### Uncertaity of Measurement - Better Work and Standards Programme

```Bangladesh BEST Programme
Uncertainty of Measurement
Nihal Gunasekara
Sri Lanka
1
What is a measurement ?
Property of something
 How heavy of an object is
 How hot of an object is
 How long it is
A measurement gives a number of that property
2
What do you need for a measurement ?
Instrument
Rulers
Stopwatches
Weighing scales
Thermometers
3
How do you report a measurement ?
 The length of table is 20 m
 The weight of the object is 3 kg
 The temperature of the sample is 50 °C
 The volume of liquid is 50 ml
Use SI units for all measurements
4
What is not a measurement ?
 Comparing two pieces of strings to see which is longer
 Comparing two liquids to see which is hotter
 Comparing height of two persons to see who is taller
5
What is uncertainty of measurement ?
The uncertainty of measurement tells us something
Uncertainty of measurement is the doubt that exists
about the result of any measurement
Can we expect accurate results from all measuring
instruments ?
A margin of doubt !!!!!
6
Definition of Uncertainty of Measurement
“ Non-negative parameter characterizing the
dispersion quantity values being attributed to a
measurand, based on the information used”
JCGM 200: 2012 BIPM 3rd Edition
7
Measurement Uncertainty
X
U
U
A range containing the true value
8
Expressing Uncertainty of Measurement
Margin of doubt about any measurement !!!!
How big is the margin ?
How bad is the doubt ?
Two numbers are needed to quantify an uncertainty
Width of the margin or interval
Confidence level
9
Error Versus Uncertainty
Error : is the difference between the “measured value”
and the” true value” of the thing being measured
Error = measured value - true value (reference value)
Uncertainty : is a qualification of the doubt about the
measurement result
10
Error Versus Uncertainty
Error can be corrected !!!!!
How ?
Apply correction form calibration certificates
But any error whose value we do not know is a
source of uncertainty !!!!
11
Why is uncertainty of measurement important?
“ We wish to make good quality measurement and to
understand the result”
ISO 17025 requirements
Calibrations & Testing laboratories shall have a procedure for
calculation of MU
Where not possible for some test methods of testing labs, the
contributing factors need to be identified and a reasonable
When estimating MU all components that contribute to MU
should be taken into account
12
Basic Statistics on Sets of Numbers
“Measure thrice, cut once- operator error”
Risk can be reduced by checking the measurement
several times !!!!
Take several measurements to obtain a value !!!!
13
Basic Statistical Calculations
To increase the amount of information of your
measurement : take several readings !!!!
Two most important statistical calculations :
 Average or arithmetic mean  Standard deviation - s
14
Getting the Best Estimate
If there is variation in readings when they are repeated
 Get the average
Best estimate for the “true” value
Mean or average value
15
How Many Readings Should you Average ?
More measurements : better estimate of true value
What is a good number ? 10
20 would give slightly better estimate than 10
16
Usual way to quantify spread is “Standard Deviation”
The standard deviation of a set of numbers tells us “about
how different the individual readings typically are from the
average of the set”
17
Calculating an Estimated Standard Deviation
Example :
Let the readings are 16, 19,18, 16, 17, 19,20,15,17, and 13
Average is 17
Find the difference between each reading and the average
ie. -1
+2
+1
-1 0 +2 +3
-2
0 -4
And square each of those
ie 1
4
1
1
0 4
9
4 0 16
Find the total and divide by n-1 (in this case n is 10)
ie. 1+4+1+1+0+4+9+4+0+16 = 40 = 4.44
9
9
Standard deviation s =
= 2.1
18
Mathematical Equation for Standard Deviation
s
 r
i
r
2
n  1
19
Where do Errors and Uncertainties come from ?
Measuring instrument - ageing effect, drift, poor readability
etc
Item being measured - ice cube in a warm room
Measurement process - measurement itself may be difficult
Imported uncertainties – instrument uncertainty
Environment – temperature, air pressure, humidity vibration
etc.
