The Expression of Uncertainty in Measurement

Bunjob SuktatJICA Uncertainty Workshop

January 16-17, 2013Bangkok, Thailand

Acceptance of the Measurement Results

Contents

• Introduction • GUM Basic Concepts• Basic Statistics• Evaluation of Measurement Uncertainty• How is Measurement Uncertainty estimated?• Reporting Result• Conclusions and Remarks

Introduction

• Guide to the Expression of Uncertainty in Measurement was published by the International Organization for Standardization in 1993 in the name of 7 international organizations

• Corrected and reprinted in 1995• Usually referred to simply as the “GUM”

International Organisations

Guide to the Expression of Uncertaintyin Measurement (1993)

BIPM - International Bureau of Weights and Measures http//: www.bipm.orgIEC - International Electrotechnical Commision http//: www.iec.chIFCC - International Federation of Clinical Chemistry http//: www.ifcc.orgIUPAP - International Union of Pure and Applied Physics http//: www.iupap.org

ISO - International Organisation for Standardisation http//: www.iso.ch

IUPAC - International Union of Pure and Applied Chemistry http//: www.iupac.org

OIML - International Organisation for legal metrology http//: www.oiml.org

Every measurement is subject to some uncertainty.

A measurement result is incomplete without a statement of the uncertainty.

When you know the uncertainty in a measurement, then you can judge its fitness for purpose.

Understanding measurement uncertainty is the first step to reducing it

Basic concepts

Introduction to GUM

• When reporting the result of a measurement of a physical quantity, it is obligatory that some quantitative indication of the quality of the result be given so that those who use it can assess its reliability.

• Without such an indication, measurement results can not be compared, either among themselves or with reference values given in the specification or standard. GUM 0.1

Stated Purposes

• Promote full information on how uncertainty statements are arrived at

• Provide a basis for the international comparison of measurement results

Benefits

• Much flexibility in the guidance• Provides a conceptual framework for

evaluating and expressing uncertainty• Promotes the use of standard terminology

and notation• All of us can speak and write the same

language when we discuss uncertainty

Uses of MU

• QC & QA in production• Law enforcement and regulations• Basic and applied research• Calibration to achieve traceability to national

standards• Developing, maintaining, and comparing

international and national reference standards and reference materials

–GUM 1.1

R1 R2

After uncertainty evaluation

R1 R2

No uncertainty evaluation

(only precision)

R1 R2

10.5

11.5

11.0

12.0

12.5

mg k

g-1

value

Are these results different?

En-score according to GUM

)( 22reflab

reflab

uu

xxEn

“Normalized” versus ...

propagated combined uncertainties

Performance evaluation:0 <|En|< 2 : good2 <|En|< 3 : warning preventive action |En|> 3 : unsatisfactory corrective action

What is Measurement?

Measurement is ‘Set of operations having the object of

determining a value of a quantity.’ ( VIM

2.1 )

Note: The operations may be performed automatically.

Basic concepts

• Measurement– the objective of a measurement is to

determine the value of the measurand, that is, the value of the particular quantity to be measured

• a measurement therefore begins with – an appropriate specification of the

measurand– the method of measurement and– the measurement procedure

GUM 3.1.1

Principles of Measurement

Method of

ComparisonResultDUT

Standard

Basic concepts

• Result of a measurement – is only an estimate of a true value and only

complete when accompanied by a statement of uncertainty.

– is determined on the basis of series of observations obtained under repeatability conditions

• Variations in repeated observations are assumed to arise because influence quantities

GUM 3.1.2

GUM 3.1.4

Gum 3.1.5

Influence quantity

• Quantity that is not the measurand but that affects the result of measurement.

Example : temperature of a micrometer used to measure length.

( VIM 2.7 )

What is Measurement Uncertainty?

• “parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand” – GUM, VIM

• Examples:– A standard deviation (1 sigma) or a multiple of it

(e.g., 2 or 3 sigma)– The half-width of an interval having a stated level

of confidence

• The uncertainty gives the limits of the range in which the “true” value of the measurand is estimated to be at a given probability..

Measurement result = Estimate ± uncertainty

(22.7 ± 0.5) mg/kg

The value is between 22.2 mg/kg and 23.2 mg/kg

Uncertainty

Measurement Error

Measured Value True Value

Real Number System

Measurement Error

Measured values are inexact observations of a true value.

The difference between a measured value and a true value is known as the measurement error or observation error.

Basic concepts

• The error in a measurement– Measured value – True value.–This is not known because:

• The true value for the measurand–This is not known–The result is only an estimate of a true value and only

complete when accompanied by a statement of uncertainty.

GUM 2.2.4

GUM 3.2.1

Random & Systematic Errors

• Error can be decomposed into random and systematic parts

• The random error varies when a measurement is repeated under the same conditions

• The systematic error remains fixed when the measurement is repeated under the same conditions

Random error

• Result of a measurement minus the mean result of a large number of repeated measurement of the same measurand.

