Treatment of Measurement Uncertainties

12.3.3.1 Introduction

The probable largest inconsistency in all measurements is the fact that, re­gardless of the instruments used and the methods applied, we never find the true value of the quantity that is being measured. It is possible to improve the

Estimate of the true value but still fall short of the ideal. This fact, which can­not be avoided, is the “First Law of Measurements.” It is a consequence of the existence of measurement errors, and is not sufficiently understood among all those involved with measuring instruments and carrying out measurement:-. Before focusing on errors and their causes, it will be of benefit to define a tew fundamental terms:1’2

True value of a quantity: This characterizes a perfectly defined quantity, in the conditions existing at the time the value was observed.

Result of a measurement: This is the value of the measured quantity obtained by measurement.

Error of measurement: This is the discrepancy between the result of measurement and the true value of the quantity.

12.3.3.2 Measurement Errors

The measurement errors are divided into two categories: systematic errors and random errors.1

Systematic error is an error which, in the course of a number of

Measurements carried out under the same conditions of a given value and quantity, either remains constant in absolute value and sign, or varies according to definite law with changing conditions.

Random error varies in an unpredictable manner in absolute value and in sign when a large number of measurements of the same value of a quantity are made under essentially identical conditions.

The origins of the above two errors are different in cause and nature. A sim ­ple example is, when the mass of a weight is less than its nominal value, a system­atic error occurs, which is constant in absolute value and sign. This is a pure systematic error. A ventilation-related example is, when the instrument factor of a Pitot-static tube, which defines the relationship between the measured pressure difference and the velocity, is incorrect, a systematic error occurs. On the other hand, if a Pitot-static tube is positioned manually in a duct in such a way that the tube tip is randomly on either side of the intended measurement point, a random error occurs. This way, different phenomena create different types of error. The (total) error of measurement usually is a combination of the above two types.

The question may be asked, What is the reason in dividing the errors into two categories? The answer is the totally different way of dealing with these different types. Systematic error can be eliminated to a sufficient degree, whereas random error cannot. The following section shows how to deal with these errors.

12.3.3.3 Systematic Errors

Systematic error, as stated above, can be eliminated—not totally, but usually to a sufficient degree. This elimination process is called “calibration.” Calibration is simply a procedure where the result of measurement recorded by an instrument is compared with the measurement result of a standard. A standard is a measuring device intended to define, to represent physically, to conserve, or to reproduce the unit of measurement in order to transmit it to other measuring instruments by comparison.1 There are several categories of standards, but, simplifying a little, a standard is an instrument with a very high accuracy and can for that reason be

Used as a reference for ordinary measuring instruments. The calibration itself is usually carried out by measuring the quantity over the whole range required and by defining either one correction factor for the whole range, for a constant sys­tematic error, or a correction curve or equation for the whole range. Applying this correction to the measurement result eliminates, more or less, the systematic error and gives the corrected result of measurement.

A primary standard has the highest metrological quality in a given field. Hence, the primary standard is the most accurate way to measure or to repro­duce the value of a quantity. Primary standards are usually complicated instru­ments, which are essentially laboratory instruments and unsuited for site measurement. They require skilled handling and can be expensive. For these reasons it is not practical to calibrate all ordinary meters against a primary standard. To utilize the solid metrological basis of the primary standard, a chain of secondary standards, reference standards, and working standards combine the primary standard and the ordinary instruments. The lower level standard in the chain is calibrated using the next higher level standard. This is called “traceability.” In all calibrations traceability along the chain should ex­ist, up to the instrument with the highest reliability, the primary standard.

The question is often asked, How often should calibration be carried out? Is it sufficient to do it once, or should it be repeated? The answer to this ques­tion depends on the instrument type. A very simple instrument that is robust and stable may require calibrating only once during its lifetime. Some funda­mental meters do not need calibration at all. A Pitot-static tube or a liquid U — tube manometer are examples of such simple instruments. On the other hand, complicated instruments with many components or sensitive components may need calibration at short intervals. Also fouling and wearing are reasons not only for maintenance but also calibration. Thus the proper calibration interval depends on the instrument itself and its use. The manufacturers recommenda­tions as well as past experience are often the only guidelines.

12.3.3.4 Estimation of Random Errors

Normal Distribution

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a ba­sic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency his­togram of a repeated measurement and the normal distribution F(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity.

The experimental sample on which the frequency histogram is based has an experimental mean M and an experimental variance s2, which are

Treatment of Measurement Uncertainties

FIGURE 12.10 Frequency histogram and normal distribution.

Where N is the sample size. When the sample approaches infinity, the experi­mental mean and the experimental variance approach the mean Fx and the variance A2 of the N(fi, a) normal distribution. Thus M is an estimate of /x, and s2 is an estimate of A2.