20
Distribution – Shape of Errors
The spread of set of values can take different
forms
Probability of occupation
Normal or Gaussian distribution
21
Uniform or Rectangular Distribution
When measurements are quite evenly spread between
the highest and lowest values a rectangular or uniform
distribution is produced
Probability of occurrence
Range
Full width
22
Triangular Distribution
Probability of occurrence
23
What is not a Measurement Uncertainty ?
Tolerances of a product
Specifications of instruments
24
How to Calculate Uncertainty of Measurement
Identify the sources of uncertainty in the measurement
Estimate the size of the uncertainty from each source
Combine individual uncertainties to give an overall
figure
25
How to Calculate Uncertainty of Measurement
Specify Measurand
Identify Uncertainty sources
STEP 1
STEP 2
Simplify by grouping the sources covered
by available data
STEP 3
Quantify grouped and remaining components
Convert components to standard uncertainties
26
How to calculate Uncertainty of measurement
Calculate the
combined standard
Uncertainty
Review and if required re-evaluate large
components
STEP 4
Calculate the Expanded Uncertainty
27
Estimation of Total Uncertainty
Type A evaluation – method of evaluating the
uncertainty by the statistical analysis of a series of
observations
Type B evaluation - uncertainty estimates by means
other than the statistical analysis of a series of observations.
28
Type B Evaluation
Category may be derived from:
Previous measurement data
Experience with or general knowledge of the
behaviour and properties of relevant materials and
instruments
Manufacture’s specifications
Data provided in calibration and other certificates
Uncertainties assigned to reference data taken from
handbooks
29
Standard Uncertainty for a Type A Evaluation
“When a set of several repeated readings has been
taken the mean and estimated standard deviation, s,
can be calculated for the set”
Fro these , the estimated standard uncertainty , u of the
mean is calculated from :
U=
30
Standard Uncertainty for Type B Evaluation
“Where the information is more scarce (in some Type B
estimates), you might be able to estimate the upper
and lower limits of uncertainty. You may then have to
assume the value is equally likely to fall anywhere in
between ie. rectangular or uniform distribution “
The standard uncertainty for rectangular distribution is
found from:
U =
“a “ is the semi range or half width between upper and
31
lower limits
Rectangular Distribution
2a
a
a
f(x)
Area enclosed by
1
2a
rectangle = 1
a
Lower limit
a  a
2
a
x
Upper limit
Best estimate
32
There are simple mathematical expressions
to evaluate the standard deviation for this.
Another such distribution we normally
encounter is the triangular distribution
a
a
Area enclosed by
f x 
Triangle=1
1
a
a
a  a
2
a
x
33
Confidence Level
Gaussian probability distribution
68%
95%
99%
-ks

+ks
Within 1s of mean k = 1
Within 2s of mean k = 2
Within 3s of mean k = 3
34
Combining Standard Uncertainties
Individual standard uncertainties calculated by Type A
and Type B evaluations can be combined validly by “root
sum of the squares”
The result is the “combined standard uncertainty”
This is represented by uc
If the Type A and Type B uncertainties are a, b, c & d,
then combined standard uncertainty is :
uc =
35
Coverage Factor
The overall uncertainty is stated at the confidence
level of 95% with the coverage factor k=2
Multiplying the combined standard uncertainty
uc by the coverage factor gives the result which is
called “ expanded uncertainty “ usually shown by
the symbol “Uc “
Uc = kuc (y)
36
Reporting Uncertainty
State the result of the measurement as :
Y = y ± U and give the units of y and U
where the uncertainty U is given with no more than two
significant digits and y is correspondingly rounded to the
same number of digits
The nominal value of 100 g mass is 100.02147 g
The expanded uncertainty is 0.00079 g
The result of measurement is expressed as 100.02147 g ± 0.00079 g
and the coverage factor k = 2
37
Statement of Uncertainty in Measurement
Calibration Certificate :
“The
reported expanded uncertainty in measurement is
stated as the standard uncertainty in measurement
multiplied by the coverage factor k = 2, which for a
normal distribution corresponds to a coverage
probability of approximately 95 %.