( VIM 3.13 )

Random Errors

• Random errors result from the fluctuations in observations

• Random errors may be positive or negative

• The average bias approaches 0 as more measurements are taken

Random error

• Presumably arises from unpredictable temporal and spatial variations

• gives rise to variations in repeated observations

• Cannot be eliminated, only reduced.

GUM 3.2.2

Systematic Errors

Mean result of a large number of repeated measurements of the same measurand minus a true value of the measurand.

( VIM 3.14 )

Systematic Errors• A systematic error is a consistent deviation in

a measurement• A systematic error is also called a bias or an

offset

• Systematic errors have the same sign and magnitude when repeated measurements are made under the same conditions

• Statistical analysis is generally not useful, but

rather corrections must be made based on experimental conditions.

Systematic error • If a systematic error arises from a recognized

effect of an influence quantity – the effect can be quantified– can not be eliminated, only reduced.– if significant in size relative to required

accuracy, a correction or correction factor can be applied to compensate

– then it is assumed that systematic error is zero.

GUM 3.2.3

Basic concepts

Systematic error • It is assumed that the result of a measurement

has been corrected for all recognised significant systematic effects

GUM 3.2.4

Measurement Error

Systematic error

Random error

Correcting for Systematic Error• If you know that a substantial systematic error exists and

you can estimate its value, include a correction (additive) or correction factor (multiplicative) in the model to account for it

• Correction - Value that , added algebraically to the uncorrected result of a measurement , compensates for an assumed systematic error

(VIM 3.15)• Correction Factor - numerical factor by which the

uncorrected result of a measurement is multiplied to compensate for systematic error.

[VIM 3.16]

Uncertainty • The result of a measurement after

correction for recognized systematic effects is still only an estimate of the value of the measurand because of the uncertainty arising;– from random effects and – from imperfect correction of the result for

systematic effects

GUM 3.3.1

Classification of effects and uncertainties

• Random effects• Unpredictable variations of influence quantities• Lead to variations in repeated measurements• Expected value : 0• Can be reduced by making many measurement

• Systematic effects• Recognized variations of influence quantities• Lead to BIAS in repeated measurements• Expected value : unknown• Can be reduced by applying a correction which

carries an uncertainty

bunjob_ajchara 33

• It is important not to confuse the terms error and uncertainty

• Error is the difference between the measured value and the “true value” of the thing being measured

• Uncertainty is a quantification of the doubt about the measurement result

• In principle errors can be known and corrected • But any error whose value we do not know is a

source of uncertainty.

Error versus uncertainty

Blunders

• Blunders in recording or analysing data can introduce a significant unknown error in the result of a measurement.

• Measures of uncertainty are not intended to account for such mistakes

GUM 3.4.7

Basic Statistics

Slide 7

Population and Sample

• Parent Population

The set of all possible measurements.

• Sample

A subset of the population - measurements actually made.

Population

Bag of Marbles

Handful of marbles from the bag

Samples

Histograms• When making many

measurements, there is often variation between readings. Histogram plots give a visual interpretation of all measurements at once.

• The x-axis displays a given measurement and the height of each bar gives the number of measurements within the given region.

• Histograms indicate the variability of the data and are useful for determining if a measurement falls outside of “specification”.

For a large number of experiment replicates the results approach an ideal smooth curve called the GAUSSIAN or NORMAL DISTRIBUTION CURVE

Characterised by:

The mean value – x

gives the center of the distribution

The standard deviation – s

measures the width of the distribution

Average

• The most basic statistical tool to analyze a series of measurements is the average or mean value :

n

xx i

“Sum of” Individual measurement

Number of measurements

5.123

5.121510

x

The average of the three values 10, 15and 12.5 is given by:

Deviation

Need to calculate an average or “standard” deviation

To eliminate the possibility of a zero deviation, we square di

xxd ii Deviation = individual value – avg value

Standard Deviation• The average amount that each measurement

deviates from the average is called standard deviation (s) and is calculated for a small number of measurements as:

1

)( 2

n

xxs i xi = each measurement

= averagen = number of measurements

Note this is called root mean square: square root of the mean of the squares

Sum of deviation squared

x

Standard Deviation

Standard DeviationFor example, calculate the standard deviation of the following measurements: 10, 15 and 12.5 (avg = 12.5)

5.22

5.12

2

)5.2()5.2(

1

5.125.125.12155.1210

1

)(

22

2222

nn

xxs i

10.0 12.5 15.0

The values deviate on average plus or minus 2.5 :12.5 ± 2.5

• Variance

• Relative standard deviation

• Percent RSD or Coefficient of Variation (CV)

x

sRSD

Other ways of expressing the precision of the data:

Variance = s2

100x

s%RSD

Standard Deviation of the Mean

n

s

The uncertainty in the best measurement is given by the standard deviation of the mean (SDOM)

Gaussian Distribution

• Given a set of repeated measurements which have random error.

