a), u(b) and u(c) are the standard
uncertainties of best estimates a, b and c
respectively obtained through Type A or Type
B evaluations.
124
sensitivity coefficient
Partial derivative with respect to input
quantities Xi of functional relationship
f
between measurand Y and input
quantities Xi on which Y depends
sensitivity coefficient formula
f
ci 
xi
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Example
The value of the resistance Rt, at the temperature t, is obtained
from equation:
Rt  R0 (1  t )
Where:
α is the temperature coefficient of the resistor in Ω / °c
t is the temperature in °c , and
R0 is the resistance in ohms at the reference temperature,
The partial differentiation of Rt with respect to t is:
Rt
 t
t
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Correlation of Input Quantities
SRef
Ref
SUUT
UUT
Difference
(Correction Refbunjob_ajchara
UUT)
Scorr
128
correlation
Consider
c  a  b 
c  a  b   a  2ab  b
2
2
2
2
If, 2ab  0
If, 2ab  0
c2  a2  b2
c2  a2  b2
c  a b
c  ab
2
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2
129
correlation coefficient
correlation coefficient, r(xi , xj) - degree
of correlation between xi , x j
r xi , x j  
Value :
u xi , x j 
u xi u x j 
.
 1  r ( xi , x j )  1
r ( xi , x j )  0.......Uncorrelated
r xi x j   0........correlated
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Uncorrelated input quantities
For uncorrelated input quantities r (xi , xj) = 0
N 1
2  ci c j u xi u x j r xi , x j   0
Then
N
i 1 j i 1
N
2
uc2  y    ciu xi 
i 1
For
c =1
i
2
2
2


uc y  u x1  u x2  u x 3  ...u x2n
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Combinations of Uncertainties
Addition/Subtraction
y  ax  bz
2
u y2
y
x
u y2
 y 
 y 
 y  y 
   u x2    u z2  2  u xz2
 z 
 x  z 
 x 
y
 a;
b
z
 a 2u x2  b 2u z2  2abu xz2
2
For independent variables, we have,  2abu xz2  0 !
u y2  a 2ux2  b2uz2
Combinations of Uncertainties
Multiplication/Division
y   axz
y 
 y
u y2   u x  u z 
z 
 x
y
y
  az ;
  ax
x
z
u y2  a 2 z 2u x2  a 2 x 2u z2  2a 2 xzuxz2
2
2
Similar arguments would apply to
the expression
x
y  a
z
 u y   ux   uz 
u xz2
         2
xz
 y  x  z 
2
2
For independent variables, we have,
 uy

 y
2
  ux   uz 
      
  x  z 
2
2
Worked example
The mass, m, of a wire is found to be 2.255 g with a
standard uncertainty of 0.032 g. The length, l, of the wire is
0.2365 m with a standard uncertainty of 0.0035 m. The
mass per unit length, m, is given by:
m
m
l
Determine the,
a) best estimate of m,
b) standard uncertainty in m.
m 2.255
m 
 9.535 g/m
l
0.2365
134
Worked example continued
The partial differentiation of µ with respect to m and l
 m
  m

u m   
u (m)    u (l ) 
 m
  l

m 1
1
-1
2
2
2
c
 
 4.2283 m
m l 0.2365
m
m
2.255
2
 2 


40
.
317
g/m
l
l
0.2365 2
u m   4.2283  0.032   40.317  0.0035
2
c
2
2
uc m   0.1955 g/m
135
correlated input quantities
For the very special case where all input
estimates are correlated r xi , x j   1
N 1
Then
2  ci c j u xi u x j r xi , x j   0
N
i 1 j i 1
The combined standard uncertainty
N
uc  y    ci u xi 
i 1
uc  y   c1u x1  c2u x2  c3u x3  ....cnu xn
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136
Correlated input quantities
Example
R1
R3
R2
R10
Rref 10 kW
1)Ri (R1,R2,R3,……,R10) each has nominal value 1000 ohms
2)Each has been calibrated by direct comparison with
negligible uncertainty
3)Standard uncertainty of Rs is u(Rs) = 100 mohms
Model equation :
Rref
 f Ri 
10
  Ri  10kW
i 1
u Rref    u Rs   10 100mW  1W
10
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i 1
137
Calculate
Expanded Uncertainty
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138
Expanded Uncertainty
• expanded uncertainty
– quantity defining an interval about the
result of a measurement that may be
expected to encompass a large fraction of
the distribution of values that could
reasonably be attributed to the measurand.
GUM 3.2.5
Expanded Uncertainty, U
The Expanded Uncertainty, U, is a simple
multiple of the standard uncertainty, given
by
U = kuc(y)
k is referred to as the coverage factor.
So we can write:
Y=yU
140
coverage factor, k
• coverage factor, k
– numerical factor used as a multiplier of
combined standard uncertainty in order to
obtain expanded uncertainty
GUM 3.2.6
Coverage factor
Coverage Factor - Confidence Interval
k
1.00
2.00
68.27%
95.45%
2.58
3.00
99.%
99.73%
Most cal labs adopt 95.45% which gives k  2
for effective degrees of freedom  30
Coverage Factor of Combined
Uncertainty
• Effective Degree of Freedom
– to determine the coverage factor of combined uncertainty, the
effective degree of freedom must be first calculated from the
Welch-Satterthwaite formula:

eff
uc 4 ( y )
 N 4
ui ( y )

i 1
i
• Based on the calculated veff, obtain the t-factor tp(veff) for the
required level of confidence p from the t-distribution table
• The coverage factor will be: kp = tp(veff)
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143
Effective number of degrees of freedom
• Example -- A steel rod was measured 4 times. The
calculated .
u A  3.5 mm and u B  2.3 mm;
uc ( y)  4.2 mm
• The effective degree of freedom:

uc  y 
4.2
 4

 6.22
4
4
u A u B
3.5  04

vA
vB
4  1
4
veff
4
• For  eff  6 @ 95% confidence level and from “student’s t”
table, we get k = 2.52
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144
Effective number of degrees of
freedom
Therefore, the expanded uncertainty U is:
U  k .uc  y 
 2.52  4.2
 11 mm.
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145
Relative standard uncertainty
Relative standard uncertainty of input estimate ,
Relative combined standard uncertainty, y

uc  y 
y
 u  xn  
 uc  y   u x1   u x2 



.........