The probability density of the normal distribution F(x) is not very useful in error analysis. It is better to use the integral of the probability density, which is the cumulative distribution function

■i — A

F(a) = | F(x)dx = —== ( E " ‘ dx. (12.3)

J — co Irjl itJ

This function represents the area under the density curve of the normal distri­bution between -» and A. This is the probability for the random variable X to obtain a value smaller or equal to A. Expressed mathematically,

P(X<a) = F(a) (12.4)

The probability for the random, normally distributed variable X to obtain a value between some limits A and B is

P(a <:X s B) = F(b) — F(a) (12.5)

This is illustrated in Fig. 12.11. As the integral in Eq. (12.3) cannot be evaluated by elementary methods, the cumulative distribution function is determined from tables.

To deal with all kinds of normal distributions of different means and vari­ances, the cumulative distribution is further normalized. This introduces a new variable U = (x — ix)/a. This operation changes a N(ju, A) distribution to a N(0, 1) distribution. From Eq. (12.3) the following is obtained:

Treatment of Measurement Uncertainties

0

Fn(°)

A

Fn(°)

A

Fn(o)

A

A

0

Fn(o)

0.00

0 5000

0.50

0.6915

1.00

0.8413

1.50

0.9332

2,00

0.9772

2.50

0.9938

0.01.

0.5040

0.51

0.6950

1.01

0.8438

1.51

0.9345

2,01

0.9778

2,51

0.9940

0.02

0.5080

0.52

0.6985

1.02

0.8461

1.52

0.9357

2.02

0.9783

2,52

0.9941

0.03

0.5120

0.53

0.7019

1.03

0.8485

1.53

0.9370

2.03

0.9788

2.53

0..9943

0.04

0.5160

0.54

0.7054

1.04

0.8508

1.54

0.9382

2.04

0.9793

2.54

0.9945

0.05

0.5199

0.55

0.7088

1.05

0.8531

1.55

0.9394

2.05

0.9798

2.55

0.9946

0.06′

0.5239

0.56

0.7123

1.06

0.8554

1.56

0.9406

2.06

0.9803

2.56

0.9948

0.07

0.5279

0.57

0.7157

1.07

0.8577

1..57

0.9418

2.07

0.9808

2.57

0.9949

0.08

0,5319

0.58

0.7190

1.08

0.8599

1.58

0.9429

2.08

0.9812

2.58

0.9951

0.09

0.5359

0.59

0.7224

1.09

0.8621

1.59

0.9441

2.09

0.9817

2.5.9

0.9952

0.1 0

0.5398

0,60

0.7257

1.10

0.8643

1.60

0.9452

2.10

0.9821

2.60

0.9953

0.1 1

0.5438

0.61

0.7291

1.11

0.8665

1.61

0.9463

2.11

0.9826

2.61

0.995.5

0.12

0.5478

0.62

0.7324

1.12

0.8686

1.62

0.9474

2.12

0.9830

2.62

0.9956

0.13

0.5517

0.63

0.7.357

1.13

0.8708

1.63

0.9484

2.13

0.9834

2.63

0.9957

0.14

0.5557

0.64

0.7389

1.14

0.8729

1.64

0.9495

2.14

0.9838

2.64

0.9959

0.1.5

0.5596

0.65

0.7422

1.15

0.8749

1.65

0.9505

2.15

0,9842

2.65

0.9960

0.16

0.5636

0.66

0.7454

1.16

0.8770

1.66

0.9515

2.16

0.9846

2.66

0.9961

0.17

0.5675

0.67

0.7486

1.17

0.8790

1.67

0.9525

2.17

0.98,50

2.67

0.9962

0.18

0.5714

0.68

0.7517

1.18

0.8810

1.68

0.9535

2.18

0.9854

2.68

0.9963

0.19

0.5753

0.69

0.7549

1.19

0.8830

1.69

0.9545

2.19

0.9857

2.69

0.9964

0.20

0,5793

0.70

0.7580

1.20

0.8849

1.70

0,9554

2.20

0.9861

2.70

0.9965

0.21

0.5832

0.71

0.7611

1.21

0.8869

1.71

0.9.564

2.21

0.9864

2,71

0.9966

0.22

0.5871.

0.72

0.7642

1.22

0.8888

1.72

0.9573

2.22

0.9868

2.72

0.9967

0.23

0.5910

0.73

0.7673

1.23

0.8907

1.73

0.9582

2.23

0.9871

2.73

0.9968

0.2.4

0.5948

0.74

0.7704

1.24

0.8925

1.74

0.9591

2.24

0.9875

2.74

0.9969

0.25

0.5987

0.75

0.7734

1.25

0.8944

1.75

0.9599

2.2.5

0.9878

2.75

0.9970

0.26

0.6026

0.76

0.7764

1.26

0.8962

1.76

0.9608

2.26

0.988!

