uncertainty of measurement nihal gunasekara sri lanka bangladesh best programme 1
TRANSCRIPT
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Uncertainty of Measurement
Nihal GunasekaraSri Lanka
Bangladesh BEST Programme
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Bangladesh BEST Programme
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
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What do you need for a measurement ?
InstrumentRulersStopwatchesWeighing scalesThermometers
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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
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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
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What is uncertainty of measurement ?
The uncertainty of measurement tells us something about its quality
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 !!!!!
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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
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Measurement Uncertainty
U
X
U
A range containing the true value
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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
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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
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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 !!!!
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Why is uncertainty of measurement important?
“ We wish to make good quality measurement and to understand the result”
ISO 17025 requirementsCalibrations & Testing laboratories shall have a procedure for calculation of MUWhere not possible for some test methods of testing labs, the contributing factors need to be identified and a reasonable estimation be madeWhen estimating MU all components that contribute to MU should be taken into account
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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 !!!!
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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
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Getting the Best Estimate
Repeated measurements give different answers
If there is variation in readings when they are repeated
Take many readings Get the average
Best estimate for the “true” value
Value of reading
Mean or average value
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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
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Standard Deviation – Spread of Readings
Repeated measurements : different readings
How widely spread the readings are ?
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”
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Calculating an Estimated Standard DeviationExample :
Let the readings are 16, 19,18, 16, 17, 19,20,15,17, and 13 Average is 17Find the difference between each reading and the average ie. -1 +2 +1 -1 0 +2 +3 -2 0 -4And square each of thoseie 1 4 1 1 0 4 9 4 0 16Find 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
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Mathematical Equation for Standard Deviation
1
2
n
rrs i
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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.
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Distribution – Shape of ErrorsThe spread of set of values can take different forms
Mean or average reading
Value of reading
Probability of occupation
Normal or Gaussian distribution
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Uniform or Rectangular Distribution
When measurements are quite evenly spread between the highest and lowest values a rectangular or uniform distribution is produced
Range
Value of reading Value of reading
Probability of occurrence
Full width
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Triangular Distribution
Probability of occurrence
Value of reading
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What is not a Measurement Uncertainty ?
Mistakes made by a operator
Tolerances of a product
Specifications of instruments
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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
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Specify Measurand
Identify Uncertainty sources
Simplify by grouping the sources covered by available data
Quantify grouped and remaining components
Convert components to standard uncertainties
STEP 1
STEP 2
STEP 3
How to Calculate Uncertainty of Measurement
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Calculate the combined standard
Uncertainty
Review and if required re-evaluate large components
Calculate the Expanded Uncertainty
STEP 4
How to calculate Uncertainty of measurement
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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.
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Type B Evaluation
Category may be derived from:
Previous measurement dataExperience with or general knowledge of the behaviour and properties of relevant materials and instrumentsManufacture’s specificationsData provided in calibration and other certificatesUncertainties assigned to reference data taken from handbooks
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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 =
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Bangladesh BEST Programme
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 lower limits
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Rectangular Distribution
f(x)
x
2a
a a
a2
1Area enclosed by
rectangle = 1
2 aa
aa
Upper limitLower limit
Best estimate
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There are simple mathematical expressions to evaluate the standard deviation for this. Another such distribution we normally encounter is the triangular distribution
a a
2 aa
a a
a
1 xf
x
Area enclosed by
Triangle=1
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Confidence Level
Gaussian probability distribution
-ks +ks68% Within 1s of mean k = 195% Within 2s of mean k = 299% Within 3s of mean k = 3
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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 =
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Coverage FactorThe 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)
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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 gThe expanded uncertainty is 0.00079 gThe result of measurement is expressed as 100.02147 g ± 0.00079 g and the coverage factor k = 2
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Bangladesh BEST Programme
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”
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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
uncertainties and address them
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Some Good Measurement Practices
Follow the manufacture’s instruction for using and maintaining instrumentsUse experienced staff and provide training Validate softwareCheck raw data by a third partyKeep good records of your measurements and calculations
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Preparation of Uncertainty BudgetsExample: Calculation of uncertainty of a balance calibrationCapacity of balance : 50 gResolution of balance : 0.1 mgMeasured max. Std. deviation : 0.0939 mgNumber of measurements :10Task : Calibration of scale value of 45 gMethod : A combination of three masses are required
Mass Value U95 (mg) k u (mg)
1 20.000088 g 0.019 2 0.0095 2 19.999995 g 0.019 2 0.0095 3 5.000030 g 0.0043 2 0.0045 Total 45.000113 g 0.0235
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Bangladesh BEST Programme
Preparation of Uncertainty Budget
Observations:1st zero reading : 0.0000 g 1st reading of standard mass : 45.0003 g2nd reading of standard mass : 45.0003 g2nd zero reading : 0.0001 g
Calculations:Mean zero reading ( zi ): 0.00005 gMean reading on standard mass ( ri ) : 45.00030 g
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Bangladesh BEST Programme
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 zi is the mean of two no-load (zero) 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)
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ss
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Uncertainty Budget
Source of uncer. (quant..)