• For the set of measurements there is a mean value.

• If the deviation from the mean for all the measurements follows a Gaussian probability distribution, they will form a “bell-curve” centered on the mean value.

• Sets of data which follow this distribution are said to have a normal (statistical) distribution of random data.

POPULATION DATA

For an infinite set of data,

n → ∞ x → µ and s → σ

population mean population std. dev.

The experiment that produces a small standard deviation is more precise .

Remember, greater precision does not imply greater accuracy.

Experimental results are commonly expressed in the form:

mean standard deviation

sx

_

22 /2)(xe2

1y σμ

πσ

The Gaussian curve whose area is unity is called a normal error curve.

µ = 0 and σ = 1

The Gaussian curve equation:

πσ 2

1= Normalisation factor

It guarantees that the area under the curve is unity

+3s-3s

+2s-2s

+1s-1s

Normal Error Curve

• 68.3% of measurements will fall within ± s of the mean.

m

xi

Rel

ativ

e fr

eque

ncy,

dN

/ N

• 95.5% of measurements will fall within ± 2s of the mean.

• 99.7% of measurements will fall within ± 3s of the mean.

EXAMPLE

Replicate results were obtained for the measurement of a resistor. Calculate the mean and the standard deviation of this set of data.

Replicate ohms

1 752

2 756

3 752

4 751

5 760

Replicate ohms

1 752

2 756

3 752

4 751

5 760

n

xx i_

2.7545

760 751 752 756 752

2

i

1n

xxs

15

754.2760754.2751754.2752754.2756754.2752 22222

4

8.52.32.28.12.2 22222

77.3 NB DON’T round a std dev. calc until the very end.

69.15

77.3

n

s

Also:

x

sRSD

100x

s%RSD

754

3.77

100754

3.77

3.77 2

754.2x 3.77s

2 Variance s

69.1

Student's t-Distribution • If the sample size is not large enough, say

n ≤ 30.• Then the distribution of is not normal.• It has a distribution called Student’s t-

distribution.

t = (x – )/(s/n).

x

Student's t-Distribution

• The Student's t-distribution was discovered by W. S. Gosset in 1908.

• He used the pseudonym ‘Student’ to avoid getting fired for doing statistics on the job!!

Student's t-Distribution

• The shape of the Student's t-distribution is very similar to the shape of the standard normal distribution.

• The Student's t-distribution has a (slightly) different shape for each possible sample size.

• They are all symmetric and unimodal.• They are all centered at 0.

Student's t-Distribution

• They are somewhat broader than normal• distribution, reflecting the additional

uncertainty resulting from using s in place of .

• As n gets larger and larger, the shape of the t-distribution approaches the standard normal.

Degrees of Freedom

• If the sample size is n, then t is said to have n – 1 degrees of freedom.

• We use df to denote degrees of freedom.

Student's t-Distributionfor 95% Confident level

Student's t-Distribution

t x s n

• When s is estimated from the sample standard deviation , s

• The distribution for the mean follows a

t- distribution with degrees of freedom, n − 1

1

)( 2

n

xxs i

x

CONFIDENCE INTERVAL

n

tsx

The confidence interval is given by:

Where t is the value of student’s t taken from the table

The confidence interval is the expression stating that the true mean, µ, is likely to lie within a certain distance from the measured mean, x

Degrees of Freedom1 3.0777 6.314 12.706 31.821 63.6572 1.8856 2.9200 4.3027 6.9645 9.9250. . . . . .. . . . . .

10 1.3722 1.8125 2.2281 2.7638 3.1693. . . . . .. . . . . .

100 1.2901 1.6604 1.9840 2.3642 2.62591.282 1.6449 1.9600 2.3263 2.5758

0.80 0.90 0.95 0.98 0.99

Use of t-Table 95% confidence interval; n = 11

112281.2 :

sx

bunjob_ajchara 67

Example:

The mercury content in fish samples were determined as follows: 1.80, 1.58, 1.64, 1.49 ppm Hg. Calculate the 50% and 90% confidence intervals for the mercury content.

n

tsx_μ

50% confidence:

t =0.765 for n-1 = 3

4

0.131 0.7651.63μ

05.01.63μ

There is a 50% chance that the true mean lies between 1.58 and 1.68 ppm Hg.