The standard uncertainty of measurement has been
determined in accordance with Guide to expression of
uncertainty in measurement (GUM) JCGM 100:2008”
38
How to Reduce Uncertainty in Measurement
 Calibrate measuring instruments
 Use calibration corrections given in the certificate
 Make your measurements traceable to International
system of units (SI)
 Confidence in measurement traceability from an
accredited laboratory (UKAS, SWEADC, NABL etc.)
 Choose the best measuring instruments for smallest
uncertainty
 Check measurements by repeating them
 Check all calculations when transferring data
 Use an uncertainty budget to identify the worst
39
Some Good Measurement Practices
Follow the manufacture’s instruction for using and
maintaining instruments
Use experienced staff and provide training
Validate software
Check raw data by a third party
Keep good records of your measurements and
calculations
40
Preparation of Uncertainty Budgets
Example:
Calculation of uncertainty of a balance calibration
Capacity of balance : 50 g
Resolution of balance : 0.1 mg
Measured max. Std. deviation : 0.0939 mg
Number of measurements :10
Task : Calibration of scale value of 45 g
Method : A combination of three masses are required
Mass
1
2
3
Total
Value
20.000088 g
19.999995 g
5.000030 g
45.000113 g
U95 (mg)
0.019
0.019
0.0043
k
2
2
2
u (mg)
0.0095
0.0095
0.0045
0.0235
41
Preparation of Uncertainty Budget
Observations:
: 0.0000 g
1st reading of standard mass : 45.0003 g
2nd reading of standard mass : 45.0003 g
: 0.0001 g
Calculations:
Mean zero reading ( zi ): 0.00005 g
Mean reading on standard mass ( ri ) : 45.00030 g
42
Preparation of Uncertainty Budget
The basic measurement model is:
Ci = Mi – (ri - zi )
Where C is the calculated correction
Mi is the calibrated value of standard mass
ri is the mean of two repeated readings
Correction : Ci = Mi – ( ri – zi )
= 45.000113 g – (45.00030 – 0.00005 ) g
= -0.000137 g
= - 0.1 mg (rounded to least count of balance)
43
Uncertainty Budget
Source of
uncer.
(quant..)
Units
Type Prob. Dis.
of
evalu.
ss
Uncer. (U
or s)
Divisor
Stand.
Uncer.
uc
Cal. Uncer.
umass
mg
B
Normal
0.0235
Resolution
uresolution
mg
B
Rect.
Repeatabili
ty
urepeatability
mg
A
Normal
1
0.0235
0.00055
0.1/2
0.02887
0.00083
0.0939
0.02972
0.00088
Sum
0.00226
Comb. std
uncer.
0.0475 mg
Cov. Fac. k
Expan.uncr
2
0.09544mg
Comparison of magnitudes of Standard Uncertainty
Components
mg
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
Std. mass Scale res.
Bal.
repeat.
Expa.
Uncer.
45
Calibration and Measurement Capability (CMC)
History
In order to enhance the harmonization in expression of
uncertainty on calibration certificates and on scope of
accreditation of calibration laboratories, ILAC approved a
resolution at its third General Assembly meeting in 1999.