2.76

0.9971

0.27

0,6064

0.77

0.7794

1.27

0.8980

1.77

0.9616

2.27

0.9884

2.77

0.9972

0.28

0.6103

0,78

0.7823

1.28

0.8997

1.78

0.9625

2.28

0.9887

2.78

0.9973

0.2.9

0.6141

0,79

0.7852

1.29

0.9015

1.79

0.9633

2.29

0.9890

2.79

0.9974

0.30

0.6179

0.80

0.7881

1.30

0.9032

1.80

0.9641

2.30

0.9893

2.80

0.9974

0.31

0.6217

0.81

0.7910

1.31

0.9049

1.81

0.9649

2.31

0.9896

2.8!

0.9975

0.32

0.62.55

0,82

0.7939

1.32

0.9066

1.82

0.9656

2.32

0.9898

2.82

0.9976

0.33

0.62.93

0.83

0.7967

1.33

0.9082

1.83

0.9664

2..33

0.9901

2.83

0 9977

0.34

0.6331

0.84

0.7995

1.34

0.9099

1.84

0.9671

2.34

0.9904

2.84

0.9977

0.35

0.6368

0.85

0.8023

1.35

0.9115

1.85

0.9678

2.35

0.9906

2.85

0.9978

0.36

0.6406

0.86

0.8051

1.36

0.9131

1.86

0.9686

2.36

0.9909

2.86

0.9979

0.37

0.6443

0.87

0.8078

1.37

0.9147

1.87

0.9693

2.37

0.9911

2,87

0.9979

0.38

0.6480

0.88

0.8106

1.38

0.9162

1.88

0.9699

2.38

0.9913

2.88

0.9980

0..39

0.6517

0.89

0.813.3

1.39

0.9177

1.89

0.9706

2.39

0.9.916

2.89

0.9981

12.3 MEASUREMENT TECHNIQUES TABLE 12.2 (continued)

A

W

A

Fn(<0

A

Fn(°)

A

Fn(°)

A

Fn(°)

A

0.40

0.6554

0.90

0.81.59

1.40

0.9192

1.90

0.9713

2.40

0.9918

2.90

0.9981

0,41

0.6591

0.91

0.8186

1.41

0.9207

1.91

0.9719

2.41

0.9920

2.91

0.9982

0.42

0.6628

0.92

0.8212

1.42

0.9222

1.92

0.9726

2.42

0.9922

2.92

0.9982

0.43

0.6664

0.93

0.8238

1.43

0.9236

1.93

0.9732

2.43

0.9925

2.93

0.9983

0.44

0.6700

0.94

0.8264

1.44

0.9251

1.94

0.9738

2.44

0.9927

2.94

0.9984

0.45

0.6736

0.95

0.8289

1.45

0.9265

1.95

0.9744

2.45

0.9929

2.95

0.9984

0.46

0.6772

0.96

0.831.5

1.46

0.9279

1.96

0.9750

2.46

0.9931

2.96

0.9985

0.47

0.6808

0.97

0.8340

1.47

0.9292

1.97

0.9756

2.47

0.9932

2.97

0.9985

0.48

0.6844

0.98

0.8365

1.48

0.9306

1.98

0.9761

2.48

0.9934

2.98

0.9986

0.4.9

0.6879

0.99

0.8389

1.49

0.9319

1.99

0.9767

2.49

0.9936

2.99

0.9986

The probabilities for a ^-distributed random variable are obtained in a similar way to those in the normal distribution:

P(t s A) = T (a, v) (12.8)

P(a < T s B) = T(b, v) — T (a, v), (12.9)

Where T is a ^-distributed random variable and T{t, v) is the corresponding cu­mulative distribution. The only difference from the normal distribution is the use of the number of degrees of freedom, which in case of a repeated measure­ment is V = N — 1 .

Confidence Limits of a Repeated Measurement

The confidence limits of a measurement are the limits between which the measurement error is with a probability P. The probability P is the confidence level and a = 1 — P is the risk level related to the confidence limits. The con­fidence level is chosen according to the application. A normal value in ventila­tion would be P = 95%, which means that there is a risk of A = 5% for the

Measurement error to be larger than the confidence limits. In applications such as nuclear power plants, where security is of prime importance, the risk level selected should be much lower. The confidence limits contain the random errors plus the “residual” of the systematic error after calibration, but not the actual systematic errors, which are assumed to have been eliminated.