 y   x   x 
x

  1   2 
 n 
2
then
2
2
2
 u  xn  
 u x1   u x2 
uc  y 
 

 
  .........  
y
x
x
x
 1   2 
 n 
2
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u  xi 
xi  
xi
2
2
146
Relative standard uncertainty
Example
The measurand: C NaOH
Description
rep
Repeatability
mKHP
Weight of KHP
PKHP
M KHP
VT
Purity of KHP
Value,x
1,0
0,3888 g
1,0
1000  mKHP  PKHP

M KHP  VT
Standard
uncertainty,
u  xi 
Relative
u  xi 
standard
xi
uncertainty,
0,0005
0,0005
0,00013g
0,00033
0,00029
0,00029
Molar mass of KHP
204,2212
gmol-1
0,0038gmol-1
0,000019
Volume of NaOH for
KHP titration
18,64 ml
0,013ml
0,0007
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147
Relative standard uncertainty
1)
Value of the measurand
C NaOH
2)
1000  0,3888 1,0 = 0,10214 mol l-1

204,2212 18,64
Combined relative standard uncertainty
uc (CNaOH )

CNaOH

 urep    u mKHP    uPKHP    uM KHP    uVT  
  
  
  


  
 rep   mKHP   PKHP   M KHP   VT 
2
2
2
2
2
0,00052  0,000332  0,000292  0,0000192  0,000702
 0,00097
uc(CNaOH) = 0,00097 X 0.10214 mol l-1 = 0,00010 mol l-1
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148
Reporting Result
Reporting
• should include
– result of measurement
– expanded uncertainty with coverage factor and
level of confidence specified
– description of measurement method and
reference standard used
– uncertainty budget
• example of uncertainty statement
e.g.The expanded uncertainty of measurement is ±
____ , estimated at a level of confidence of
approximately 95% with a coverage factor k =
____.
Reporting Result
• It usually suffices to quote uc(y) and U [as well as the standard
uncertainties u(xi) of the input estimates xi] to at most two significant
digits, although in some cases it may be necessary to retain
additional digits to avoid round-off errors in subsequent calculations.
• In reporting final results, it may sometimes be appropriate to round
uncertainties up rather than to the nearest digit. For example, uc(y) =
10,47 m might be rounded up to 11 m.
• However, common sense should prevail and a value such as u(xi) =
28,05 kHz should be rounded down to 28 kHz.
• Output and input estimates should be rounded to be consistent with
their uncertainties.
GUM 7.2.6
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151
Reporting Conventions
• 1000 (30) mL
– Defines the result and the (combined) standard
uncertainty
• 1000 +/- 60 mL
– Defines the result and the expanded uncertainty
(k=2)
• 1000 +/- 60 mL at 95% confidence level.
– Defines the expanded uncertainty at the specified
confidence interval
The 9-steps GUM Sequence
1. Define the measurand
2. Build the model equation
3. Identify the sources of uncertainty
4. (If necessary) Modify the model
5. Evaluate of the input quantities and calculate the value
of the result
6. Calculate the value of the measurand (using the
equation model)
7.Calculate the combined standard uncertainty of the
result
8. Calculate the expanded uncertainty (with a selected k)
9. Report result
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153
Conclusions and Remarks
Some Important Practical Consequences
… or a little common sense with errors!
1. When several (independent) errors are to be added, addition in quadrature
is much more realistic than addition.
2. If one error ie less than one quarter of another error in the addition then
the smaller error may be realistically ignored.
3. There is little point in spending much time estimating small errors –
concentrate on the large errors!
4. The experimental procedure should minimise the dominant errors, This implies
that these must be identified and estimated (usually in a pilot run) before the
final data is taken.
5. Try to bring the precision of each variable to a common level, if possible, by
repeated measurements.
Basic concepts
“…The evaluation of uncertainty is neither
a routine task nor a purely mathematical
one; it depends on detailed knowledge
of the nature of the measurand and of
measurement…”
GUM 3.4.8
Bibliography and acknowledgement
ISO (1993) Guide to the Expression of Uncertainty in Measurement (Geneva,
Switzerland: International Organisation for Standardisation).
NIST Technical Note 1297 (1994) Guidelines for Evaluating and Expressing the
Uncertainty of NIST Measurement Results.
M 3003, The Expression of Uncertainty and Confidence in Measurement, published
by UKAS
EA-4/02 - December 1999• Expression of the Uncertainty of Measurement in
Calibration
EURACHEM / CITAC Guide: Traceability in Chemical
Measurement - A guide to achieving comparable results in chemical
measurement 2003
Assessment of Uncertainties of Measurement for Calibration and Testing
Laboratories - Second Edition , c R R Cook 2002
Published by National Association of Testing Authorities, Australia
ACN 004 379 748 ISBN 0-909307-46-6
158