Find x = 1.63

s = 0.131

n

tsx_μ

90% confidence:

t = 2.353 for n-1 = 3

4

0.1312.3531.63 μ

15.01.63 μ

There is a 90% chance that the true mean lies between 1.48 and 1.78 ppm

1.63

1.68

1.48

1.58

1.78

90%

50%

131.0

63.1

s

x

Evaluation of Measurement Uncertainty

bunjob_ajchara 70

Terms specific to the GUM

• Standard uncertainty, – the uncertainty of the result of a measurement

expressed as a standard deviation

• Type A evaluation (of uncertainty) – method of evaluation of uncertainty by the statistical

analysis of a series of observations

• Type B evaluation (of uncertainty) – method of evaluation of uncertainty by means other than

the statistical analysis of series of observations

GUM 2.3.1

GUM 2.3.2

GUM 3.2.3

Terms specific to the GUM

• Combined standard uncertainty – the standard deviation of the result of a

measurement when the result is obtained from the values of a number of other quantities.

– It is obtained by combining the individual standard uncertainties (and covariances as appropriate), using the law of propagation of uncertainties, commonly called the "root-sum-of-squares" or "RSS method.

GUM 2.3.4

Terms specific to the GUM

• expanded uncertainty – quantity defining an interval about the

result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand.

• coverage factor, k– numerical factor used as a multiplier of

combined standard uncertainty in order to obtain expanded uncertainty

GUM 3.2.5

GUM 3.2.6

Process of Uncertainty Estimation

• Specify Measurand

• Identify all Uncertainty Sources

• Quantify Uncertainty Components

• Calculate Combined Uncertainty

bunjob_ajchara 75

Specify the Measurand

The measurand?

GUM 1.2

Measurand = particular quantity subject to

measurement

[VIM 2.6 / GUM B.2.9]

Example: the conventional mass of a 1kg

weight.

Measurement Model

• Define the measurand – the quantity subject to measurement

• Determine a mathematical model, with input quantities, X1,X2,…,XN, and (at least) one output quantity,Y.

• The values determined for the input quantities are called input estimates and are denoted by x1,x2,…,xN.

• The value calculated for the output quantity is called the output estimate and denoted by y.

78

Identify all Uncertainty Sources

2. How is MU estimated?

ISO/IEC 17025

• 5.4.7.2– attempt to identify all the components of

uncertainty• 5.4.7.3

– All uncertainty components which are of importance shall be taken into account

Sources of uncertainty

ISO/IEC 17025 5.4.7.3 Note 1: Some sources contributing to the uncertainty:

– reference standards– reference materials– methods– equipment– environmental conditions – properties and condition of the item to be tested – the operator

81

Sources of MU• Incomplete definition of the measurand• Imperfect realisation of the definition of the measurand• Non-representative sampling• Effects of environmental conditions on the measurement• Personal bias in reading analogue instruments• Finite instrument resolution or discrimination threshold• Inexact values of measurement standards• Inexact values of constants obtained from external sources• Approximations incorporated into the measurement• Variations in repeated observations under apparently

identical conditions

GUM 3.3.2

2. How is MU estimated?

Causes for uncertainty

Measurement results

Measuring instrument

Measurement standard

Measuring methods

Measurer

Number of measurements

Measurement environment

Calibration certificate

Secular change

Manufacturer’s specificationResolution

Dispersions in repetition

Peculiarities in readout

Sources of error and uncertainty indimensional calibrations

• Reference standards and instrumentation• Thermal effects• Elastic compression• Cosine errors• Geometric errors

UKAS M3003 Dec 1999

bunjob_ajchara 83

Sources of error and uncertainty inelectrical calibrations

• Instrument Calibration• Secular Stability• Measurement Conditions• Interpolation of calibration

data• Resolution• Layout of apparatus• Thermal emfs

• Loading and lead impedance

• RF mismatch errors and uncertainty

• Directivity• Test port match• RF Connector

repeatability

UKAS M3003 Dec 1999

bunjob_ajchara 84

Sources of error and uncertainty inmass calibrations

bunjob_ajchara 85

• Reference weight calibration• Secular stability of reference weights• Weighing machine / weighing process• Air buoyancy effects• Environment

UKAS M3003 Dec 1997

86

Quantify Uncertainty Components

2. How is MU estimated?

The Measurement Model

• Usually the final result of a measurement is not measured directly, but is calculated from other measured quantities through a functional relationship

• This is called function a “measurement model”• The model might involve several equations, but

we’ll follow the GUM and represent it abstractly as a single equation:

),...,,( 21 NXXXfY

Input and Output Quantities

• In the generic model Y = f(X1,…,XN), the measurand is denoted by Y

• Also called the output quantity• The quantities X1,…,XN are called input

quantities• The value of the output quantity

(measurand) is calculated from the values of the input quantities using the measurement model

Input and Output Estimates

• When one performs a measurement, one obtains estimated values x1,x2,…,xN for the input quantities X1,X2,…,XN

• These estimated values may be called input estimates

• The calculated value for the output quantity may be called an output estimate

Measurement model

A measurand Y can be determined from N inputs quantities X1, X2, X3 … XN

The model is written abstractly as Y=f(X1,X2,…,XN) where X1,X2,…,XN are input quantities and Y is the output quantity