ILAC and BIPM have signed a MOU to harmonize the
terminology, namely the “Best Measurement Capability
(BMC)” used on the scope of accreditation of calibration
laboratories with the “Calibration and Measurement
Capability (CMC)” of CIPM MRA
This document was effective November 2011
46
Calibration and Measurement Capability (CMC)
The scope of accreditation of an accredited laboratory
shall include CMC expressed in terms of:
Measurand
Calibration/measurement/performance method
Measurement range
Uncertainty of measurement
47
Calibration and Measurement Capability (CMC)
In the formulation of CMC:
“The smallest uncertainty of measurement that can be
expected to be achieved by a laboratory during a
calibration or measurement”
“The uncertainty covered by the CMC shall be expressed
as the expanded uncertainty having a specific coverage
probability of approximately 95%”
48
Calibration and Measurement Capability (CMC)
In the formulation of CMC :
“ Take the notice of the performance of the “best existing
device” which is available for a specific category of
calibrations”
Consideration should also be given to “repeatability of
measurement”
49
Calibration and Measurement Capability (CMC)
Example:
SWEDAC
Measured
Quantity
Method of
Calibration
Range
Calibration and
Measurement
Capability( ±)
Calibration of
weighing balance
MM/MA/01
0 to 200 g
0.01 mg
0.10 mg
Performance
test of
laboratory oven
MM/TE/01
50 to 250 °C
1 °C
0.2 °C
One mark
pipette
MM/VO/01
0 to 200 ml
0.001 ml
50
Examples
Example 1 : Determination of uncertainty of the mass 1000 g
Reference mass standard used : uncertainty given in the calibration
certificate is 0.005 g at 95% confidence level
Resolution of the balance : 0.001 g
51
Example of uncertainty calculation
Determine the weight of 1kg
Observation
Value of test mass
1
2
3
4
5
6
7
8
9
10
1000.143
1000.144
1000.144
1000.146
1000.146
1000.146
1000.144
1000.143
1000.145
1000.145
Mean Value
: 1000.1446 g
Standard deviation
: 0.0011 g
Estimated Standard deviation of mean : 0.0011/√10=0.00035 g
52
Uncertainty Budget
Source of
uncer.
(quant.)
Units
Type
of
Eval.
Pro. Dist.
Uncer. (U
or s)
Divisor
Stand.
Uncer.
uc
Cal. Uncer.
umass
mg
B
Normal
5
Resolution
uresolution
mg
B
Rect.
Repeatabili
ty
urepeatability
mg
A
Normal
2
2.5
6.25
0.5x
0.4082
0.1666
1.1
0.35
0.1225
Sum
6.539
Comb.std
uncer.
2.55 mg
Cov. Fac. k
Exp. uncer.
2
5.1 mg
53
Presentation of Results
The result is reported as:
The value of the test mass = 1000.145 g
Expanded uncertainty = ± 0.005 g with k=2 at 95%
confidence level
or
The value of test mass is 1000.145 g ± 0.005 g with
k=2 at 95% confidence level
54
Preparation of Uncertainty Budget
Example 2:
Calibration of an oven at 100 °C
Reference thermometer : Calibrated set of TC, uncertainty given in
the calibration certificate is 0.5 °C at 95% confidence level
Digital thermometer with a resolution of 0.1 °C
Test oven used with a resolution of 1 °C
The standard deviation of 10 readings obtained at 100 °C is 0.6 °C
55
Uncertainty Budget
Source of
uncer.
(quant.)
Units
Type
of
Eval.
Pro. Dist.
Uncer. (U
or s)
Divisor
Stand.
Uncer.
uc
Cal. Uncer.
utc
°C
B
Normal
0.5
Dig. Ther.
uresolution
°C
B
Rect.
Dig. Ther.
urepeatability
°C
A
Dig. Ther.
U cjc
°C
Test Oven
uresolution
°C
2
0.25
0.0625
0.1/2
0.0289
0.00084
Normal
0.6
0.190
0.0361
B
Rect.
0.2
0.1156
0.0134
B
Rect.
1/2
0.289
Sum
Co. Std. u
0.0835
0.1963
0.44 °C
Cov. Fac. k
2
Exp. uncer.
0.9 °C
56
Comparison of Magnitudes of Standard
Uncertainty Components
°c
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Dig. Th. Dig. Th. Dig.Th. Dig. Th. Tes. Ov. Exp. Un.
