Mesurement of Uncertainty - Abdelouahhab Salih

8/13/2019 Mesurement of Uncertainty - Abdelouahhab Salih

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The practice of the uncertainty in

measurement

Abd el o u a h h a b SAL I H

Professor of the Higher education

in Mechanical engineering : ENSETRABAT

P r esi d en t o f t h e Team o f Resea r ch I M E

Con su l t a n t : A N PME

Au d i t o r I SO170 25 a t t a ched t o t h e M CI

F u n d er o f C2M T (M o r r o ca n c en t er o f

M et r o l o g y a n d t h e New Tech n o l o g i es)

0 6 61 45 02 0 2

a . s a l i h@yahoo . f r


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Etape 1y=f(x 1 ,x 2,...x n)

Etape 2

Identify Sources

Quantify the constituents

In type A and type B

Etape 3

Etape 4

Expanded uncertainetyU = k uc

c

(y)

noProcessModlisable ?

yes

Numerisation ?1. Analytic

GUM

2. Numerical

Monte-Carlo

Etape 3Generate M

realizations of Y

Etape 2Probability

distributions of xi

3. Synthetic

Mesurement incertainty evaluation

1

2

noyes

yes


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Yes

Yes

intralaboratoire

Mthod

no

uc=saptitude

no

1

Standard

deviation sR is

known ?

Partic ipation in the

aptitude test ?

no

Yes

Method is just ?

Bias validated by the

aptitude test

Bias validated by the

RM

Yes

22

refRC usu += Etape 4Expanded uncertainety

U = k uc

c

(y)

Modify the Method

Or fix a tolerance

no

Modify the Method

Method is apte ?


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I. Introduction

II. Analytical method based on the Guide for the

expression of the uncertainty of measure

(GUM).Stage 1 : Specify the mesurand Y =f(X1; : : : ;XN)

Stage 2 : Identify the sources of uncertainty

Stage 3 : Quantify the constituents of uncertaintyStage 4 : Calculate the expanded uncertainty.

III. Practical applications in the calibrations / tries

of laboratories.

Contents

Analytic Method


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I. IntroductionII. Numerical method based on the method

Monte Carlo.Stage 1 : Specify the mesurand Y =f(X1; : : : ;XN)

Stage 2 : probability distributions for the Xi

Stage 3 : Generated number M of Monte Carlo trials

Stage 4 : Estimate y of Y and calculate expanded

uncertainety.

III. Practical applications.

IV. Comparison with the method GUM.

Contents

Monte-carlo method


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I. Of the interest of the synthetics methodsII. The comparisons interlaboratories

III. Application of the standards ISO 5725 in the calculation

of the uncertainties in the tries of laboratories.Stage 1 : Acquerir the results

Stage 2 : Calculate the averages

Stage 3 : Calculate the average of the averages

Stage 4 : Calculate the standard deviation of every laboratory si

Stage 5 : Cochran test

Stage 6 : Grubbs testStage 7 : Calculate the variance of Repeatability sr

Stage 8 : Calculate the variance of Reproducibility sR

Stage 9 : Calculate the expanded uncertainty U=k.uc

IV. Exploitation of the results of intercomparison.

Contents

Synthetic method : ISO 5725


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I. Application of the standard ISO / TS 19036

II. Intra-laboratory

Stage 1 : Protocol of the essay intralaboratoire

Stage 2 : The 5M of the process of measurement

Stage 3 : Harvest of the data(xiA, xiB)

Stage 4 : Calculation of the transformed functions (yiA, yiB)

Stage 5 : Calculation of the standard deviation of reproducibility sRStage 6 : Calculation of the expanded uncertainty U

Stage 7 : Exploitation of the results.

Contents

Synthetic method : ISO/TS 19036


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1. Definition of measurement uncertainty

measurement uncertainty :

Non-negative parameter characterizing the

dispersion of the quantity values being

attributed to a measurand, based on the

information used.

I. INTRODUCTION


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Systematic Random

true quantity

value

XiX

measurement result

U= k.uc

1. Measurement uncertainty


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2. Use of the mesurement uncertainty

Declaration of conformity with a technical specification(verification)

Comparison of several mesurement results

(aptitude tries)

Confirmation of the choice of mesurement method(validation of the methods)

The choice of a measuring instrument for an expressed(need capability )


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Part 1 :

Analytic methodGUM


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II. Analytical method in 4 stages

Identify the sources of

uncertainty

Specify the mesurandY =f(X1; : : : ;XN)

Quantify the constituents of

uncertainty type A and type B

Calculate the expanded uncertainty

1st stage

2nd stage

3rd stage

4th stage


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cas

1st stage : Specify the mesurandY =f(X1; : : : ;XN) .