Developing a Measurement model

• Decide what is the measurand Y– the quantity subject to measurement

• Decide what are the quantities X1, …, XN influencing the measurement– observed quantities, applied corrections,

material properties, etc• Decide the relationship between Y and X1, …, XN

– the model of the measurement

bunjob_ajchara 91

Example: CALBRATION OF A HAND-HELD DIGITAL MULTIMETER AT 100 V DC

The error of indication EX of the DMM to be calibrated is obtained from

whereVi X - voltage, indicated by the DMM (index i means indication),VS - voltage generated by the calibrator,δ VI X - correction of the indicated voltage due to the finite resolution

of the DMM,δ VS - correction of the calibrator voltage due to

(1) drift since its last calibration,(2) deviations resulting from the combined effect of offset,

non-linearity and differences in gain,(3) deviations in the ambient temperature,(4) deviations in mains power,(5) loading effects resulting from the finite input resistance

of the DMM to be calibrated.EA-4/02:1999

bunjob_ajchara 93

An estimate of Y, denoted by y, is obtained from x1, x2, x3 … xN, the estimates of the input quantities X1, X2, X3 … XN,

Measurement model

Represent each input quantity Xi by1. Best estimate xi as mean of distribution, and2. Standard uncertainty u(xi) as s.d. of distribution

bunjob_ajchara 94

Measurement Model

For each input quantity1. Obtain knowledge of that quantity2. Assign a probability distribution to each quantity

consistent with that knowledgeOften a Gaussian (normal) or a rectangular distribution

Classification of uncertainty components

• Type A components: those that are evaluated by statistical analysis of a series of observations

• Type B components: those that are evaluated by other means– Both based on probability distributions– standard uncertainty of each input estimate is

obtained from a distribution of possible values of input quantity: both based on the state of our knowledge

– Type A founded on frequency distributions – Type B founded on a priori distributions

96

Type A evaluations of uncertainty are based on the statistical analysis of a series

of measurements.

Type A evaluations of uncertainty

bunjob_ajchara 97

Type A Evaluation of Standard Uncertainty

• For component of uncertainty arising from random effect

• Applied when multiple independent observations are made under the same conditions

• Data can be from repeated measurements, control charts, curve fit by least-squares method etc

• Obtained from a probability density function derived from an observed frequency distribution (usually Gaussian

Type A Evaluation

Arithmetic mean• Best estimate of the expected value of a

input quantity -

n

1kkq

n

1q

Type A Evaluation

n

1k

2__

kk qq1n

1)s(q

Experimental standard deviation

Distribution of the quantity

Type A Evaluation

Experimental standard deviation of the mean• spread of the distribution of the means -

n

)s(q)qs( k

__

Type A Evaluation

• Type A standard uncertainty

• degrees of freedom

)()( qsxu i

1ni

Example

A digital multimeter is used to measure a high value resistor and the following readings are recorded.

The standard uncertainty, u, is therefore 0.008 83 kΩ.

Type A Evaluation

• For a well-characterized measurement under statistical control, a pooled experimental standard deviation Sp that characterizes the measurement may be available. – The value of a measurand q is

determined from n independent observations and

– The standard uncertainty is nsqu p)(

Pooled Experimental Standard Deviation

A previous evaluation of the repeatability of measurement process (10 comparisons between standard and unknown) gave an experimental standard deviation

mgWs R 7.8

mg

n

WsWsWu R

RR 0.53

7.8

If 3 comparisons between standard and unknown were made this time (using 3 readings on the unknown weight), this is the value of n that is used to calculate the standard uncertainty of the measurand

Type A Evaluation Example:

Type B Evaluation of Standard Uncertainty

• Evaluation of standard uncertainty is usually based on scientific judgment using all relevant information available, which may include:– previous measurement data,– experience with, or general knowledge of the

behavior and property of relevant materials and instruments,

– manufacturer's specifications,– data provided in calibration and other reports,

and– uncertainties assigned to reference data taken

from handbooks.GUM 4.3.1

March 2006 Slide 106

Type B Evaluations• Normal distribution:

• Examples:– expanded uncertainties from a calibration

certificate

-4 -3 -2 -1 0 1 2 3 4

68%

99.7%

95%

k

Uu i

i

where Ui is the expanded uncertainty of the contribution and k is the coverage factor (k = 2 for 95% confidence).

mgmg

k

Uxu

mgkg

i 52

10

10999999.0

Type B Evaluations

Normal distribution Example A calibration certificate reports the measured value of a nominal 1kg OIML weight class F2 at approximately 95% confidence level as:

“It is likely that the value is somewhere in that range”

Rectangular distribution is usually described in terms of: the average value and the range (±a)Certificates or other specification give limits where the value could be,without specifying a level of confidence (or degree of freedom).