Since the confidence limits of a repeated measurement are based on the dispersion of the measurement result, they usually are presented as symmetri­cal limits:

Dx = ±t(ol, v)sx, (12.10)

Where K((x, v) is a coefficient depending on the sample size (or number of de­grees of freedom) and on the risk level chosen, and sx is the experimental stan­dard deviation of the measurement result. If the measurement is repeated several times (n 30), the confidence limits for the experimental mean are

Dx = ±u(a)^f=, (12.11)

Jn

Where T(a, v) is a value of the variable of the ^-distribution and V = N ~ 1 is the number of the repeated measurements minus 1.

Where the coefficient U(a) is a value of the variable of the N(0, 1) distribution. If the measurement is repeated only few times (n < 30} the confidence limits are evaluated using Student’s distribution:

подпись: where the coefficient u(a) is a value of the variable of the n(0, 1) distribution. if the measurement is repeated only few times (n < 30} the confidence limits are evaluated using student’s distribution: Treatment of Measurement UncertaintiesConfidence Limits of a Single Measurement

Usually there is no opportunity to repeat the measurements to determine the experimental variance or standard deviation. This is the most common sit­uation encountered in field measurements. Each measurement is carried out only once due to restricted resources, and because field-measured quantities are often unstable, repetition to determine the spread is not justified. In such cases prior knowledge gained in a laboratory with the same or a similar meter and measurement approach could be used. The second alternative is to rely on the specifications given by the instrument manufacturer, although instrument manufacturers do not normally specify the risk level related to the confidence limits they are giving.

Confidence Limits of Combined Measurements

Frequently the value of the quantity of interest has to be determined indi­rectly. For example, the determination of the efficiency of any system is based on the measurement of several quantities and some equation relating the mea­sured quantities X, and the “final” quantity Y under consideration. When the confidence limits of the different measured quantities are known, and the rela ­tionship Y = F(Xj) is known, an estimate for the “cumulated” confidence lim­its ±dy of the “final” quantity can be determined from

Treatment of Measurement Uncertainties

Where N is the number of the measured quantities X;, and Dxt are the confidence limits of these quantities. This equation is derived from the law of combination of standard errors. Hence, all the confidence limits should correspond to the same value of risk level.

12.3.3.5 Instrument Performance1’2

The performance of a measuring instrument can be expressed in several ways. The Precision or Accuracy describes the instrument performance in a general and qualitative sense. Thus, these expressions cannot be characterized using numbers.

The Bias error is a quantity that gives the total systematic error of a mea­suring instrument under defined conditions. As mentioned earlier, the bias should be minimized by calibration. The Repeatability error consists of the confidence limits of a single measurement under certain conditions. The Inac­curacy or Error of indication is the total error of the instrument, including the

Systematic error (bias) as well as the random (repeatability) error. The measur­ing instrument Drift is the variation in the metrological properties, including the bias error of an instrument during a long period of time. Because of drift, repeated calibration of an instrument is required; where the time interval be­tween the calibration is the shorter, the larger the drift.

12.3.3.6 Target Values

In the case of measuring airflow, temperature, and other performance-related quantities of new or renovated air-handling systems, the commissioning engineer attempts to determine if the system target values are met. To meet a target value ex­actly is not the goal. The objective is to reach a value between some specified toler­ances. The tolerances are often placed symmetrically on both sides of the nominal target value, but they may be nonsymmetrical or even single-sided. Single-sided tol­erances are often used for efficiencies, contaminant concentrations, and other quantities where a maximum or minimum value is of concern. The usual proce­dure is to approve the value if the result of measurement is between the tolerances or 011 the correct side of a single-sided tolerance value. This, however, is not the correct procedure from the technical measurement aspect. Even if the measurement results seem to have a proper value, the true value of the quantity may be outside of the specified range. For this reason, the demand must be set in such a way, that the true value of the measured quantity fulfills the requirements. This can be achieved by demanding that the measurement result, with its confidence limits, is between or on the correct side of the tolerance values. A more precise expression for this demand is

^xm + dx (12 ;|4)

X, ^xm-dx,

Where X* and Xj are the upper and lower tolerance values, Xm is the measure­ment result, and Dx is the confidence limit value. (Fig. 12.12). Even if the con­dition (12.14) is fulfilled, the risk always exists that the true value is out of the range. This risk can, however, be limited at an appropriate level by choosing the confidence level (risk level) in a proper way.

Figure 12.12 demonstrates that the larger the confidence limits of mea­surement are, the closer to the target value the measurement result must be for symmetrical tolerances. A consequence of this is that if the confidence limits of the measurement are larger than the tolerance intervals JxmXT, it is impos­sible to ensure that the true value is inside the desired interval. In such a case, a more precise instrumentation must be chosen.

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