U tc
Ures
Urep
U cjc
Ures
57
Sensitivity Coefficients
Sensitivity coefficient converts all uncertainty components
to the same unit as the measurand
Ex. The standard uncertainty due temperature( u1 ):0.05 °C
The standard uncertainty in the bridge (u2 ) : 0.001 Ω
The standard uncertainty in diameter ( u3 ) : 0.01mm
Combined standard uncertainty Uc =
58
Sensitivity Coefficients
The general formula for the sensitivity coefficient is:
Where : ci is the sensitivity coefficient for component xi
y the measurand is a function of xi
is the partial derivative of yi with respect to xi
“The partial derivative gives the slope of the curve that results
when the function yi, the measurand, is plotted for the
appropriate range of xi values”
59
Preparation of Uncertainty Budget
Example 3:
Measurement of resistivity of a rod using the following equation
Where :
R
l
A
d
is the rod resistance in ohms
is the length of the rod in meters
is the cross sectional area of the rod in m
is the diameter of the rod in m
60
Uncertainty Budget
Input Data :
Distance between knife degrees : 1.00003 m , unce. ± 0.01 mm, 95% CL
Measured diameter of the rod : 6.001 mm
No. of measurements of diameter : 10
Estimated std. dev. Of diameter : 0.25 µm
Micrometer uncertainty
: ±3 µm at 95% CL
Measurement Data :
Mean resistance : 604.44 µΩ
No. of resistance measurements : 5
Estimated std. dev. : 0.3 µΩ
Bridge reading uncertainty : ±1 µΩ
Rod temperature : 20 ± 0.05 °C
61
Uncertainty Components and their Evaluation
Rod diameter uncertainty ud
Type A evaluation:
The sensitivity coefficient c is obtained by differentiating the model
equation for ρ with respect to d, thus
62
Uncertainty Components and their Evaluation
Micrometer uncertainty u
m
Micrometer uncertainty is 3 µm,
um = U/k
= 3.0/2 µm
63
Uncertainty Components and their Evaluation
Rod length uncertainty u
l
Uncertainty value supplied is 0.01 mm
Standard uncertainty ul is calculated as :
ul = U/k = 0.01/2 mm
The sensitivity coefficient ci is calculates as :
64
Uncertainty Components and their Evaluation
Resistance uncertainty u
R
Uncertainty of resistance includes several terms
a.Repeatability uncertainty urdg
Type A evaluation is
Sensitivity coefficient crdg is given by
65
Uncertainty Components and their Evaluation
b.Bridge reading uncertainty ub is given by :
( Assume rectangular distribution)
Sensitivity coefficient is as in the previous case :
66
Uncertainty Component and their Evaluation
C. Resistance temperature uncertainty u
T
The model equation has not included a term for temperature but
the resistance varies with temperature as:
The model equation can be written as :
Differentiate this equation with respect to t then we get:
67
Uncertainty Components and their Calculations
As per data supplied the possible temperature variation is 0.05 °C
Uncertainty due to temperature variation is :
Sensitivity coefficient is given by : cT =
68
Uncertainty Budget
Source of
uncer.
(quant.)
U Typ. Pro.
nit of
Dist.
Ev.
Uncer.
(U or s)
Div.
Stand.
Uncer.
uc
Sen.
Coff.
ci
x uc
=
ui (y)
Rod dia.
ud
m
B
Nor.
7.
91e-8
1
7.1e-8
5.7xe-6
4.5e-13
2.03e-25
Mi. Ca. um
m
A
Nor.
3.0e-6
2
1.5e-6
5.7xe-6
8.7e-12
7.61e-23
Length ul
m
A
Nor.
1e-5
2
0.5e-5
-1.7e-8
8e-14
6.45e-27
Res. U rdg
Ω
B
Nor.
1.34e-7
1
1.34e-7
2.83e-5
3.8e-12
1.44e-23
Bri. Ca. Ub
Ω
A
Rec.
1e-6
5.77e-7
-2.83e-5
1.6e-11
2.67e-22
R. Tem. ut ° C
B
Rec.
5e-2
2.89e-2
6.7e-11
1.9e-12
3.71e-24
Sum
Std. Un.
3.62e-22
1.9e-11 Ωm
k
Exp.
Un.