Rectangular distribution

1/2a

2a(= a)

X

The value is between the limits

The expectation aa

axy

a a

A B

P=1/2a

Range = 2a ,

Semi-range = Range /2 = a

Rectangular distribution

3

3332

1

;

3/2

1

2

1;

2

1

2332

322

222

a

aaa

a

aAaB

Xa

dxxaa

P

dxxPPdxx

B

A

B

A

B

A

B

A

Rectangular distribution

Example• From the previous example, if the

Maximum Permissible Error (MPE) according to OIML class F2 (±16 mg) is used; then

mgmga

u B 23.93

16

3

Example of Rectangular distribution

Example of Rectangular distribution

• A Handbook gives the value of coefficient of linear thermal expansion of pure copper at 20

and the error in this value should not exceed,

assuming rectangular distribution

the standard uncertainty is:

CCu /1052.16 620

C /1040.0 6

Cxu

arangesemi

i

/1023.03

1040.0

1040.0 ;

66

6

C

Handbook

Example of Rectangular distribution

• Manufacturer’s Specifications A voltmeter used in the measurement process has the

accuracy of ± 1 % of full scale on 100 V. range

semi - range ( a ) = 1 V

V 6.03

V 1

3

axu i

Example of Rectangular distribution

•If the resolution of the digital device is δx, the value of X can lie with equal probability anywhere in the interval X - δx /2 to X + δx /2 and thus described by a rectangular probability distribution with the width δx

x0.29

)(

2 range -Semi

32

xxu

Range

i

x

x

1 2 3 4 56

4

Resolution of a digital indication

Example of Rectangular distribution

•Digital indication•A digital balance having capacity of 210g and the least significant digit 10 mg. The standard uncertainty contributed by this balance is:

mg 2.9 32

01,0

gxu i

Example of Rectangular distribution

HysteresisThe indication of instrument may differ by a fixed and known amount according to whether successive reading are rising or falling.If the range of possible readings from that is dx

x

ixxu

0.29 32

March 2006 Slide 117

U-shaped distribution• When the measurement result

has a higher likelihood of being some value above or below the median than being at the median.

2i

i

au

• Examples:– Mismatch (VSWR)– Distribution of a sine

wave

-2ai -ai 0 ai 2ai

-2ai -ai 0 ai 2ai

Example of U-Shaped distribution

• A mismatch uncertainty associated with the calibration of an RF power sensor has been evaluated as having semi-range limits of 1.3%. Thus the corresponding standard uncertainty will be

bunjob_ajchara 118

UKAS M3003

6/1as

Distribution used when it is suggested that values near the centre of range are more likely than near to the extremes

Assumed standard deviation:

2a (=a)

1/a

X

axy

Triangular distribution

Values close to x are more likely than near the boundaries

Example: A tensile testing machine is used in a testing laboratory where the air temperature can vary randomly but does not depart from the nominal value by more than 3°C. The machine has a large thermal mass and is therefore most likely to be at the mean air temperature, with no probability of being outside the 3°C limits. It is reasonable to assume a triangular distribution, therefore the standard uncertainty forits temperature is:

Example of Triangular distribution

In case of doubt, use the rectangular distribution

UKAS M3003

Which is better A or B?

It should be recognized that a Type B evaluation of a standard uncertainty can be as reliable as a Type A evaluation, especially in a measurement situation where a Type A evaluation is based on a comparatively small number of statistically independent observation.

GUM 4.3.2

Calculate

Combined Standard Uncertainty

bunjob_ajchara 123

• Components of standard uncertainty of measurand y=f(x1,x2,x3……xN) are combined using the “ Law of Propagation of Uncertainty” or “Root Sum of Square :RSS”

),()()(2

),()()(2

1

1 1

1

1 1

2

jijij

N

i

N

iji

2N

1iii

jiji

N

i

N

ij jii

2N

1i

2

ic

N21

xxrxuxucc)u(xc

xxrxuxux

f

x

f)(xu

x

f(y)u

),...,x,xf(xy

jiji xxxxr , quantiesinput between n correlatio theis ,

combined standard uncertainty

ii x

ftcoefficienysensitivitc

theis

124

The relationship between the measurand, Y, and A, B and C is written most generally as Y = f(A,B,C).

Combined Standard Uncertainty, uc

222

)()()(

cuc

fbu

b

fau

a

fyuc

u(a), u(b) and u(c) are the standard uncertainties of best estimates a, b and c respectively obtained through Type A or Type B evaluations.

bunjob_ajchara 126

sensitivity coefficient

Partial derivative with respect to input quantities Xi of functional relationship between measurand Y and input quantities Xi on which Y depends

sensitivity coefficient formula

ii x

fc

f

bunjob_ajchara 127

Example

The value of the resistance Rt, at the temperature t, is obtained from equation:

Where:α is the temperature coefficient of the resistor in Ω / °ct is the temperature in °c , and R0 is the resistance in ohms at the reference temperature,

The partial differentiation of Rt with respect to t is:

)1(0 tRRt

tt

Rt

bunjob_ajchara 128

Correlation of Input Quantities

Ref

UUT

SRef

SUUT

Scorr

Difference (Correction Ref-UUT)

bunjob_ajchara 129

correlation Consider

bacbac

bacbac

abab

bababac

bac

02 If, 02 If,

2

22

222222

2222

bunjob_ajchara 130

correlation coefficient

correlation coefficient, r(xi , xj) - degree of correlation betweenji xx ,

.