Units Type of evalu.
Prob. Dis. Uncer. (U or s)
Divisor Stand. Uncer. uc
Cal. Uncer. umass
mg B Normal 0.0235 1 0.0235 0.00055
Resolution uresolution
mg B Rect. 0.1/2 0.02887 0.00083
Repeatability
urepeatability
mg A Normal 0.0939 0.02972 0.00088
Sum 0.00226
Comb. std uncer.
0.0475 mg
Cov. Fac. k 2
Expan.uncr 0.095 mg
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Bangladesh BEST Programme
Comparison of magnitudes of Standard Uncertainty Components
Std. m
ass
Scale
res.
Bal. re
peat.
Expa.
Uncer.
0
0.02
0.04
0.06
0.08
0.1
mg
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Calibration and Measurement Capability (CMC)HistoryIn 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
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Bangladesh BEST Programme
Calibration and Measurement Capability (CMC)
The scope of accreditation of an accredited laboratory shall include CMC expressed in terms of:
MeasurandCalibration/measurement/performance methodMeasurement rangeUncertainty of measurement
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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%”
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Bangladesh BEST Programme
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”
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Calibration and Measurement Capability (CMC)Example:
SWEDAC
Measured Quantity
Method of Calibration
Range Readability 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
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Bangladesh BEST Programme
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
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Example of uncertainty calculation
Determine the weight of 1kg
Mean Value : 1000.1446 gStandard deviation : 0.0011 gEstimated Standard deviation of mean : 0.0011/√10=0.00035 g
Observation Value of test mass
123456789
10
1000.1431000.1441000.1441000.1461000.1461000.1461000.1441000.1431000.1451000.145
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Bangladesh BEST Programme
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 2 2.5 6.25
Resolution uresolution
mg B Rect. 0.5x 0.4082 0.1666
Repeatability
urepeatability
mg A Normal 1.1 0.35 0.1225
Sum 6.539
Comb.std uncer.
2.55 mg
Cov. Fac. k 2
Exp. uncer. 5.1 mg
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Bangladesh BEST Programme
Presentation of Results
The result is reported as: The value of the test mass = 1000.145 gExpanded uncertainty = ± 0.005 g with k=2 at 95% confidence level orThe value of test mass is 1000.145 g ± 0.005 g with k=2 at 95% confidence level
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Bangladesh BEST Programme
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 °CTest oven used with a resolution of 1 °C
The standard deviation of 10 readings obtained at 100 °C is 0.6 °C
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Bangladesh BEST Programme
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 2 0.25 0.0625
Dig. Ther. uresolution
°C B Rect. 0.1/2 0.0289 0.00084
Dig. Ther.urepeatability
°C A Normal 0.6 0.190 0.0361
Dig. Ther.U cjc
°C B Rect. 0.2 0.1156 0.0134
Test Ovenuresolution
°C B Rect. 1/2 0.289Sum
Co. Std. u
0.08350.19630.44 °C
Cov. Fac. k 2
Exp. uncer. 0.9 °C
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Dig. Th. U tc
Dig. Th. Ures
Dig.Th. Urep
Dig. Th. U cjc
Tes. Ov. Ures
Exp. Un.0
0.10.20.30.40.50.60.70.80.9
1
Comparison of Magnitudes of Standard Uncertainty Components
°c
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Bangladesh BEST Programme
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 =
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Bangladesh BEST Programme
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”
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Bangladesh BEST Programme
Preparation of Uncertainty BudgetExample 3: Measurement of resistivity of a rod using the following equation
