uncertainty of measurement nihal gunasekara sri lanka

1

Uncertainty of Measurement

Nihal GunasekaraSri Lanka

Bangladesh BEST Programme

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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

4


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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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

a21

Area 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

a1

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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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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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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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


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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Comparison of magnitudes of Standard Uncertainty Components

Std. m

ass

Scale re

s.

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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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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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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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 calculationDetermine 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 mass123456789

10

1000.1431000.1441000.1441000.1461000.1461000.1461000.1441000.1431000.1451000.145


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Uncertainty Budget

Source of uncer. (quant.)

Units Type of Eval.

Pro. Dist. Uncer. (U or s)


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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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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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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Uncertainty Budget


Units Type of Eval.

Pro. Dist. Uncer. (U or s)


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

58


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 =

59


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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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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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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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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Micrometer uncertainty um

Micrometer uncertainty is 3 µm, um = U/k = 3.0/2 µm

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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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Resistance uncertainty uR

Uncertainty of resistance includes several terms

a.Repeatability uncertainty urdg

Type A evaluation is

Sensitivity coefficient crdg is given by

66


b.Bridge reading uncertainty ub is given by :

( Assume rectangular distribution)

Sensitivity coefficient is as in the previous case :


67


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:

68


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 =

69


Uncertainty Budget


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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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

71



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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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

73


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


Uncertainty Budget

Source of uncer. (quant..)

Units Type of evalu.

Prob. Dis. Uncer. (U or s)


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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U (TC) U (dig. Ther.)

U (Rept.) Exp. Uncer.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

°C

76



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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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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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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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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Sensitivity coefficient:

81


Uncertainty Budget


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

82


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


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

84


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

85

m

kkqq m 1

__ 1

Type A Evaluation

Mean

Standard Deviation 1)(m)( 2m

1k

__

kq qqs


86

Standard Deviation of the Mean (SDOM)

mqss q__

Standard Uncertainty

mssu q

A q__

Type A Evaluation


87

y = f(x1,x2,x3,……xn)

............)()()( 22

2

21

22

1

2

xUxfxU

xfyU c

)(...... 2

2

nn

xUxf

Combined Standard Uncertainty

Law of Propagation of Uncertainties


88

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 %)


89

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

90

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

91

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

92

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 ( )

93

Presentation of Results


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

94


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