1.Physical law : Y = f (X1, X2,,Xn) Where(X1, X2,,Xn ) input quantities Y output quantitie .

Ex : =M/V, P=F/S , P=gh+v2/2, PV=nRT,

Qv=V/t=Cte P, Qm=V/t

2. Empirical law ++= onsinterractieffectyY


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1st stage (2/10)

Mesurement = Comparison

Inconnue

Comparateur talon

traabilitTracability

Unknown

StandardComparator


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1st stage (3/10)

Definition of mesurand

To define the mesurand exactly

is an indispensable operation

Concentration in HR% ?


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1st stage (4/10)

Examples of definitions of mesurands

1) Distance entre le centre de la face

suprieure de la cale et le plan sur

lequel elle est adhre, 20 C et

en posit ion verticale.

2) Distance entre les deux centres

des faces de la cale, 20 C, la

cale tant en posit ion horizontale.

3) Distance entre deux plans parallles, 20 C, la cale tant

en posit ion horizontale.

Distance between the center and superior

face and the plan in 20C and in vertical

position

Distance between both centers of the

faces, in 20C in horizontal position

Distance between two parallel plans in20C in horizontal position


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1st stage (5/10)

Definition of mesurand

To define with enough details the mesurande it is :

Avoid wasting time with the user of the mesurement

result of a bad understanding

Do not introduce causes of uncertainties bound to a"vague" definition of what we want to measure

Choose a process of measure adapted to the mesurand


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1st stage (6/10)

Mathematic model

Mesurement method

Operating mode

Mathematic model


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1st stage (7/10)

Example of writing of the mathematical model

Mesurement method: Measure of the temperature of thewater ( t ) contained in one be toffee-nosed by means ofa thermometer with dilation of liquid.

Operating mode : To place the thermometer and toimmerse it until the line locates, wait 2 minutes then read

(l), apply the correction of calibration indicated in the

certificate (+ C), begin again a the next time of theoperation, the announced mesurement result is the

average of both obtained values.

Modle mathmatique ?


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1st stage (8/10)

Example of writing of the mathematical model

Mathermatic model :

CllClCl

t 22

2121


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1st stage (9/10)

Other example of mathematical model

(volume measurement of water with a balance)

Where M is the mass read on the balance and t

is the temperature read on the thermometer C. is the coefficient of expansion of the glass.

W is the density of the water.

a is the density of the air.

= Ct

aw

MV 2011


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1st stage (10/10)Application : process of mesurement mass

Write the mathematical model of the

determination of the mass of an object, whose

density we know, by means of a simple weighing on

a calibrated balance of laboratory, with standards

masses (OIML)


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Do not focus its attention on the instrument, but beinterested in the process which obtaining of mesurementresult.

The uncertainty characterizes the result not the instrument

In the process are going to intervene : Instruments, standards,

The operators,

The mesurement method and the operating mode, The environment of the measurement (temperature, pressure),

The moderate object,

2nd stage : Identify the sources of uncertainty

(1/3)


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2nd stage (2/3)

ISHIKAWA

We use collectively the diagram of 5M

MEANS METHOD

MATERIAL

(SUBJECT)

MIDDLE :

ENVIRONMENT

Uncertainty

WORKFORCE

(OPERATOR)

2nd stage (3/3)


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Time of flow (oilcans, pipettes,)

Calibation

Repeatability

type of Liquid

Resolution

ExcentrationTemperature

Humidity HR%

Incertainety

Qualification

Skillful -hability

Parallaxe

Temperature

Derive

Forme

Coefficient of expansion

Thickness of the line

Pressure atm.

Exemple : Uncertainty of calibration of the glass of the laboratory

2nd stage (3/3)

Repetability

3rd stage : Q tif th tit t f


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Bu2

Au2u

C

2 +=

Standard uncertainty

Type A method

Type B method

3rd stage : Quantify the constituents ofuncertainty type A and type B

uC

uB

uA


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Two methods :

Type A : Evaluation by means of statisticalmethods

Type B : Evaluation by the other means

The evaluations of type A are based on

probability distributions while the evaluationsof type B are based on laws in priori


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1. Type A method : (Random errors)

uA = Dependent measurements

uA=

1 cas :

uA =

n

2 cas :

uA

=r

12

if =0

Insufficient resolution3 cas :

0

0,250,5

0,751

1,251,5

1,75

22,252,5

2,753

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

/ n

n

Independents measurements


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3rd stageType A method : (Random errors)

Evaluation of the repeatability of mesurement process. Themeasures are dependent.