,,

ji

jiji xuxu

xxuxxr

1)(1 :Value , ji xxr

correlatedxxr

edUncorrelatxxr

ji

ji

0........

.......0),(

bunjob_ajchara 131

Uncorrelated input quantities

For uncorrelated input quantities r (xi , xj) = 0

2N

1ii

2 c

ic xuyu

0,2 Then 1

1 1

jijij

N

i

N

iji xxrxuxucc

2222 ... 321 nxxxxyc uuuuu

For ci =1

Combinations of Uncertainties

Addition/Subtraction

For independent variables, we have,

222222

222

2

2

2

2

;

2

xzzxy

xzzxx

y

abuubuau

bz

ya

x

y

uz

y

x

yu

z

yu

yu

bzaxy

22222zxy ubuau

! 02 2 xzabu

Combinations of Uncertainties

Multiplication/Division

For independent variables, we have,

Similar arguments would apply tothe expression

xz

u

z

u

x

u

y

u

xzuauxauzau

axz

yaz

x

y

uz

yu

x

yu

axzy

xzzxy

xzzxy

zxy

2222

222222222

22

2

2

;

222

z

u

x

u

y

uzxy

z

xay

134

Worked example

The mass, m, of a wire is found to be 2.255 g with a standard uncertainty of 0.032 g. The length, l, of the wire is 0.2365 m with a standard uncertainty of 0.0035 m. The mass per unit length, , is given by:

Determine the,a) best estimate of ,b) standard uncertainty in .

g/m 535.92365.0

255.2

l

m

l

m

135

Worked example continued

22

2 )()(

lul

mum

uc

1-m 2283.42365.0

11

lm

222

g/m 317.402365.0

255.2

l

m

l

222 0035.0317.40032.02283.4 cu

g/m 1955.0cu

The partial differentiation of µ with respect to m and l

bunjob_ajchara 136

For the very special case where all input estimates are correlated

The combined standard uncertainty

0,2 Then 1

1 1

jijij

N

i

N

iji xxrxuxucc

1, ji xxr

correlated input quantities

nxnxxxc

N

iiic

ucucucucyu

xucyu

....321 321

1

bunjob_ajchara 137

Correlated input quantities

Example

1)Ri (R1,R2,R3,……,R10) each has nominal value 1000 ohms

2)Each has been calibrated by direct comparison with negligible uncertainty

3)Standard uncertainty of Rs is u(Rs) = 100 mohms

Model equation :

R1 R2 R3 R10 Rref10 kW

110010

10

10

1

10

1

mRuRu

kRRfR

isref

iiiref

CalculateExpanded Uncertainty

bunjob_ajchara 138

Expanded Uncertainty

• expanded uncertainty – quantity defining an interval about the

result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand.

GUM 3.2.5

140

Expanded Uncertainty, U

Y = y U

The Expanded Uncertainty, U, is a simple multiple of the standard uncertainty, given by

U = kuc(y)

k is referred to as the coverage factor.

So we can write:

coverage factor, k

• coverage factor, k– numerical factor used as a multiplier of

combined standard uncertainty in order to obtain expanded uncertainty

GUM 3.2.6

Coverage factor

Most cal labs adopt 95.45% which gives k 2 for effective degrees of freedom 30

Coverage Factor - k

Confidence Interval

1.00 68.27%

2.00 95.45%

2.58 99.%

3.00 99.73%

bunjob_ajchara 143

Coverage Factor of Combined Uncertainty

• Effective Degree of Freedom– to determine the coverage factor of combined uncertainty, the

effective degree of freedom must be first calculated from the Welch-Satterthwaite formula:

• Based on the calculated veff, obtain the t-factor tp(veff) for the required level of confidence p from the t-distribution table

• The coverage factor will be: kp = tp(veff)

N

i i

i

c

eff yu

yu

1

4

4

)(

)(

bunjob_ajchara 144

• Example -- A steel rod was measured 4 times. The calculated .