2
3.8e-11 Ωm
69
Comparisons of Magnitudes of Standard Uncertainty Components
40
35
30
pΩm 25
20
15
10
5
0
Rod. Dia.Ud
Mic. Cal. Um Len. Ul Res. Urdg Brg. Ub R.tem. Ut Exp. Un U
70
Preparation of Uncertainty Budget
Example 4: Temperature measurement using a TC
A digital thermometer with a Type K TC was used to measure
the temperature inside a chamber at 500 °C
Specification of digital thermometer:
Resolution :0 .1 °C
Measurement accuracy : ±0.6 °C
TC calibration certificate provides :
Uncertainty is ± 2.0 °C at 95% confidence level
Correction at 500 °C is 0.5 °C
71
Preparation of Uncertainty Budget
Measure temp. (T) = Displayed temp. + Correction
Calculation of uncertainty components
Urept - standard uncertainty in the repeatability of the measured results
Udig -standard uncertainty in the digital thermometer
Utc - standard uncertainty in the thermocouple
72
Preparation of Uncertainty Budget
Measurement record:
Measurement
1
2
3
4
5
6
7
8
9
10
Temperature °C
500.1
500.0
501.1
499.9
4 99.9
500.0
500.1
500.2
499.9
500.0
Mean value
is 500.02
Standard deviation s is 0.103 °C
Standard deviation of mean SDOM is 0.03 °C (Type A)
73
Uncertainty Budget
Source of
uncer.
(quant..)
Units
Type Prob. Dis.
of
evalu.
ss
Uncer. (U
or s)
Divisor
Stand.
Uncer.
uc
Cal. Uncer.
utc
°C
B
Normal
1
Cal. Uncer.
udig
°C
B
Rect.
Repeat.
Urep.
°C
A
Normal
2
0.5
0.25
0.6
0.349
0.1223
0.103
0.326
0.0011
Sum
0.3734
Comb. std
uncer.
Cov. Fac. k
Expan.uncr
0.61 °C
2
1.2 °C
74
1.4
1.2
1
°C
0.8
0.6
0.4
0.2
0
U (TC)
U (dig.
Ther.)
U (Rept.) Exp. Uncer.
75
Preparation of Uncertainty Budget
Example 5: Calibration of 250 ml volumetric flask
A balance with a resolution of 1 mg is used for the calibration
Uncertainty of balance is ± 1 mg
Weight of volumetric flask is 200.001g
First measurement
: 449.822 g Measured temperature : 20.2 °C
Second measurement : 450.055 g Measured temperature : 20.1 °C
Third measurement
: 449.892 g Measured temperature : 20.2 °C
76
Preparation of Uncertainty Budget
The volume at 20 °C is given by :

1
V20  ( RL  RE  * 
 w  a
  a 
 * 1 
 * 1   t  20
  b 
Z values are given in Tables B6, B7 and B8 in ISO 4787 : 2010 for
different types of glass at common air pressure Vs temperature
77
Preparation of Uncertainty Budget
Measurement
First
Weight of water (g)
249.821
Second
250.054
Third
249.891
Mean value
249.922
Std. deviation
0.1195
SDOM (Type A )
0.06899 g
78
Preparation of Uncertainty Budget
First measurement
Second measurement
Third measurement
Volume at 20 °C ml
250.55
250.78
250.65
Average volume is 250.66 ml at 20 °C
Uncertainties :
Std. uncertainty of weighing process U1 = 0.06899 g
Weighing uncertainty U2 = cer. Value/2 = 0.0005 g
Balance resolution U3 = half inet./1.7321 = 0.00029 g
79
Preparation of Uncertainty Budget
Sensitivity coefficient:
80
Uncertainty Budget
Source of
uncer.
(quant.)
Uni
t
Typ. Pro.
of
Dist.
Ev.
Uncer. (U
or s)
Repeatabi
lity U1
g
B
Nor.
0.1195
Calibr. U2
g
A
Nor.
0.001
Resolu. U3
g
A
Rec.
0.0005
Div.
Stand.
Uncer.
uc
2
Sen.
Coff.
ci
x uc
=
ui (y)
0.06899
1.003
0.0692
0.4789e-2
0.0005
1.003
0.0005
0.2e-6
0.00028
1.003
0.00028
7.84e-4
Sum
Std. Un.
0.004868
0.0697
k
Exp.
Un.
2
0.14 ml
81
Uncertainty Budget
0.16
0.14
0.12
ml
0.1
0.08
0.06
0.04
0.02
0
Rep.U1
Cal. U2
Res. U3
Exp. Un.