Observations :

90,040 mm 90,044 mm 90,049 mm 90,046 mm 90,041 mm 90,054mm 90,056 mm 90,052 mm 90,063 mm 90,060 mm

( ) ( ) m9,71

1 10

1

2=

=

=ii xx

nxs


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2. Type B method : (Systematic errors)

2cas : empirical law : Experiment

uB=

1 cas : physical law Law of propagation

of uncertainty

++= onsinterractieffectyYNot of interractions

ii xayY +=

Law of propagation of

uncertainty

uBuA

, /n or r/12

L f i f i


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Law of propagation of uncertainty

= +==

+=1

1 1

22

1

222 ....2.n

i

n

ij

jiijji

n

i

iiB uurccucu

= +==

+

=

1

1 11

2

2

2 )().().,(.2)(.n

i

n

ij

jiji

ji

n

i

i

i

B xuxuxxrx

f

x

fxu

x

fu

nCorrelatioxxr ji :1),(1 Where

We put the sensibility coefficienti

i x

f

c

=


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physic law : Particular case: law in the

form of product (chemists)

=

=

n

i

Bxi

xiuyYu

1

2)(

*)(

=

=

n

i

B

xi

xiu

y

Yu

1

2)()(

Absolute uncertainty

Relative uncertainty

2 hypotheses : The function is a product

Factors are independent


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3rd stageQuantification of the standard uncertainty u(xi)

An evaluation of the standard uncertainty is made by

a scientific judgment based on all the available

information which can understand :

Results of previous measurements

The experience or the behavior of materials and used instruments

Specifications of the manufacturer

Data supplied by certificates of calibration and the other

documents The uncertainty assigned to reference values


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Two informations are necessary :

Range

Form

3rd stageQuantification of the standard uncertainty u(xi)


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3rd stageProbability density functions (PDF)

Gaussian distribution : range : 2a 99,73% 3

variance a2 / 9

standard deviation a / 3

-a a


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Rectangular distributions : range 2a

variance a2 / 3

standard deviation a / 3

-a a

3rd stageForms of usual probability distributions


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Arc sine ( en U ):

Range 2a

variance a2 / 2

standard deviation 2/a

a-a

3rd stageForms of usual probability distributions


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Distribution

a prioriMethod of

calculation

Real case

RectangularDigital resolution, hystrsis,

Instrument is conform to a class,

Homogeneity and stability of an

middle of comparison,

Arc sine Temperature of an environment,

Gaussian

distributionUncertainty calculated from the

method spc

3/au =

2/au =

3/au =


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One micrometer in 1/1000 is in accordance with the

class I. Its e.m.t. is 7m.

We suppose that the corresponding uncertaintyfollows a rectangular distributions :

u (class) = 7 / 1,732 = 4,04 m

3rd stageExample 1


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A manometer with dial in a resolution of 0,05 bar,

We suppose that the corresponding uncertainty

follows a rectangular distributions :

u (resolution) = 0,025 / 1,732 = 0,014 bar

3rd stageExample 2


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A manometer with dial in a resolution of 0,05 bar,


follows a rectangular distributions :

u (resolution) = 0,025 / 1,732 = 0,014 bar

3rd stageExample 2


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A laboratory is regulated in temperature in 2 C


follows a arc sine distributions :

u (temperature ) = 2 / 1,414 = 1,41 C

3rd stageExample 3


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A laboratory is regulated in temperature in 2 C


follows a arc sine distributions :

u (temperature ) = 2 / 1,414 = 1,41 C

3rd stageExample 3


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A glass of laboratory presents an error ofindication of +0,03 ml, its certificate ofcalibration mentions an uncertainty of

0,08 ml to 2 uncertainties-types :We suppose that the corresponding uncertaintyfollows a rectangular distributions

u (correction) = 0,08 / 2 = 0,04 ml

3rd stageExample 4


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Several operators realize a measure of strengthwith a spring dynamometer.

The maximal distance from the values observed bythe diverse operators is 3 N.

u (oprateur) = 1,5 / 3 = 0,5 N

3rd stageExample 5


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Several operators realize a measure of strengthwith a spring dynamometer.

The maximal distance from the values observed bythe diverse operators is 3 N.

u (oprateur) = 1,5 / 3 = 0,5 N

3rd stageExample 5

4th stage : Calculate the expanded uncertainty


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U = k. uck coverage factor.

According to the GUM the standard ENV 13005.

For an interval of 95,45 % confidence this factor is

equal to 2.

The final result :

Y = y U

4th stage : Calculate the expanded uncertainty

Unit


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Applications of calculation of the

uncertainties

analytical method

Application 1 :

Mesure of Concentration of cadmium

Documents

Mesurement of Uncertainty - Abdelouahhab Salih