• The effective degree of freedom:

• For @ 95% confidence level and from “student’s t” table, we get k = 2.52

mmyu

mmummu

c

BA

2.4)(

; 3.2 and 5.3

22.6

014

5.3

2.4

44

4

44

4

B

B

A

A

ceff

vu

vu

yuv

6eff

Effective number of degrees of freedom

bunjob_ajchara 145

Effective number of degrees of freedom

. 11

4.22.52

.

mm

yukU c

Therefore, the expanded uncertainty U is:

bunjob_ajchara 146

Relative standard uncertainty

Relative standard uncertainty of input estimate ,

Relative combined standard uncertainty, y

i

ii x

xux

y

yuc

22

2

2

2

1

1 .........

n

nc

x

xu

x

xu

x

xu

y

yu

22

2

2

2

1

1

2

.........

n

nc

x

xu

x

xu

x

xu

y

yu

then

bunjob_ajchara 147

Relative standard uncertainty

Example The measurand:

TKHP

KHPKHPNaOH VM

PmC

1000

Description Value,x Standard uncertainty,

Relative standard uncertainty,

rep Repeatability 1,0 0,0005 0,0005

Weight of KHP 0,3888 g 0,00013g 0,00033

Purity of KHP 1,0 0,00029 0,00029

Molar mass of KHP 204,2212 gmol-1

0,0038gmol-1 0,000019

Volume of NaOH for KHP titration

18,64 ml 0,013ml 0,0007

KHPm

KHPP

KHPM

TV

ixu

i

i

x

xu

bunjob_ajchara 148

Relative standard uncertainty1) Value of the measurand

= 0,10214 mol l-1

2) Combined relative standard uncertainty

uc(CNaOH) = 0,00097 X 0.10214 mol l-1 = 0,00010 mol l-1

22222)(

T

T

KHP

KHP

KHP

KHP

KHP

KHP

NaOH

NaOHc

V

Vu

M

Mu

P

Pu

m

mu

rep

repu

C

Cu

00097,0

00070,0000019,000029,000033,00005,0 22222

64,182212,204

0,13888,01000

NaOHC

Reporting Result

Reporting • should include

– result of measurement – expanded uncertainty with coverage factor and

level of confidence specified– description of measurement method and

reference standard used– uncertainty budget

• example of uncertainty statemente.g.The expanded uncertainty of measurement is ±

____ , estimated at a level of confidence of approximately 95% with a coverage factor k = ____.

Reporting Result• It usually suffices to quote uc(y) and U [as well as the standard

uncertainties u(xi) of the input estimates xi] to at most two significant digits, although in some cases it may be necessary to retain additional digits to avoid round-off errors in subsequent calculations.

• In reporting final results, it may sometimes be appropriate to round uncertainties up rather than to the nearest digit. For example, uc(y) = 10,47 m might be rounded up to 11 m.

• However, common sense should prevail and a value such as u(xi) = 28,05 kHz should be rounded down to 28 kHz.

• Output and input estimates should be rounded to be consistent with their uncertainties.

bunjob_ajchara 151

GUM 7.2.6

Reporting Conventions

• 1000 (30) mL– Defines the result and the (combined) standard

uncertainty

• 1000 +/- 60 mL– Defines the result and the expanded uncertainty

(k=2)

• 1000 +/- 60 mL at 95% confidence level.– Defines the expanded uncertainty at the specified

confidence interval

The 9-steps GUM Sequence1. Define the measurand

2. Build the model equation

3. Identify the sources of uncertainty

4. (If necessary) Modify the model

5. Evaluate of the input quantities and calculate the value of the result

6. Calculate the value of the measurand (using the equation model)

7.Calculate the combined standard uncertainty of the result

8. Calculate the expanded uncertainty (with a selected k)

9. Report resultbunjob_ajchara 153

Conclusions and Remarks

Some Important Practical Consequences

… or a little common sense with errors!

1. When several (independent) errors are to be added, addition in quadrature is much more realistic than addition.

2. If one error ie less than one quarter of another error in the addition then the smaller error may be realistically ignored.

3. There is little point in spending much time estimating small errors – concentrate on the large errors!

4. The experimental procedure should minimise the dominant errors, This implies that these must be identified and estimated (usually in a pilot run) before the final data is taken.

5. Try to bring the precision of each variable to a common level, if possible, by repeated measurements.

Basic concepts

“…The evaluation of uncertainty is neither a routine task nor a purely mathematical one; it depends on detailed knowledge of the nature of the measurand and of measurement…”

GUM 3.4.8

158

ISO (1993) Guide to the Expression of Uncertainty in Measurement (Geneva, Switzerland: International Organisation for Standardisation).

NIST Technical Note 1297 (1994) Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results.

M 3003, The Expression of Uncertainty and Confidence in Measurement, published by UKAS

EA-4/02 - December 1999• Expression of the Uncertainty of Measurement in Calibration

EURACHEM / CITAC Guide: Traceability in ChemicalMeasurement - A guide to achieving comparable results in chemical measurement 2003

Assessment of Uncertainties of Measurement for Calibration and Testing Laboratories - Second Edition , c R R Cook 2002Published by National Association of Testing Authorities, AustraliaACN 004 379 748 ISBN 0-909307-46-6

Bibliography and acknowledgement