82
Estimation of Standard Uncertainty
Modeling of the measurement process
The measurands are the particular quantities subject to a measurement
Only one mesurand or output quantity Y that depends upon number of
input quantities Xi
Y= f(X1,X2,X3,……Xn)
Y-
measurement result
X1,X2,X3,……Xn - input values
f - functional relationship
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Estimation of Standard Uncertainty
An estimation of the measurand Y, the output estimate
denoted by y, is obtained from the previous equation using
input estimates xi for the values of input quantities Xi as
y = f ( x1, x2, x3,………xn )
The uncertainty of measurement of input estimates are
determined by :
 Type A evaluation
 Type B evaluation
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Type A Evaluation
__
Mean
q
Standard Deviation
1

m
sq 
m
q
k 1
k
m
__
2
(

)
 q q (m  1)
k 1
k
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Type A Evaluation
Standard Deviation of the Mean (SDOM)
sq
__
s

q
m
Standard Uncertainty
u s
A
__
q

s
q
m
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Combined Standard Uncertainty
y = f(x1,x2,x3,……xn)
Law of Propagation of Uncertainties
2
 f 
 f
 U 2 ( x1 )  
U c2 ( y )  
 x1 
 x2
2

 U 2 ( x2 )  ............

2
 f  2
 U ( xn )
......  
 xn 
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Expanded Uncertainty & Coverage Factor
U = k .uc (y)
U- Expanded Uncertainty
Uc (y)- Combined Standard Uncertainty
k- Coverage factor , obtained from the t-distribution corresponding to
the level of confidence desired (95 %)
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Reporting Results
Results are reported in a
“ Calibration Certificate”
or
“Test Report”
Information to be included:
Name and address of laboratory, and the location
Unique identification of test report or calibration certificate
Identification of each page
Name and address of the customer
Description of item, including capacity or range, resolution, serial
number, manufacture and model number, any identification number
etc.
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Reporting of Results
Date of performance of test or calibration
Identification of method used
Environmental conditions
Uncertainty of measurement
Traceability of measurement including reference standards used
 eg. “Set of accuracy class E2 traceable to Primary standards
maintained at Bangladesh Standards and Testing Institution (BSTI)
– certificate number……….”
Name (s), function(s) and signature(s) or equivalent
identification of person(s) authorizing the test or calibration
certificate
Recommendation of re-calibration should not be included
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Presentation of Results
Example : Calibration of Volumetric Glassware
METHOD OF CALIBRATION
The volumetric flask was calibrated generally in accordance with the
method manual Ref. No MM/VO/01 – Calibration of volumetric glassware
by the gravimetric method,
TEST EQUIPMENT USED
Description
Model
Precision Balance
BP 221 S
Liquid in Glass Thermometer
Manufacture
Sartorius
-
-
Digital Pressure Gauge
Model 370
Setra
Liquid :
Deionised Water
Capacity
220g
-10 to 52C
600 to 1100mbar
Resolution
0.1 mg
0.1C
0.01 mbar
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Presentation of Results
Calibration of Results
Nominal capacity
(ml)
Volume at reference
temperature of 20oC
(ml)
Expanded
Uncertainty
U (ml)
100
99.87
0.08
The measurement results can be varied  U
The reported expanded uncertainty in measurement is stated as the standard
uncertainty in measurement multiplied by the coverage factor k = 2, which for a normal
distribution corresponds to a coverage probability of approximately 95 %.
The standard uncertainty of measurement has been determined in accordance with
Guide to expression of uncertainty in measurement (GUM) JCGM 100:2008
Note: The user is obliged to have the flask re-calibrated at appropriate intervals
Authorized by
Authorized Signatory
Designation
Test Performed by
Name
Designation
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page ( ) of ( )
Presentation of Results
NOTE :
Temperature effect
When the temperature at which the glassware is used (t2) differs from the
reference temperature (t1=200C), the corresponding
volume change can be calculated via the following equation.
Where :
is the volume change due to temperature change
is the cubical thermal expansion coefficient of the material by which
is the temperature change
Material
Fused Silica (Quarts)
Borosilicate Glass
Soda-Lime Glass
Coefficient of Cubical Thermal
Expansion
OC-1 *10-6
1.6
9.9
27
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