Where : R is the rod resistance in ohms l is the length of the rod in meters A is the cross sectional area of the rod in m d is the diameter of the rod in m
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Bangladesh BEST Programme
Uncertainty Budget
Input Data :Distance between knife degrees : 1.00003 m , unce. ± 0.01 mm, 95% CLMeasured diameter of the rod : 6.001 mmNo. of measurements of diameter : 10Estimated std. dev. Of diameter : 0.25 µmMicrometer uncertainty : ±3 µm at 95% CL
Measurement Data :Mean resistance : 604.44 µΩNo. of resistance measurements : 5Estimated std. dev. : 0.3 µΩBridge reading uncertainty : ±1 µΩRod temperature : 20 ± 0.05 °C
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Bangladesh BEST Programme
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
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Bangladesh BEST Programme
Uncertainty Components and their Evaluation
Micrometer uncertainty um
Micrometer uncertainty is 3 µm, um = U/k = 3.0/2 µm
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Bangladesh BEST Programme
Uncertainty Components and their Evaluation
Rod length uncertainty ul
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 :
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Bangladesh BEST Programme
Uncertainty Components and their Evaluation
Resistance uncertainty uR
Uncertainty of resistance includes several terms
a.Repeatability uncertainty urdg
Type A evaluation is
Sensitivity coefficient crdg is given by
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b.Bridge reading uncertainty ub is given by :
( Assume rectangular distribution)
Sensitivity coefficient is as in the previous case :
Uncertainty Components and their Evaluation
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Uncertainty Component and their EvaluationC. Resistance temperature uncertainty uT
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:
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Bangladesh BEST Programme
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 =
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Bangladesh BEST Programme
Uncertainty Budget
Source of uncer. (quant.)
Unit
Typ. of Ev.
Pro. Dist.
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
SumStd. Un.
3.62e-221.9e-11 Ωm
kExp. Un.
23.8e-11 Ωm
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Bangladesh BEST Programme
Rod. Dia.Ud
Mic. Cal. Um
Len. Ul Res. Urdg Brg. Ub R.tem. Ut Exp. Un U
0
5
10
15
20
25
30
35
40
pΩm
Comparisons of Magnitudes of Standard Uncertainty Components
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Bangladesh BEST Programme
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 °CMeasurement accuracy : ±0.6 °C
TC calibration certificate provides :Uncertainty is ± 2.0 °C at 95% confidence levelCorrection at 500 °C is 0.5 °C
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Bangladesh BEST Programme
Preparation of Uncertainty Budget
Measure temp. (T) = Displayed temp. + Correction
Calculation of uncertainty components
Urept - standard uncertainty in the repeatability of the measured resultsUdig -standard uncertainty in the digital thermometer Utc - standard uncertainty in the thermocouple
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Preparation of Uncertainty BudgetMeasurement record: Measurement Temperature °C 1 500.1 2 500.0 3 501.1 4 499.9 5 4 99.9 6 500.0 7 500.1 8 500.2 9 499.9 10 500.0
Mean value is 500.02 Standard deviation s is 0.103 °C Standard deviation of mean SDOM is 0.03 °C (Type A)
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ss
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Uncertainty Budget
Source of uncer. (quant..)
Units Type of evalu.
Prob. Dis. Uncer. (U or s)
Divisor Stand. Uncer. uc
Cal. Uncer. utc
°C B Normal 1 2 0.5 0.25
Cal. Uncer.udig
°C B Rect. 0.6 0.349 0.1223
Repeat.Urep.
°C A Normal 0.103 0.326 0.0011
Sum 0.3734
Comb. std uncer.
0.61 °C
Cov. Fac. k 2
Expan.uncr 1.2 °C
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Bangladesh BEST Programme
U (TC) U (dig. Ther.)
U (Rept.) Exp. Uncer.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
°C
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Bangladesh BEST Programme
Preparation of Uncertainty Budget
Example 5: Calibration of 250 ml volumetric flask
A balance with a resolution of 1 mg is used for the calibrationUncertainty of balance is ± 1 mgWeight of volumetric flask is 200.001gThree readings are obtained:
First measurement : 449.822 g Measured temperature : 20.2 °CSecond measurement : 450.055 g Measured temperature : 20.1 °CThird measurement : 449.892 g Measured temperature : 20.2 °C
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Bangladesh BEST Programme
Preparation of Uncertainty Budget
The volume at 20 °C is given by :
201*1*1
*(20
tRRVb
a
awEL
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
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Bangladesh BEST Programme
Preparation of Uncertainty Budget
Measurement Weight of water (g) First 249.821
Second 250.054
Third 249.891
Mean value 249.922
Std. deviation 0.1195
SDOM (Type A ) 0.06899 g
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Bangladesh BEST Programme
Preparation of Uncertainty Budget
Volume at 20 °C mlFirst measurement 250.55Second measurement 250.78 Third measurement 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
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Preparation of Uncertainty Budget
Sensitivity coefficient:
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Bangladesh BEST Programme
Uncertainty Budget
Source of uncer. (quant.)
Unit
Typ. of Ev.
Pro. Dist.
Uncer. (U or s)
Div. Stand. Uncer. uc
Sen. Coff.ci
x uc
=ui (y)
Repeatability U1
g B Nor. 0.1195 0.06899 1.003 0.0692 0.4789e-2
Calibr. U2 g A Nor. 0.001 2 0.0005 1.003 0.0005 0.2e-6
Resolu. U3 g A Rec. 0.0005 0.00028 1.003 0.00028 7.84e-4
SumStd. Un.
0.0048680.0697
kExp. Un.
20.14 ml
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Bangladesh BEST Programme
Uncertainty Budget
Rep.U1 Cal. U2 Res. U3 Exp. Un.0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
ml
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Estimation of Standard Uncertainty
Modeling of the measurement process
Y= f(X1,X2,X3,……Xn) Y- measurement result
X1,X2,X3,……Xn - input values
f - functional relationship
Bangladesh BEST Programme
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
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Bangladesh BEST Programme
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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m
kkqq
m 1
__ 1
Type A Evaluation
Mean
Standard Deviation 1)(m)( 2m
1k
__
kq qqs
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Standard Deviation of the Mean (SDOM)
mqs
s q__
Standard Uncertainty
mssu
q
A q__
Type A Evaluation
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y = f(x1,x2,x3,……xn)
............)()()( 22
2
21
2
2
1
2
xUx
fxU
x
fyU c
)(...... 2
2
n
n
xUx
f
Combined Standard Uncertainty
Law of Propagation of Uncertainties
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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 %)
Bangladesh BEST Programme
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Reporting ResultsBangladesh BEST Programme
Results are reported in a “ Calibration Certificate” or “Test Report”Information to be included:Name and address of laboratory, and the locationUnique identification of test report or calibration certificateIdentification of each page Name and address of the customerDescription of item, including capacity or range, resolution, serial number, manufacture and model number, any identification number etc.Condition of received
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Reporting of ResultsBangladesh BEST Programme
Request received dateDate of performance of test or calibrationIdentification of method usedEnvironmental conditionsUncertainty of measurementTraceability 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 certificateRecommendation of re-calibration should not be included
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Presentation of ResultsBangladesh BEST Programme
Example : Calibration of Volumetric GlasswareMETHOD OF CALIBRATIONThe 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 USEDDescription Model Manufacture Capacity Resolution
Precision Balance BP 221 S Sartorius 220g 0.1 mg
Liquid in Glass Thermometer - - -10 to 52C 0.1C
Digital Pressure Gauge Model 370 Setra 600 to 1100mbar 0.01 mbar
Liquid : Deionised Water
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Presentation of ResultsBangladesh BEST Programme
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 UThe 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 Test Performed by
Authorized Signatory Name Designation Designation page ( ) of ( )
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Presentation of Results
Bangladesh BEST Programme
Material Coefficient of Cubical Thermal Expansion
OC-1 *10-6
Fused Silica (Quarts) 1.6Borosilicate Glass 9.9Soda-Lime Glass 27
NOTE : Temperature effectWhen 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
the glassware is made is the temperature change
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Bangladesh BEST Programme